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zWCbc|usjOz4BQ7?_9QG0entiGKt{NM9trMiF(fDngX4i*_pf=lyPDbATe;G$U3V7M zw(|uyp}`R=IN8;5a)NHw06U9VyHl;~Xzc{rU>@sjdBiEf|8~_t2=U(^tzD7>_YGX# T*UogowDG_O(DCytXej 1?_ + +*A:* *1. Закон Амонтона – Кулона (аналитический вид)* + +Для скольжения: + +$ +F_"тр" = mu N +$ + +Для предельного трения покоя: + +$ +F_"тр.п"^"max" = mu N +$ + +где: + +- $F_"тр"$ — сила трения, +- $N$ — сила нормальной реакции поверхности, +- $mu$ — коэффициент трения (покоя или скольжения). + +*2. От чего зависит коэффициент трения покоя* + +- От *материала и состояния поверхностей* (сталь по льду, дерево по асфальту и т.д.). +- От *шероховатости* и чистоты поверхностей. +- От *наличия смазки* (уменьшает $mu$). +- Слабо зависит от площади соприкосновения (в отличие от интуитивного ожидания). + +*3. Может ли коэффициент трения покоя быть больше 1?* + +- $mu > 1$ означает, что сила трения больше силы нормального давления. +- Это возможно для очень шероховатых или «липких» поверхностей (резина по сухому асфальту, специальные покрытия). +- Например, для хорошей автомобильной шины на сухом асфальте $mu approx 1.0 – 1.2$. + +#line(length: 100%) + +*Q:* _*25*. Как при помощи наклонной плоскости можно определить коэффициент трения покоя? Что называют углом трения покоя? Нарисуйте качественный график зависимости силы трения, которая действует на тело, от угла наклона плоскости в границах от 0° до 90°._ + +*A:* Как определить коэффициент трения покоя с помощью наклонной плоскости + +Кладём брусок на гладко меняющий наклон стол. Медленно увеличиваем угол $alpha$ до момента *начала скольжения*. При «срыве» выполняется условие равновесия по касательной: + +$ +m g sin alpha^* = mu_s m g cos alpha^* arrow.double mu_s = tan alpha^* +$ + +Где $alpha^*$ — угол, при котором брусок только начинает скользить. + +Что такое угол трения покоя + +$ +alpha^* = arctan mu_s +$ + +Это угол наклона плоскости, при котором максимальная сила трения покоя равна касательной составляющей веса и начинается скольжение. + +Качественный график $F_"тр"(alpha)$ при $alpha in[0 degree, 90 degree]$ + +- До срыва: $F_"тр" = m g sin alpha$ (растёт линейно по синусу) до $alpha^*$. +- После срыва (движение): сила становится силой *трения скольжения* $F_"тр" = mu_k m g cos alpha$ и убывает с ростом $alpha$ ($mu_k lt.eq mu_s$). + +Я построил наглядный график (по оси $y$ — $F_"тр" / m g$) с «переломом» в точке $alpha^* = arctan mu_s$: + +#align(center)[#image("assets/3.png")] + +#line(length: 100%) + +*Q:* _*26*. Начертите график зависимости силы сухого трения скольжения от относительной скорости движения. Как можно объяснить эту зависимость?_ + +*A:* + +#align(center)[#image("assets/4.png")] + +*Особенности зависимости* + +- Сила трения скольжения практически *не зависит от скорости*: + +$ +F_"тр" approx mu_k N +$ +- Поэтому на графике она изображается как почти горизонтальная линия. +- В реальных условиях при очень малых скоростях возможны колебания (stick-slip), а при очень больших — небольшое уменьшение силы из-за разогрева и образования смазочного слоя. + + +#line(length: 100%) + +*Q:* _*27*. Запишите уравнения движения тела при его равноускоренном скольжении по шероховатой наклонной плоскости._ + +*A:* Кладём ось $x$ вдоль плоскости, вверх по наклону. Нормальная реакция $ N = m g cos alpha$. +Сила трения скольжения $F_"тр" = mu_k N = mu_k m g cos alpha$. + +1) Скатывание вниз (вдоль $-x$): тело движется вниз + +Ускорение вдоль оси $x$ (со знаком!): + +$ +a = -g(sin alpha - mu_k cos alpha). +$ + +Если считать вниз положительным направлением, модуль ускорения: + +$ +a_arrow.b = g(sin alpha - mu_k cos alpha). +$ + +Уравнения движения (для произвольных $x_0,v_0$ в принятом направлении): + +$ +v(t)= v_0 + a t, space.quad x(t) = x_0 + v_0 t + 1/2 a t^2 . +$ + +2) Бросок вверх по плоскости (тело скользит вверх) + +Обе силы по касательной (проекция веса $m g sin alpha$ и трение $mu_k m g cos alpha$) направлены вниз, поэтому + +$ +a = -g(sin alpha+mu_k cos alpha), +$ + +а уравнения движения те же по форме: + +$ +v(t)=v_0 + a t, space.quad x(t)= x_0 + v_0 t + 1/2 a t^2 . +$ + +Время до остановки: $t_"ст" = -frac(v_0, a) = frac(v_0, g(sin alpha + mu_k cos alpha))$. + +#line(length: 100%) + +*Q:* _*28*. Запишите закон Кулона для трения качения. Проанализируйте, от каких факторов зависит эта сила? В чем заключается физический смысл коэффициента трения качения?_ + +*A:* *Закон Кулона для трения качения* + +Сила трения качения пропорциональна силе нормальной реакции и обратно пропорциональна радиусу катящегося тела: + +$ +F_"кач" = frac(M_"кач", R) = frac(mu_"кач" N, R) +$ + +где: + +- $F_"кач"$ — сила трения качения, +- $M_"кач" = mu_"кач" N$ — момент сопротивления качению, +- $mu_"кач"$ — коэффициент трения качения (имеет размерность *длины*), +- $R$ — радиус катящегося тела, +- $N$ — сила нормальной реакции опоры. + +*От чего зависит сила трения качения* + +- От *величины силы нормальной реакции* (массы тела). +- От *радиуса катящегося тела* (чем больше радиус, тем меньше сопротивление качению). +- От *коэффициента трения качения $mu_"кач"$*, который определяется: + + - свойствами материалов (твёрдость, упругость); + - степенью деформации поверхностей (колесо и дорога, шарик и подшипник); + - качеством смазки. + +*Физический смысл коэффициента трения качения* + +$mu_"кач"$ — это *плечо силы нормальной реакции*, т.е. расстояние от линии действия силы нормального давления до геометрической точки контакта. + +- Чем больше деформация поверхностей → тем больше $mu_"кач"$. +- Поэтому коэффициент трения качения измеряется в *метрах*, в отличие от коэффициента трения скольжения (безразмерного). + +#line(length: 100%) + +*Q:* _*29*. Объясните возникновение силы трения качения. Какую роль при этом играют пластичность и упругое последействие? Могла бы возникнуть сила трения качения, если бы тело, которое катится, и поверхность были абсолютно упругими?_ + +*A:* *1. Причина возникновения силы трения качения* + +При качении соприкасающихся тел (шар, колесо, цилиндр) в точке касания возникает *не точечный, а площадной контакт*. + +- Из-за деформации катящегося тела и (или) поверхности линия действия силы нормальной реакции *смещается вперёд* относительно вертикали. +- Это смещение создаёт момент сопротивления качению — и именно он проявляется как *сила трения качения*. + +*2. Роль пластичности и упругого последействия* + +- *Пластичность* (необратимая деформация) приводит к тому, что часть материала остаётся «смятой» после контакта → это увеличивает плечо силы и, значит, трение качения. +- *Упругое последействие* (запаздывание восстановления формы после снятия нагрузки) также вызывает смещение реакции вперёд, усиливая сопротивление качению. + +*3. Абсолютно упругий случай* + +Если бы и катящееся тело, и поверхность были *совершенно упругими и недеформируемыми*, контакт происходил бы в одной идеальной точке. + +- Смещения реакции не было бы. +- Следовательно, *сила трения качения отсутствовала бы*. + +#line(length: 100%) + +*Q:* _*30*. Материальная точка массой 𝑚 движется по окружности радиуса $R$ с угловым ускорением $beta$. Можно ли по этим данным определить действующую на точку силу? Дайте ответ и приведите необходимые пояснения._ + +*A:* *полную силу определить нельзя* — данных недостаточно. + +Почему: при движении по окружности ускорение точки + +$ +arrow(a) = arrow(beta) times arrow(r) + arrow(omega) times (arrow(omega) times arrow(r)) +$ + +Следовательно, + +$ +arrow(F) = m arrow(a) = m (arrow(beta) times arrow(r) + arrow(omega) times (arrow(omega) times arrow(r))), +$ + +и нужна ещё *угловая скорость $omega$* (или $v$) для нормальной составляющей. + +Что можно сказать по данным: + +- Тангенциальная часть определяется: $F_tau = m beta R$ (по касательной; направление по знаку $beta$). +- Нормальная часть: $F_n = m omega^2 R$ — *неопределима без $omega$*. +- Частные случаи: + - в момент пуска из покоя ($omega = 0$) $arrow.double arrow(F) = m beta R arrow(tau)$; + - зная $omega$ (или $v$), $|arrow(F)| = m R sqrt(beta^2 + omega^4)$ и $tan phi = frac(F_tau, F_n) = frac(beta, omega^2)$ (угол $phi$ между $arrow(F)$ и нормалью). + +#line(length: 100%) + +*Q:* _*31*. Можно ли утверждать полную идентичность протекания некоторого явления или процесса во всех инерциальных системах отсчета? Подтвердите сделанный вывод примерами._ + +*A:* Да. Согласно *принципу относительности Галилея (Ньютона)*, *законы механики имеют одинаковый вид во всех инерциальных системах отсчёта*. + +Это означает, что процессы в разных ИСО протекают одинаково, и *по самим законам движения* невозможно выделить «привилегированную» систему. + +*Пояснение* + +- В ИСО одинаково выполняются законы Ньютона. +- Следовательно, полная механическая картина явления (траектории, силы, ускорения, уравнения движения) будет тождественна, если учесть относительные скорости систем. + +*Примеры* + +1. *Бросок тела вверх в поезде, движущемся равномерно* + + - В системе, связанной с поездом, мяч подлетает и возвращается в руку. + - В системе Земли — мяч движется по наклонной траектории, но относительно пассажира результат одинаков. + +2. *Опыт Галилея с падающими телами* + + - Камень падает одинаково на палубе равномерно движущегося корабля и на неподвижной земле. + +3. *Маятник Фуко в равномерно движущейся лаборатории* + + - Колебания маятника будут происходить так же, как если бы лаборатория покоилась. + +#line(length: 100%) + +*Q:* _*32*. В чем заключается принцип относительности Галилея?_ + +*A:* *Принцип относительности Галилея* + +Все механические явления во *всех инерциальных системах отсчёта* протекают одинаково. +Законы механики имеют *одинаковый вид* во всех инерциальных системах, поэтому никакими механическими опытами, проведёнными внутри такой системы, невозможно определить, находится ли она в состоянии покоя или равномерного прямолинейного движения. + +*Примеры* + +- В равномерно движущемся поезде предметы ведут себя так же, как и в покоящемся (яблоко падает вертикально в руку, маятник качается одинаково). +- На корабле, идущем с постоянной скоростью по гладкой воде, механические эксперименты дают те же результаты, что и на неподвижном. + +#line(length: 100%) + +*Q:* _*33*. Приведите примеры, когда реальные объекты можно рассматривать как системы материальных точек. Какова природа сил взаимодействия между точками?_ + +*A:* *1. Когда реальные объекты можно рассматривать как системы материальных точек* + +Реальные тела состоят из большого числа частиц, и в механике их часто упрощают, представляя как *систему материальных точек*. Примеры: + +- *Газ* → как совокупность большого числа молекул (каждая молекула — материальная точка). +- *Жидкость* → при изучении движения частиц в гидродинамике (каждый элементарный объём жидкости — система точек). +- *Население планеты или города* → при моделировании транспортных потоков или движения толпы. +- *Звёзды в галактике* → каждая звезда рассматривается как материальная точка в задаче о движении галактики. +- *Рой спутников* или движущихся тел → при анализе их гравитационного взаимодействия. + +*2. Природа сил взаимодействия между точками* + +- *Гравитационные силы* — притяжение всех тел, имеющих массу (действуют на большие расстояния). +- *Электромагнитные силы* — взаимодействия заряженных частиц и атомов (определяют упругость, трение, сопротивление и т.п.). +- *Сильное и слабое взаимодействия* — действуют на уровне элементарных частиц и ядер (удерживают протоны и нейтроны в ядре, вызывают радиоактивные превращения). + +#line(length: 100%) + +*Q:* _*34*. Что понимают под аддитивностью массы? Какими опытами подтверждается аддитивность массы?_ + +*A:* *Аддитивность массы* — это свойство массы быть *суммой масс частей системы*: + +$ +m_"сист" = m_1 + m_2 + dots + m_n +$ + +если тела движутся с малыми скоростями (в нерелятивистской механике) и не учитывать превращения энергии в массу. + +*2. Физический смысл* + +- Масса сложной системы равна массе её компонентов. +- Это свойство делает массу удобной мерой количества вещества. + +*3. Опыты, подтверждающие аддитивность массы* + +1. *Взвешивание смеси тел* + + - Масса двух тел, помещённых вместе на весы, равна сумме масс каждого по отдельности. + +2. *Опыт Лавуазье (закон сохранения массы)* + + - При химических реакциях (например, горении) масса продуктов равна массе исходных веществ, если учесть все выделившиеся газы. + +3. *Механическое сложение* + + - Соединение грузов на одной чаше весов → показания равны сумме масс. + +#line(length: 100%) + +*Q:* _*35*. Что называют центром масс механической системы? Запишите формулы для нахождения радиуса - вектора и координат центра масс системы материальных точек._ + +*A:* *Центр масс механической системы* — это воображаемая точка, положение которой описывает «среднее» распределение массы в системе. + +- При движении система ведёт себя так, *как если бы вся её масса была сосредоточена в центре масс*, а на него действовала равнодействующая всех внешних сил. + +*Формулы для системы материальных точек* + +1. *Радиус-вектор центра масс* + +$ +arrow(R) = frac(sum_(i = 1)^n m_i arrow(r)_i, sum_(i = 1)^n m_i) +$ + +где: + +- $m_i$ — масса $i$-й точки, +- $arrow(r)_i$ — радиус-вектор $i$-й точки, +- $arrow(R)$ — радиус-вектор центра масс. + +2. *Координаты центра масс* + Если заданы координаты точек $(x_i, y_i, z_i)$: + +$ +x_c = frac(sum_(i = 1)^n m_i x_i, sum_(i = 1)^n m_i), space.quad +y_c = frac(sum_(i = 1)^n m_i y_i, sum_(i = 1)^n m_i), space.quad +z_c = frac(sum_(i = 1)^n m_i z_i, sum_(i = 1)^n m_i) +$ + +#line(length: 100%) + +*Q:* _*36*. Что называют импульсом системы материальных точек?_ + +*A:* *Импульс системы материальных точек* — это векторная сумма импульсов всех точек системы: + +$ +arrow(P) = sum_(i = 1)^n arrow(p)_i = sum_(i = 1)^n m_i arrow(v)_i +$ + +где: + +- $arrow(p)_i = m_i arrow(v)_i$ — импульс $i$-й точки, +- $m_i$ — масса точки, +- $arrow(v)_i$ — её скорость. + +*Свойства* + +- Импульс системы равен произведению её полной массы на скорость центра масс: + +$ +arrow(P) = M arrow(V)_"ц.м.", space.quad M = sum_(i = 1)^n m_i +$ + +- Изменение импульса системы определяется действием *внешних сил* (внутренние силы взаимно компенсируются по третьему закону Ньютона). + +#line(length: 100%) + +*Q:* _*37*. Опираясь на третий закон Ньютона, покажите, что в замкнутой системе сумма внутренних сил равна нулю._ + +*A:* Пусть в замкнутой системе из $N$ материальных точек внутренняя сила, с которой $j$-я точка действует на $i$-ю, равна $arrow(F)_(i j)$. По третьему закону Ньютона + +$ +arrow(F)_(i j) = -arrow(F)_(j i). +$ + +Суммарная внутренняя сила по системе: + +$ +arrow(F)_"вн" = sum_(i = 1)^N sum_(j = 1", " j eq.not i)^N arrow(F)_(i j). +$ + +Сгруппируем попарно действия и противодействия: + +$ +arrow(F)_"вн" = sum_(i < j) (arrow(F)_(i j) + arrow(F)_(j i)) = sum_(i < j) (arrow(F)_(i j) - arrow(F)_(i j)) = arrow(0). +$ + +Итак, в замкнутой системе (внешние силы отсутствуют) сумма *внутренних* сил равна нулю. Отсюда следует + +$ +frac(d arrow(P), d t) = sum arrow(F)_"внеш" + sum arrow(F)_"вн" = arrow(0), +$ + +то есть импульс системы $arrow(P)$ сохраняется. + + +#line(length: 100%) + +*Q:* _*38*. Сформулируйте закон сохранения импульса системы материальных точек. Приведите примеры применения закона сохранения импульса системы._ + +*A:* *Закон сохранения импульса системы материальных точек* + +В замкнутой системе (где отсутствуют внешние силы или их равнодействующая равна нулю) *векторный импульс сохраняется во времени*: + +$ +arrow(P) = sum_(i = 1)^n m_i arrow(v)_i = "const" +$ + +или + +$ +frac(d arrow(P), d t) = arrow(F)_"внеш" = 0 arrow.double arrow(P) = "const". +$ + +*Примеры применения закона* + +1. *Удар тел* + + - При абсолютно упругих и неупругих ударах сумма импульсов тел до удара равна сумме импульсов после удара. + +2. *Реактивное движение (ракета)* + + - Импульс системы «ракета + истекающие газы» сохраняется, поэтому ракета получает движение в противоположную сторону относительно струи газов. + +3. *Выстрел оружия* + + - Пуля получает вперёд импульс, ружьё — назад (отдача). + +4. *Разлет осколков при взрыве* + + - Сумма импульсов всех осколков равна импульсу системы до взрыва. + +5. *Движение по воде или льду* + + - Человек, толкая лодку от берега, придаёт ей импульс, а сам получает равный и противоположный (лодка уплывает, человек отталкивается). + +#line(length: 100%) + +*Q:* _*39*. Каким образом необходимо выбрать начало координат системы отсчета для того, чтобы импульс механической системы был равен нулю?_ + +*A:* Импульс системы выражается через скорость центра масс: + +$ +arrow(P) = M arrow(V)_"ц.м.", space.quad M = sum_(i = 1)^n m_i . +$ + +Следовательно, чтобы $arrow(P) = 0$, необходимо, чтобы скорость центра масс системы была равна нулю: + +$ +arrow(V)_"ц.м." = 0. +$ + +Как этого достичь? + +Нужно выбрать *систему отсчёта с началом координат в центре масс и неподвижную относительно него* (т.е. систему отсчёта, связанную с центром масс). + +В такой системе: + +$ +arrow(P) = 0, +$ + +и импульс системы в целом равен нулю, хотя отдельные точки могут двигаться относительно центра масс. + +#line(length: 100%) + +*Q:* _*40*. Покажите, что для незамкнутых систем импульс может сохраняться неизменным относительно некоторых направлений. Приведите примеры._ + +*A:* Из уравнения для системы $frac(d arrow(P), d t) = arrow(F)_"внеш"$ следует, что для любого направления с единичным вектором $hat(n)$ + +$ +frac(d, d t) (arrow(P) dot hat(n))=arrow(F)_"внеш" dot hat(n). +$ + +Значит, *проекция импульса на $hat(n)$ сохраняется*, если *проекция равнодействующей внешних сил на $hat(n)$ равна нулю*. + +Примеры + +1. *Бросок тела в однородном поле тяжести.* + $arrow(F)_"внеш" = m arrow(g)$ вертикальна ⇒ горизонтальные компоненты импульса $(P_x,P_y)$ *сохраняются*. + +2. *Столкновение тел на горизонтальной гладкой поверхности.* + Внешние силы (вес и реакции опор) вертикальны ⇒ *горизонтальный импульс системы* во время удара *сохраняется*. + +3. *Тело на гладкой наклонной плоскости.* + Перпендикулярно плоскости: $N - m g cos alpha = 0$ ⇒ проекция внешней силы на нормаль нулевая ⇒ *импульс по нормали* сохраняется (остаётся нулём). Вдоль плоскости действует $m g sin alpha$ ⇒ там импульс не сохраняется. + +4. *Заряженная частица в однородном магнитном поле $arrow(B)$.* + Сила $arrow(F) = q, arrow(v) times arrow(B) perp arrow(B)$ ⇒ $arrow(F) dot hat(B) = 0$ ⇒ *проекция импульса на направление $arrow(B)$* постоянна. + +#line(length: 100%) + +*Q:* _*41*. Сформулируйте закон сохранения импульса механической системы, используя скорость движения центра масс системы._ + +*A:* *Запишем связь импульса с центром масс* + +Импульс системы из $n$ точек: + +$ +arrow(P) = sum_(i = 1)^n m_i arrow(v)_i = M arrow(V)_"ц.м.", +$ + +где + +- $M = sum_(i = 1)^n m_i$ — масса системы, +- $arrow(V)_"ц.м."$ — скорость центра масс. + +*Формулировка закона* + +*Импульс механической системы сохраняется, если равнодействующая внешних сил равна нулю.* + +В терминах центра масс это означает: + +- Если $sum arrow(F)_"внеш" = 0$, то + +$ +arrow(V)_"ц.м." = "const", space.quad arrow(P) = M arrow(V)_"ц.м." = "const". +$ + +*Физический смысл* + +- Центр масс замкнутой системы либо покоится, либо движется прямолинейно и равномерно. +- Даже если внутри системы происходят столкновения, взрывы, деформации — *движение центра масс не меняется*. + +#line(length: 100%) + +*Q:* _*42*. Сформулируйте второй закон Ньютона для системы материальных точек. Поясните, почему в изменении импульса играют роль только внешние силы._ + +*A:* *Второй закон Ньютона для системы материальных точек* + +Для системы из $n$ точек суммарная производная импульса по времени равна равнодействующей *внешних сил*, действующих на систему: + +$ +frac(d arrow(P), d t) = sum_(i = 1)^n arrow(F)_i^"внеш", +space.quad "где " arrow(P) = sum_(i = 1)^n m_i arrow(v)_i. +$ + +*Почему в изменении импульса играют роль только внешние силы?* + +1. *Внутренние силы подчиняются 3-му закону Ньютона:* + + - Если на $i$-ю точку действует сила от $j$-й $arrow(F)_(i j)$, то на $j$-ю — сила $arrow(F)_(j i) = -arrow(F)_(i j)$. + - При суммировании по системе эти силы *взаимно компенсируются*. + +2. *Остаются только внешние силы:* + + - Вклад во изменение импульса системы дают лишь силы, действующие извне. + - Поэтому траектория центра масс и закон сохранения импульса зависят именно от внешних сил. + +#line(length: 100%) + +*Q:* _*43*. Запишите формулу движения тела переменной массы. Покажите, что уравнение движения тела переменной массы представляет собой второй закон Ньютона в его общей форме._ + +*A:* Уравнение движения тела переменной массы (Мещерского) + +$ +m frac(d arrow(v), d t) = arrow(F)_"внеш" + arrow(u)_"отн" (-frac(d m, d t)) +$ + +где + +- $m(t)$ — масса тела, $arrow(v)(t)$ — скорость тела, +- $arrow(F)_"внеш"$ — равнодействующая внешних сил, +- $arrow(u)_"отн"$ — скорость отделяющейся/присоединяющейся массы *относительно тела* (вектор из тела к струе). + Принято: при *истечении* массы $dot(m) < 0$, тогда тяга $arrow(T) = arrow(u)_"отн" (-dot(m))$. + +Эквивалентная форма: + +$ +frac(d, d t)(m arrow(v)) = arrow(F)_"внеш" + arrow(u)_"отн" dot(m) +$ + +(здесь $dot(m) > 0$ трактуется как *приток* массы к телу, $dot(m) < 0$ — отток.) + +Почему это — второй закон Ньютона в общей форме + +Общая формулировка второго закона: + +$ +arrow(F)_"внеш" = frac(d arrow(p), d t), space.quad arrow(p) = m arrow(v). +$ + +Для *открытой* системы «тело» обменяется массой со средой, поэтому помимо изменения собственного импульса $(d(m arrow(v)))/(d t)$ нужно учесть импульс уносимой/притекающей массы. Баланс импульса за $d t$ даёт именно добавочный член $arrow(u)_"отн" dot m$. Отсюда: + +$ +frac(d arrow(p), d t) = arrow(F)_"внеш" + arrow(u)_"отн" dot m, +$ + +что и есть применение $arrow(F)_"внеш" = (d arrow(p)) / (d t)$ к открытой системе. + +Частный случай $ dot(m) = 0 arrow.double m dot(arrow(v)) = arrow(F)_"внеш"$ возвращает привычную форму $arrow(F) = m arrow(a)$. + +#align(center)[= _Механическая работа и энергия_] + +*Q:* _*1*. Запишите формулу для расчета работы постоянной силы._ + +*A:* Формула для работы постоянной силы: + +$ +A = arrow(F) dot arrow(s) = F s cos(alpha), +$ + +где: + +- $arrow(F)$ — постоянная сила, +- $arrow(s)$ — перемещение точки приложения силы, +- $(alpha)$ — угол между направлением силы и перемещением. + +Если сила направлена вдоль перемещения $alpha = 0$, то $A = F s$. +Если перпендикулярно $alpha = 90 degree$, то $A = 0$. + +#line(length: 100%) + +*Q:* _*2*. Что можно сказать относительно работы центростремительной силы? Чему равна работа силы, которая действует на тело, движущееся по окружности с ускорением?_ + +*A:* 1. *Центростремительная сила* всегда направлена к центру окружности, а перемещение тела в данный момент — по касательной. + → угол между $arrow(F)_"цс"$ и $arrow(s)$ равен $90 degree$. + +2. Работа силы: +$ +A = F s cos alpha, +$ + при $alpha = 90 degree$ получаем $A = 0$. + +*Вывод:* центростремительная сила работы не совершает. Она лишь изменяет направление скорости, но не её модуль. + +3. *Если на тело, движущееся по окружности, действует ускорение вдоль траектории (тангенциальное ускорение),* то появляется касательная составляющая силы $F_tau$. + + - Именно эта сила и совершает работу, изменяя скорость тела. + - Тогда работа равна: +$ +A = F_tau s = m a_tau s. +$ + +#line(length: 100%) + +*Q:* _*3*. В каких случаях элементарную работу постоянной силы следует обозначать $delta A$, а в каких $d A$?_ + +*A:* Отличный вопрос, это как раз про *математическую строгость записи*. + +- *$delta A$* (дельта-A) используют, когда речь идёт об *элементарной работе силы* в физике: +$ +delta A = arrow(F) dot d arrow(r) = F d r cos alpha. +$ + Здесь важно, что работа — не полный дифференциал некоторой функции, а *в общем случае* зависит от траектории. Поэтому пишут «$delta$» вместо «$d$». + +- *$d A$* пишут тогда, когда работа может быть представлена как полный дифференциал некоторой функции состояния (например, если сила потенциальная и можно ввести потенциальную энергию). В таком случае: +$ +d A = -d U, +$ +где $U$ — потенциальная энергия. + +#line(length: 100%) + +*Q:* _*4*. Как рассчитывается работа переменной силы? Как изобразить элементарную и полную работу на графике зависимости тангенциальной составляющей силы от пути?_ + +*A:* Ключевые формулы и графическая интерпретация: + +- Элементарная работа переменной силы вдоль траектории: + $ + delta A = F_tau (s) d s = arrow(F) d arrow(r). + $ + +- Полная работа на участке $s in [s_1, s_2]$: + $ + A=integral_(s_1)^(s_2) F_tau (s) d s. + $ + +На графике $F_tau (s)$ площадь под кривой и есть работа: положительные участки дают положительный вклад, участки ниже оси — отрицательный. Прямоугольник иллюстрирует элементарную работу $delta A$ при малом $d s$. + +#align(center)[#image("assets/6.png")] + +#line(length: 100%) + +*Q:* _*5*. Что называют мощностью? Каким образом можно найти работу, если известна мощность механизма? В каких случаях мощность можно выразить формулой $N = arrow(F) dot arrow(v)$?_ + +*A:* *Мощность* — это физическая величина, равная скорости совершения работы: +$ +N = frac(d A, d t). +$ + +2. Как найти работу через мощность + +Если известна мощность механизма: + +- при постоянной мощности: +$ +A = N dot t, +$ +- при переменной мощности: +$ +A = integral_(t_1)^(t_2) N(t) space d t. +$ + +3. Формула через силу и скорость + +$ +N = arrow(F) dot arrow(v) = F v cos alpha, +$ +где $alpha$ — угол между направлением силы и скорости. + +Эта запись справедлива в тех случаях, когда работа совершается *силой, приложенной к движущейся точке*, у которой есть мгновенная скорость $arrow(v)$. + +#line(length: 100%) + +*Q:* _*6*. Чему равна кинетическая энергия свободно падающего тела в момент падения на Землю, если в средней точке пути его потенциальная энергия равна $U$?_ + +*A:* + +1. *Закон сохранения энергии* для свободного падения: +$ +E_"полная" = E_k + E_p = "const". +$ + +2. В верхней точке: $E_k=0$, $E_p = U_"нач"$. + В нижней точке (в момент падения): $E_p=0$, значит +$ +E_k^"низ" = U_"нач". +$ + +3. Нам дано, что *в средней точке пути* потенциальная энергия равна (U). + +- На середине пути по высоте потенциальная энергия в 2 раза меньше, чем начальная: +$ +U = 1/2 U_"нач". +$ +- Значит, начальная энергия: +$ +U_"нач" = 2 U. +$ + +4. Следовательно, кинетическая энергия внизу: +$ +E_k^"низ" = U_"нач" = 2 U. +$ + +*Ответ:* при падении на Землю кинетическая энергия тела равна +$ +E_k = 2 U. +$ + + + +#line(length: 100%) + +*Q:* _*7*. Какие силы называются консервативными? Перечислите, какие из известных вам сил являются консервативными. Какие системы называются консервативными? Какие силы называются диссипативными?_ + +*A:* *Консервативные силы* — это силы, работа которых *не зависит от траектории*, а определяется только начальными и конечными положениями точки. +Для них можно ввести *потенциальную энергию* $U$, и выполняется: +$ +delta A = - d U. +$ +Эквивалентное условие: циркуляция силы по замкнутому контуру равна нулю. + +2. Примеры консервативных сил + +- сила тяжести, +- сила упругости (Гука), +- кулоновская сила (электростатическое взаимодействие), +- силы в центральных полях, где энергия зависит только от расстояния до центра. + +3. Консервативные системы + +*Консервативная система* — система, в которой действуют только консервативные силы. +Для такой системы сохраняется полная механическая энергия: +$ +E = E_k + U = "const". +$ + +4. Диссипативные силы + +*Диссипативные силы* — силы, которые необратимо рассеивают механическую энергию (обычно в тепло), и для них нельзя ввести потенциальную энергию. +Примеры: + +- сила трения скольжения, +- сила сопротивления воздуха или жидкости, +- вязкое трение. + +#line(length: 100%) + +*Q:* _*8*. Что называют кинетической энергией тела? Получите формулу для подсчета кинетической энергии материальной точки, движущейся поступательно. Запишите формулу, которая устанавливает связь между кинетической энергией и импульсом тела._ + +*A:* *Кинетическая энергия* — это часть механической энергии, которая характеризует движение тела и равна работе всех сил, сообщивших телу данное состояние движения. + +2. Вывод формулы для материальной точки + +Пусть на точку массы $m$ действует сила $arrow(F)$, вызывающая перемещение $d arrow(r)$. +Элементарная работа: +$ +delta A = arrow(F) dot d arrow(r). +$ + +По II закону Ньютона: $arrow(F) = m frac(d arrow(v), d t)$. +Тогда +$ +delta A = m frac(d arrow(v), d t) dot arrow(v) d t = m arrow(v) dot d arrow(v). +$ + +Интегрируя от $v = 0$ до $v$: +$ +A = integral_0^v m v, d v = frac(m v^2, 2). +$ + +Таким образом: +$ +E_k = frac(m v^2, 2). +$ + +3. Связь с импульсом + +Импульс $arrow(p) = m arrow(v)$. +Подставим: +$ +E_k = frac(m v^2, 2) = frac(p^2, 2 m). +$ + +#line(length: 100%) + +*Q:* _*9*. Докажите теорему об изменении кинетической энергии механической системы и объясните, почему эта теорема справедлива только для равнодействующей всех сил, приложенных к системе._ + +*A:* Доказательство (система из $N$ материальных точек $m_k$): + +Кинетическая энергия: +$ +T=sum_(k = 1)^N frac(m_k v_k^2, 2). +$ +Дифференцируем: +$ +d T = sum_(k=1)^N m_k arrow(v)_k dot d arrow(v)_k +=sum_(k=1)^N arrow(F)_k dot arrow(v)_k d t +=sum_(k=1)^N arrow(F)_k dot d arrow(r)_k, +$ + +где $arrow(F)_k = m_k (d arrow(v)_k) / (d t)$ — *полная* сила на $k$-ю точку (все внешние + все внутренние, включая силы связей). Интегрируя по движению от 1 до 2, получаем теорему о работе–изменении кинетической энергии: +$ +Delta T = sum_(k = 1)^N integral_1^2 arrow(F)_k dot d arrow(r)_k = A_"всех сил" = A_"внеш" + A_"внутр" +$ + +Почему теорема «работа = изменение $T$» справедлива *только для равнодействующей всех сил*: + +- В правой части стоит суммарная работа *всех* сил, действующих на систему. Если взять работу лишь части сил (например, одной внешней), то недостающий вклад остальных сил, вообще говоря, не равен нулю. Тогда +$ +Delta T = A_"эта сила" + A_"прочие силы", +$ + +и равенство $Delta T = A_"эта сила"$ верно лишь в специальных случаях (когда $A_"прочие силы" = 0$: идеальные связи без работы, отсутствуют другие силы, или их работа взаимно компенсируется). +- В общем случае внутренние силы вносят ненулевую работу (их попарная сумма по модулю взаимно противоположна, но их *мощность* $ arrow(F)_(i j) dot (arrow(v)_i - arrow(v)_j)$ не обязана обнуляться), поэтому без их учёта равенство нарушится. +- Частные полезные формы: + + - Если силы связей идеальны (работы нет), а внутренние силы консервативны, то $Delta T = A_"внеш"$ и $Delta( T + U_"внутр") = A_"внеш, неконсерват."$. + - Для поступательного движения точки: $Delta T = integral arrow(F)_"рез" dot d arrow(r)$. + - Для системы: разложение Кёнига $T = T_"цм" + T_"отн"$; работа равнодействующей *внешних* сил меняет $T_"цм"$, а внутренние и связи — $T_"отн"$. + +Итак, строгое равенство «работа = изменение кинетической энергии» получается именно при суммировании работы *всех* сил (равнодействующей в смысле их суммарной работы по траекториям точек системы). + +#line(length: 100%) + +*Q:* _*10*. Что называют потенциальной энергией механической системы? От чего она зависит? Объясните, почему потенциальная энергия может быть установлена только с точностью до некоторой постоянной. Как выбирается эта постоянная?_ + +*A:* *Потенциальная энергия механической системы* — это часть механической энергии, которая определяется положением тел в поле *консервативных сил*. +Математически: +$ +U(M) = -integral_(M_0)^M arrow(F) dot d arrow(r), +$ +где $M_0$ — выбранная точка отсчёта. + +2. От чего зависит + +- от координат (положения) точки или системы в пространстве; +- от взаимных расстояний между телами (для сил притяжения/отталкивания); +- от деформации упругих элементов (например, пружины). + +То есть она зависит *не от пути движения*, а только от конфигурации системы. + +3. Почему задаётся с точностью до постоянной + +Работа консервативных сил выражается разностью значений потенциальной энергии: +$ +A_(1 arrow 2) = U_1 - U_2. +$ +Абсолютное значение $U$ не имеет физического смысла, важна только её разность. +Поэтому добавление произвольной константы $C$ не меняет физических результатов: +$ +U'(M) = U(M) + C. +$ + +4. Как выбирается постоянная + +Константу выбирают *условно*, в зависимости от удобства: + +- в поле тяжести Земли обычно берут $U=0$ на уровне земли или в выбранной плоскости; +- в законе всемирного тяготения и кулоновском взаимодействии — на бесконечности ($U(infinity) = 0$); +- для упругой пружины — в положении равновесия, когда деформация равна нулю. + +#line(length: 100%) + +*Q:* _*11*. Что называют потенциалом гравитационного поля? Запишите формулу работы гравитационных сил при движении материальной точки, применяя понятие потенциала._ + +*A:* *Потенциал гравитационного поля* в некоторой точке пространства — это величина, равная потенциальной энергии единичной массы, помещённой в эту точку: +$ +phi(arrow(r)) = frac(U, m). +$ + +Для поля тяготения массы $M$ на расстоянии $r$: +$ +phi(r) = -frac(G M, r), +$ +где $G$ — гравитационная постоянная. + +2. Связь с потенциальной энергией + +Потенциальная энергия массы $m$ в поле: +$ +U = m phi. +$ + +3. Работа гравитационных сил через потенциал + +При перемещении материальной точки из точки 1 в точку 2: +$ +A_(1 arrow 2) = U_1 - U_2 = m (phi_1 - phi_2). +$ + +#line(length: 100%) + +*Q:* _*12*. Как зависит потенциальная энергия гравитационного взаимодействия двух материальных точек от расстояния между ними?_ + +*A:* Для двух материальных точек массами $m_1$ и $m_2$ на расстоянии $r$ между ними: + +$ +U(r) = -frac(G m_1 m_2, r), +$ + +где $G$ — гравитационная постоянная. + +Свойства зависимости: + +- Потенциальная энергия *обратно пропорциональна расстоянию*: чем больше $r$, тем меньше по модулю $U$. +- Знак отрицательный, потому что силы тяготения притягивающие, и при сближении тел потенциальная энергия уменьшается (становится более отрицательной). +- При $r arrow infinity$: +$ +U(infinity) = 0, +$ + что принято как условие выбора нуля потенциальной энергии. +- При $r arrow 0$ $U arrow -infinity$. + +#line(length: 100%) + +*Q:* _*13*. Для произвольной консервативной силы получите формулу, которая устанавливает связь консервативной силы с потенциальной энергией._ + +*A:* Ключевой результат для консервативной силы $arrow F$ и потенциальной энергии $U$: + +$ +delta A = arrow(F) dot d arrow(r) = -d U arrow.double arrow(F) = -gradient U +$ + +В проекциях: +$ +F_x= -frac(diff U, diff x), space.quad +F_y= -frac(diff U, diff y), space.quad +F_z= -frac(diff U, diff z). +$ + +Вывод: из определения потенциальной энергии $U($*$r$*$)$ как величины, чья разность равна работе с противоположным знаком, +$ +A_(1 arrow 2) = U_1 - U_2, +$ + +получаем $delta A = arrow(F) dot d arrow(r) = - d U$. Сравнивая коэффициенты при независимых $d x, d y, d z$, имеем $arrow(F)=-gradient U$. + +Частный случай (радиально-симметричное поле $U(r)$): +$ +arrow(F)(r) = -frac(d U, d r) hat(r). +$ + + +#line(length: 100%) + +*Q:* _*14*. Будет ли при движении планеты по эллипсу оставаться постоянной ее полная механическая энергия? Будет ли меняться кинетическая энергия планеты?_ + +*A:* + +1. Полная механическая энергия + +Планета движется в гравитационном поле Солнца. Гравитационная сила — *консервативная*, внешних сил нет. +Значит, для планеты выполняется закон сохранения механической энергии: +$ +E = E_k + U = "const". +$ +То есть *полная механическая энергия при движении по эллипсу остаётся постоянной*. + +2. Кинетическая энергия + +Кинетическая энергия зависит от скорости: +$ +E_k = frac(m v^2, 2). +$ +А по второму закону Кеплера скорость планеты на эллиптической орбите переменна: + +- в перигелии скорость максимальна, +- в афелии — минимальна. + +Значит, *кинетическая энергия меняется*: она возрастает при приближении к Солнцу и убывает при удалении. + +#line(length: 100%) + +*Q:* _*15*. Сформулируйте и запишите аналитическую форму закона сохранения механической энергии для замкнутой системы, в которой действуют консервативные и диссипативные силы._ + +*A:* В замкнутой системе с консервативными и диссипативными силами *полная энергия сохраняется*, а *механическая энергия* $E_"м"= T + U$ убывает на величину работы диссипативных сил (переходит во внутреннюю энергию, тепло). + +Главные формулы + +- Интегрально: +$ +E_"м,2"-E_"м,1" = A_"нк", +space.quad A_"нк" lt.eq 0, +$ + где $A_"нк"$ — работа неконсервативных (диссипативных) сил на участке движения. + +- Дифференциально (мощность диссипации): +$ +frac(d, d t)(T + U) = P_"нк" lt.eq 0. +$ + +- Эквивалентно через внутреннюю энергию (для замкнутой адиабатной системы): +$ +Delta(T + U) + Delta E_"вн" = 0 +space.quad arrow.double.l.r.long space.quad +Delta E_"м" = -Delta E_"вн". +$ + +- Частный случай (только консервативные силы): +$ +T_2 + U_2 = T_1 + U_1. +$ + +Смысл: диссипативные силы совершают отрицательную работу, уменьшая $T +U $; убывшая механическая энергия появляется как рост $E_"вн"$ (нагрев и т.п.). + + +#line(length: 100%) + +*Q:* _*16*. По какому признаку делятся удары на абсолютно упругие и абсолютно неупругие?_ + +*A:* + +- *Абсолютно упругий удар* — это удар, при котором *сохраняется как импульс, так и механическая энергия (кинетическая)* системы тел. После удара тела разлетаются, не теряя суммарной $E_k$. + +- *Абсолютно неупругий удар* — удар, при котором сохраняется только *импульс*, а часть кинетической энергии теряется (переходит во внутреннюю энергию, тепло, деформацию). При таком ударе тела после удара движутся *совместно* как одно целое. + +*Признак деления:* по тому, сохраняется ли полная кинетическая энергия системы тел после удара. + +#line(length: 100%) + +*Q:* _*17*. Выведите формулу для работы неупругих сил при центральном неупругом ударе шаров и проанализируйте ее. Как следует поступить, чтобы вся кинетическая энергия тел, которые участвуют в столкновении, пошла на их деформацию?_ + +*A:* Работа неупругих сил при центральном ударе + +Пусть два шара масс $m_1, m_2$ движутся вдоль одной прямой с начальными скоростями $u_1, u_2$ и после удара имеют скорости $v_1, v_2$. + +Сохраняется импульс: +$ +m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2, +$ +и введём коэффициент восстановления $e in [0,1]$: +$ +v_2 - v_1 = -e (u_2 - u_1). +$ + +Потеря кинетической энергии (а значит, работа неупругих сил над системой) +$ +A_"неупр" = Delta K = K_"после" - K_"до" = -1/2 mu (1 - e^2) (u_1 - u_2)^2 lt.eq 0 +$ +где $mu = frac(m_1 m_2, m_1 + m_2)$ — приведённая масса. + +При абсолютно неупругом ударе ($e=0$): +$ +A_"неупрг" = -1/2 mu (u_1 - u_2)^2, space.quad v = frac(m_1 u_1 + m_2 u_2, m_1 + m_2) "(движение вместе)" +$ + +Анализ формулы + +- Потери тем больше, чем больше относительная скорость $|u_1 - u_2|$, чем ближе $e$ к нулю и чем больше $mu$. +- Максимально возможная потеря при данном ($u_1 - u_2$) достигается при $e=0$; она равна всей *относительной* кинетической энергии в системе центра масс: $K_"отн" = 1/2 mu(u_1 - u_2)^2$. + +Как сделать, чтобы вся кинетическая энергия ушла в деформацию? + +Это возможно, только если: + +1. удар абсолютно неупругий: $e=0$ (тела «слипаются»), и +2. начальный импульс системы равен нулю: $m_1 u_1 + m_2 u_2=0$ (центр масс покоится). + +Тогда после удара $v=0$ и +$ +K_"после" = 0, space.quad A_"неупр" = -K_"до", +$ +то есть *вся* начальная кинетическая энергия переходит в деформацию/тепло. +Если $m_1 u_1 + m_2 u_2 eq.not 0$, неизбежно остаётся «транспортная» энергия движения центра масс, которую внутренними (неупругими) силами погасить нельзя. + + +#line(length: 100%) + +*Q:* _*18*. Опишите, что происходит с телами при абсолютно упругом ударе, на какие два этапа делится процесс столкновения? Как изменяется потенциальная и кинетическая энергия тел в процессе столкновения?_ + +*A:* 1. Абсолютно упругий удар + +При абсолютно упругом ударе тела после столкновения *не теряют* суммарной механической энергии: + +- *импульс системы* сохраняется, +- *кинетическая энергия* системы тоже сохраняется. + То есть энергия не уходит на тепло и необратимые деформации, а только «перераспределяется» между телами. + +2. Два этапа процесса столкновения + +Процесс можно разделить на *два последовательных этапа*: + +1. *Сжатие (деформация):* + + - При сближении тел их скорости уменьшаются (относительное движение тормозится). + - Кинетическая энергия частично превращается в потенциальную энергию упругой деформации (например, упругого сжатия). + +2. *Разжатие (восстановление формы):* + + - Накопленная потенциальная энергия возвращается в кинетическую. + - Тела разлетаются, и в конце процесса потенциальная энергия снова равна нулю, а вся энергия — кинетическая. + +3. Изменение энергий + +- *В процессе удара:* + + - $E_k$ уменьшается на этапе сжатия и увеличивается на этапе разжатия. + - $U_"пот"$ возрастает при сжатии и падает до нуля при разжатии. + +- *Итог:* + + - До удара: энергия полностью кинетическая. + - В момент наибольшей деформации: часть кинетической энергии превращена в потенциальную (у тел минимальная относительная скорость). + - После удара: потенциальная энергия снова равна нулю, суммарная кинетическая энергия совпадает с начальной, но скорости тел перераспределены. + +#line(length: 100%) + +*Q:* _*19*. Выведите формулы для подсчета скоростей тел после удара при центральном абсолютно упругом ударе._ + +*A:* Центральный абсолютно упругий удар (одномерно, до: $u_1, u_2$; после: $v_1, v_2$): + +$ +v_1 &= frac(m_1-m_2, m_1+m_2), u_1 + frac(2m_2, m_1+m_2), u_2, \ +v_2 &= frac(2m_1, m_1+m_2), u_1 + frac(m_2 - m_1, m_1 + m_2), u_2 . +$ + +Эквивалентная форма (полезна для проверок): +$ +m_1u_1+m_2u_2=m_1v_1+m_2v_2,\qquad v_2-v_1=-(u_2-u_1). +$ + +Частные случаи: + +- $u_2 = 0$: $v_1 = frac(m_1 - m_2, m_1 + m_2)u_1, v_2 = frac(2m_1, m_1 + m_2)u_1$. +- $m_1 = m_2$: $v_1 = u_2, v_2 = u_1$ (обмен скоростями). +- $m_2 arrow infinity, u_2=0$: $v_1 arrow -u_1$ (упругое отражение от «жёсткой стены»). + + +#line(length: 100%) + +*Q:* _*20*. При каком условии тело, которое участвует в столкновении, отскакивает от другого тела без потери кинетической энергии? Какой импульс при этом получает первое тело?_ + +*A:* *Условие без потери $E_k$:* абсолютно упругий удар о неподвижное «бесконечно массивное» тело (идеально жёсткую стену). Тогда модуль скорости сохраняется: после удара нормальная компонента скорости меняет знак, касательная — та же. + +*Импульс, полученный первым телом (масса $m$, до удара скорость $arrow(v)$):* + +$ +arrow(J) = Delta arrow(p) = m(arrow(v)_"после" - arrow(v)_"до") = -2m (arrow(v) dot hat(n)) hat(n), +$ + +где $hat(n)$ — единичная нормаль к стене в точке удара. +По модулю: $J = 2 m|v_n|$. Если удар строго лобовой ($arrow(v) || hat(n)$), то $arrow(J) = -2 m arrow(v)$. + + +#pagebreak() + +#align(center)[= _Движение АТТ. Неинерциальные системы отсчета._] + +*Q:* _*1*. Что называют моментом силы относительно точки? Относительно оси вращения? Покажите, что момент силы не изменяется при перемещении силы вдоль линии действия._ + +*A:* Момент силы относительно точки + +Векторный момент силы $arrow(F)$ относительно точки $O$: +$ +arrow(M)_O = arrow(r) times arrow(F) +$ +где $arrow(r)$ — радиус-вектор от $O$ к точке приложения силы. По модулю (в плоскости): +$ +|arrow(M_O)| = F d_perp, +$ + +$d_perp$ — плечо (перпендикуляр от $O$ к линии действия силы). + +Момент силы относительно оси + +Проекция моментного вектора на единичный вектор оси $hat(e)$: +$ +M_"оси" = hat(e) dot (arrow(r) times arrow(F)) = F d_(perp ", оси") +$ +где $d_(perp ",к оси")$ — кратчайшее расстояние от оси до линии действия силы (с учётом знака по правилу правой руки). + +Неизменность при переносе силы вдоль линии действия + +Пусть точка приложения силы переносится с $A$ в $B$ по её линии действия. Тогда + +$ +arrow(r)_B = arrow(r)_A + lambda hat(s), space.quad hat(s) || arrow(F). +$ + +Моменты относительно $O$: +$ +arrow(M)_O^((B)) = arrow(r)_B times arrow(F) = (arrow(r)_A + lambda hat(s)) times arrow(F) = arrow(r)_A times arrow(F) + lambda (hat(s) times arrow(F)) = arrow(r)_A times arrow(F) = arrow(M)_O^((A)), +$ + +поскольку $hat(s) || arrow(F) arrow.double hat(s) times arrow(F) = arrow(0)$. +Следовательно, *момент силы не меняется при её сдвиге вдоль собственной линии действия* (принцип эквивалентности/переносимости). Это верно и для момента относительно оси, т.к. берётся проекция одного и того же $arrow(M)_O$. + +#line(length: 100%) + +*Q:* _*2*. Какая из составляющих силы, действующей под углом к оси вращения, вызывает вращение тела? Чему равен момент силы, параллельной оси вращения?_ + +*A:* + +- *Вращение вызывает тангенциальная составляющая* силы — перпендикулярная радиусу $arrow(r)$ и оси. +- Момент относительно оси: +$ +arrow(M) = arrow(r) times arrow(F), space.quad M = r, F_tau = r F sin phi, +$ + где $phi$ — угол между $arrow(r)$ и $arrow(F)$. +- *Если сила параллельна оси вращения*, её момент относительно этой оси равен *нулю*: +$ +M_"о оси" = (arrow(r) times arrow(F)) dot hat(e)_"оси" = 0 space.quad "при " arrow(F) || hat(e)_"оси". +$ + + +#line(length: 100%) + +*Q:* _*3*. Что называют парой сил?_ + +*A:* *Пара сил* — это система из двух сил, которые: + +- равны по модулю, +- параллельны и направлены в противоположные стороны, +- линии их действия не совпадают. + +Основные свойства пары сил: + +- Равнодействующая пары сил всегда равна нулю → поступательного движения не вызывает. +- Но пара сил создаёт *момент*, который вызывает вращение тела: +$ +M = F dot d, +$ + где $F$ — величина одной из сил, $d$ — плечо пары (расстояние между линиями действия сил). + +#line(length: 100%) + +*Q:* _*4*. Сформулируйте закон всемирного тяготения. В каких случаях аналитическое выражение этого закона для двух материальных точек справедливо и для тел, которые имеют конечные размеры?_ + +*A:* + +1. Закон всемирного тяготения (Ньютон) + +Две материальные точки массы $m_1$ и $m_2$ притягиваются друг к другу с силой: + +$ +F = G frac(m_1 m_2, r^2), +$ + +где: + +- $G approx 6,67 dot 10^(-11), "H·м"^2/"кг"^2$ — гравитационная постоянная, +- $r$ — расстояние между массами, +- сила направлена вдоль линии, соединяющей эти массы. + +2. Когда формула применима к телам конечных размеров + +Для тел с протяжённой массой закон остаётся справедливым, если: + +- *Тела сферически симметричны* (однородные шары или оболочки): + тогда можно считать, что вся масса сосредоточена в центре шара. + +- *Расстояние между телами велико* по сравнению с их размерами: + в этом случае тоже можно приближённо считать тела точечными массами. + +#line(length: 100%) + +*Q:* _*5*. Зависят ли периоды обращения планет вокруг Солнца от их масс? Каким был бы период обращения Луны вокруг Земли, если бы масса Луны была вдвое больше?_ + +*A:* + +Зависит ли период от массы планеты? + +По третьему закону Кеплера для двух тел +$ +T = 2 pi sqrt(frac(a^3, G(M + m))), +$ +где $M$ — масса центрального тела, $m$ — масса спутника/планеты. +Если $m << M$ (планеты вокруг Солнца), то + +$ +T approx 2 pi sqrt(frac(a^3, G M)), +$ +и период *практически не зависит* от массы самой планеты. + +Если массу Луны удвоить + +Полагая ту же орбиту (тот же $a$) и неизменную массу Земли, +$ +T' = T sqrt(frac(M_(plus.big)+m_"Л", M_(plus.big)+2 m_"Л")) + approx T times 0,994. +$ + +Численно: вместо $T approx 27,32$ суток получим $T' approx 27,16$ суток — период стал бы на $tilde 0,6%$ короче. + +#line(length: 100%) + +*Q:* _*6*. Какие опыты позволяют сделать заключение, что инертная и гравитационные массы пропорциональны между собой? В чем суть этих опытов?_ + +*A:* + +1. Суть вопроса + +Инертная масса $m_i$ — мера сопротивления тела ускорению ($F = m_i a$). +Гравитационная масса $m_g$ — мера силы, с которой тело участвует в гравитационном взаимодействии ($F_g = G (m_g M) / r^2$). + +Эксперименты показывают, что $m_i prop m_g$, т.е. отношение $m_g/m_i$ одинаково для всех тел. + +2. Классические опыты + +а) Галилео Галилей (Пиза, XVII в.) + +- Сбрасывал разные тела с наклонной плоскости и башни. +- Ускорение падения оказалось одинаковым для всех тел независимо от массы и материала. +- Вывод: ускорение не зависит от массы, значит $m_g prop m_i$. + +б) Ньютон (Маятники) + +- Сравнивал периоды колебаний маятников с разными гирями. +- Если бы $m_g$ и $m_i$ были разными, периоды различались бы. +- На опыте периоды совпадают. + +в) Этвёш (конец XIX в.) + +- Проводил эксперименты с *крутильными весами*: сравнивал ускорения падения разных материалов (дерево, платина, медь). +- Сверхточные измерения показали, что различий нет в пределах погрешности ($(Delta a)/a < 10^(-9)$). + +3. Современные проверки + +- С помощью спутников и лазерных интерферометров продолжают уточнять эквивалентность масс (основа *принципа эквивалентности* в общей теории относительности). +- Подтверждения получены с точностью лучше чем $10^(-13)$. + +#line(length: 100%) + +*Q:* _*7*. Как определить массу Земли, если известна гравитационная постоянная?_ + +*A:* + +1. Закон всемирного тяготения + +Для тела массы $m$ на поверхности Земли сила тяжести равна: +$ +F = G frac(M_(plus.big) m, R_(plus.big)^2), +$ +где + +- $M_(plus.big)$ — масса Земли, +- $R_(plus.big)$ — радиус Земли, +- $G$ — гравитационная постоянная. + +2. Связь с ускорением свободного падения + +Сила тяжести равна также $F = m g$. +Приравняем: +$ +m g = G frac(M_(plus.big) m, R_(plus.big)^2). +$ + +Сокращаем $m$: +$ +g = G frac(M_(plus.big), R_(plus.big)^2). +$ + +3. Формула для массы Земли + +$ +M_(plus.big) = frac(g R_(plus.big)^2, G). +$ + +4. Подстановка чисел + +- $g approx 9,81 , "м"/("с"^2)$, +- $R_(plus.big) approx 6,37 dot 10^6 , "м"$, +- $G approx 6,67 dot 10^(-11) , ("Н·м"^2)/("кг"^2)$. + +$ +M_(plus.big) approx frac(9.81 dot (6.37 dot 10^6)^2, 6.67 dot 10^(-11)) approx 5.97 dot 10^(24), "кг". +$ + +#line(length: 100%) + +*Q:* _*8*. Какую физическую величину называют напряженностью поля? Изобразите графическую зависимость модуля напряженности поля тяготения от расстояния до точки, которая создает поле. Какие поля называют центральными, однородными?_ + +*A:* Разберём подробно: + +1. Напряжённость поля + +*Напряжённость гравитационного поля* в точке пространства — это сила, действующая на единичную массу, помещённую в эту точку: + +$ +arrow(g) = frac(arrow(F), m) = G frac(M, r^2) hat(r), +$ + +где $M$ — масса, создающая поле, $r$ — расстояние до центра. +Размерность: м/с². + +2. Зависимость от расстояния + +Модуль напряжённости: +$ +g(r) = G frac(M, r^2). +$ + +Это *обратно-квадратичная зависимость*: при увеличении расстояния в 2 раза напряжённость уменьшается в 4 раза. + +3. Центральные и однородные поля + +- *Центральное поле* — поле, в котором силовые линии направлены радиально к центру (или от центра), а напряжённость зависит только от расстояния $r$. + Пример: поле тяготения Земли (вне её поверхности), кулоновское поле точечного заряда. + +- *Однородное поле* — поле, в котором напряжённость во всех точках одинакова по величине и направлению. + Пример: приближённо поле тяжести у поверхности Земли; электрическое поле между обкладками заряжённого плоского конденсатора. + +#line(length: 100%) + +*Q:* _*9*. Сформулируйте законы Кеплера. Обоснуйте вывод этих законов. Как связаны законы Кеплера с законом всемирного тяготения?_ + +*A:* + +1. Законы Кеплера + +1. *Первый закон (закон орбит):* + Каждая планета движется вокруг Солнца по эллипсу, в одном из фокусов которого находится Солнце. + +2. *Второй закон (закон площадей):* + Радиус-вектор, проведённый от планеты к Солнцу, за равные промежутки времени описывает равные площади. + (Иными словами, планета движется быстрее вблизи Солнца и медленнее вдали от него.) + +3. *Третий закон (закон периодов):* + Квадраты периодов обращения планет относятся как кубы больших полуосей их орбит: +$ +frac(T_1^2, a_1^3) = frac(T_2^2, a_2^3) = "const". +$ + +2. Обоснование + +- *1-й закон* следует из решения задачи двух тел в ньютоновской механике: при центральной силе $F prop 1/r^2$ траектория — коническая кривая (эллипс, парабола, гипербола). +- *2-й закон* отражает сохранение углового момента $arrow(L) = m arrow(r) times arrow(v)$, что выполняется для центральных сил. +- *3-й закон* выводится из равенства центростремительной силы и силы тяготения: +$ +frac(m v^2, r) = G frac(M m, r^2). +$ + Отсюда $T^2 tilde a^3$. + +3. Связь с законом всемирного тяготения + +Закон Кеплера — это *эмпирический результат* (на основе наблюдений Тихо Браге). +Закон Ньютона всемирного тяготения дал *теоретическое объяснение* этим законам: + +- форма орбит (1-й закон), +- сохранение площади (2-й закон), +- зависимость периода от радиуса (3-й закон). + +#line(length: 100%) + +*Q:* _*10*. При помощи каких физических законов можно доказать, что в соответствии с первым законом Кеплера планеты движутся по плоским эллиптическим траекториям?_ + +*A:* + +1. Что утверждает первый закон Кеплера + +Орбита планеты вокруг Солнца — *эллипс в плоскости*, в одном фокусе которого находится Солнце. + +2. Какими законами это доказывается + +1. *Закон сохранения момента импульса* +$ +arrow(L) = m arrow(r) times arrow(v) = "const". +$ + Так как сила тяготения центральная ($ arrow(F) || arrow(r)$), момент силы равен нулю. + Следовательно, вектор $arrow(L)$ постоянен и задаёт одно направление. + Значит, движение всегда происходит *в одной плоскости*, перпендикулярной $arrow(L)$. + +2. *Второй закон Ньютона + закон всемирного тяготения* +$ +m dot(dot(arrow(r))) = -frac(G M m, r^2) hat(r). +$ + Это уравнение движения в центральном поле $prop 1/(r^2)$. + Математическое решение этого уравнения (через полярные координаты и интеграл энергии) показывает, что траектория — *коническое сечение*: +$ +r(phi) = frac(p, 1+e cos phi), +$ + где $e$ — эксцентриситет. + +- При $0 lt.eq e < 1$ — это *эллипс*. +- При $e = 0$ — окружность. + +3. Вывод + +Используя: + +- *закон сохранения момента импульса* (движение в плоскости), +- *второй закон Ньютона* + закон тяготения $F prop 1/(r^2)$ (траектория — коническое сечение), + +мы доказываем, что планеты действительно движутся по *плоским эллиптическим орбитам* (1-й закон Кеплера). + +#line(length: 100%) + +*Q:* _*11*. Выведите формулу, по которой определяется первая космическая скорость._ + +*A:* Для круговой орбиты у поверхности (радиус $R$, масса планеты $M$) требуемая центростремительная сила обеспечивается гравитацией: +$ +frac(m v^2, R) = G frac(M m, R^2) arrow.double v=sqrt(frac(G M, R)). +$ + +Эквивалентно, через $g = (G M)/(R^2)$: +$ +v=sqrt(g R). +$ + +Для Земли ($R approx 6.37 dot 10^6 "м", g approx 9.81 "м/с"^2$): +$ +v_1 approx 7.9 "км/с". +$ + +#line(length: 100%) + +*Q:* _*12*. Известно, что по мере увеличения радиуса орбиты скорость искусственного спутника Земли уменьшается. Означает ли это, что при запуске спутника на орбиты большего радиуса двигатели ракеты должны совершать меньшую работу?_ + +*A:* + +1. Скорость спутника на орбите + +Для радиуса орбиты $r$: +$ +v = sqrt(frac(G M, r)), +$ +и действительно, чем больше $r$, тем меньше $v$. + +2. Энергия спутника на орбите + +Полная механическая энергия спутника: +$ +E = E_k + E_p = frac(m v^2, 2) - frac(G M m, r). +$ + +Подставляем $v$: +$ +E = -frac(G M m, 2 r). +$ + +3. Работа при переходе на орбиту + +Чтобы перевести спутник с радиуса $r_1$ на орбиту $r_2 > r_1$, нужно увеличить его энергию: +$ +Delta E = E_2 - E_1 = -frac(G M m, 2r_2) + frac(G M m, 2r_1). +$ +Так как $r_2 > r_1$, $|E_2| < |E_1|$, следовательно, $Delta E > 0$. + +Значит, при переходе на более далёкую орбиту нужно *добавить энергию*, несмотря на то что конечная скорость меньше. + +4. Вывод + +Нет, двигатели не совершают меньшую работу. Наоборот, чтобы вывести спутник на более высокую орбиту, нужно затратить *больше работы*, чем на низкую орбиту, потому что необходимо преодолеть гравитационное притяжение Земли и увеличить полную энергию спутника. + +#line(length: 100%) + +*Q:* _*13*. Дайте определение третьей космической скорости. Проанализируйте все факторы, которые влияют на величину этой скорости._ + +*A:* + +1. Определение + +*Третья космическая скорость* $v_3$ — минимальная скорость, которую нужно сообщить телу у поверхности Земли, чтобы оно смогло *навсегда покинуть Солнечную систему*, преодолев не только земное, но и солнечное притяжение. + +Это скорость ухода за пределы Солнечной системы. + +2. Факторы, которые учитываются + +1. *Притяжение Земли.* + Чтобы покинуть Землю, телу нужно сначала достичь второй космической скорости: +$ +v_2 = sqrt(frac(2 G M_(plus.big), R_(plus.big))) approx 11,2 "км/с". +$ + +2. *Притяжение Солнца.* + Находясь на орбите Земли вокруг Солнца, тело движется вместе с Землёй со скоростью +$ +v_"Земли" approx 29,8 "км/с". +$ + Чтобы покинуть Солнечную систему с орбиты Земли, нужно увеличить скорость относительно Солнца до скорости убегания с расстояния $r = 1 "а.е."$: +$ +v_"убег. от Солнца" = sqrt(frac(2 G M_(dot.circle), r)). +$ + +3. *Сложение скоростей.* + Для ухода из Солнечной системы удобно разогнаться в том же направлении, куда движется Земля, тогда требуется наименьший «добавок» скорости. + +3. Формула третьей космической скорости + +Приближённо: +$ +v_3 = sqrt(v_2^2 + v_"убег. от Солнца"^2 - v_"Земли"^2). +$ + +Численно: +$ +v_3 approx 16,7 "км/с". +$ + +4. Итог + +Величина третьей космической скорости определяется: + +- массой и радиусом Земли (через $v_2$), +- массой Солнца и расстоянием Земли до Солнца (через $v_"убег. от Солнца"$), +- орбитальной скоростью Земли (за счёт которой «помогает» движение планеты). + +#line(length: 100%) + +*Q:* _*14*. Что называется моментом инерции материальной точки относительно оси? От чего зависит момент инерции тела? Какую роль он играет во вращательном движении?_ + +*A:* + +1. Момент инерции материальной точки + +*Момент инерции материальной точки относительно оси вращения*: +$ +I = m r^2, +$ +где + +- $m$ — масса точки, +- $r$ — расстояние от точки до оси вращения. + +Это мера «инертности» точки во вращательном движении. + +2. Момент инерции тела + +Для протяжённого тела момент инерции вычисляется суммированием (интегрированием): +$ +I = sum_i m_i r_i^2 space.quad "или" space.quad I = integral r^2 d m. +$ +Он зависит от: + +- массы тела, +- распределения массы относительно оси (чем дальше масса от оси, тем больше $I$), +- выбора оси вращения. + +3. Роль в вращательном движении + +Момент инерции играет такую же роль, как масса при поступательном движении: + +- *В динамике вращения:* +$ +M = I alpha, +$ + где $M$ — момент силы, $alpha$ — угловое ускорение. + +- *В кинетической энергии вращения:* +$ +E_k = frac(1, 2) I omega^2. +$ + +#line(length: 100%) + +*Q:* _*15*. Запишите известные вам формулы для вычисления моментов инерции однородных сплошных тел правильной геометрической формы (шар, куб, прямой круговой конус)._ + +*A:* Хорошо, соберём основные формулы для *однородных сплошных тел* (масса $M$), характерный размер $R$ или $a, h$): + +1. Шар + +- Сплошной однородный шар радиуса $R$, ось через центр: + $ + I = frac(2, 5) M R^2. + $ + +2. Куб + +- Куб со стороной $a$, ось через центр, параллельная ребру: +$ +I = frac(1, 6) M a^2. +$ +- Если ось через центр, перпендикулярно грани (например, через центр куба вдоль диагонали): +$ +I = frac(1, 6) M a^2 space.quad ("ось по любой координатной оси"). +$ + +3. Прямой круговой конус + +- Однородный конус массы $M$, высоты $h$ и основания радиуса $R$, ось вращения проходит через вершину и центр основания (ось симметрии): +$ +I = frac(3, 10) M R^2. +$ + +#line(length: 100%) + +*Q:* _*16*. Сформулируйте теорему Гюйгенса-Штейнера. Что произойдет с моментом инерции тела, если ось перемещать параллельно самой себе, отдаляясь от центра инерции?_ + +*A:* + +1. Теорема Гюйгенса–Штейнера (теорема о параллельных осях) + +*Момент инерции тела относительно любой оси, параллельной оси, проходящей через центр масс, равен*: + +$ +I = I_c + M d^2, +$ + +где + +- $I_c$ — момент инерции относительно параллельной оси через центр масс, +- $M$ — масса тела, +- $d$ — расстояние между осями. + +2. Следствие + +Если ось перемещать параллельно самой себе, отдаляясь от центра масс: + +- $I$ будет *увеличиваться* пропорционально квадрату расстояния $d$. +- Чем дальше ось от центра масс, тем больше момент инерции. + +#line(length: 100%) + +*Q:* _*17*. Найдите момент инерции стержня массой 𝑚 и длиной $l$, относительно оси, которая проходит на расстоянии $2/3$ от его конца перпендикулярно стержню._ + +*A:* Ответ: $I = frac(1, 9) m l^2$. + +Короткий вывод: для однородного стержня +$ +I_"цм" = frac(1, 12)m l^2. +$ +Ось проходит на расстоянии $frac(2, 3) l$ от конца, значит её расстояние от центра масс: +$ +a = |frac(2, 3)l - frac(1, 2)l| = frac(1, 6)l. +$ +По теореме Гюйгенса–Штейнера: +$ +I=I_"цм" + m a^2 +=frac(1, 12)m l^2 + m(frac(1, 6) l)^2 +=frac(1, 12)m l^2 + frac(1, 36)m l^2 +=frac(1, 9)m l^2. +$ + +#line(length: 100%) + +*Q:* _*18*. Найдите момент инерции диска массой 𝑚 и радиусом 𝑅 относительно оси, которая перпендикулярна плоскости диска и проходит через его край._ + +*A:* $I_"край" = frac(3, 2) m R^2$ + +Короткий вывод: для сплошного диска +$ +I_"цм" = frac(1, 2)m R^2. +$ + +Ось у края параллельна оси через центр и отстоит на расстояние $d = R$. +По теореме Гюйгенса–Штейнера: +$ +I = I_"цм" + m d^2 += frac(1, 2)m R^2 + m R^2 += frac(3, 2)m R^2. +$ + +#line(length: 100%) + +*Q:* _*19*. Сформулируйте основной закон динамики вращательного движения твердого тела._ + +*A:* *Основной закон динамики вращательного движения твёрдого тела* формулируется так: + +Угловое ускорение твёрдого тела прямо пропорционально действующему на него моменту сил и обратно пропорционально моменту инерции тела относительно оси вращения. + +Математическая запись: +$ +M = I alpha, +$ +где + +- $M$ — суммарный момент сил относительно оси вращения, +- $I$ — момент инерции тела относительно той же оси, +- $alpha$ — угловое ускорение. + +#line(length: 100%) + +*Q:* _*20*. Что называется моментом импульса относительно точки? Относительно оси?_ + +*A:* + +1. Момент импульса относительно точки + +*Момент импульса материальной точки относительно точки $O$:* +$ +arrow(L)_O = arrow(r) times arrow(p), +$ +где + +- $arrow(r)$ — радиус-вектор точки относительно $O$, +- $arrow(p) = m arrow(v)$ — импульс точки, +- «$times$» — векторное произведение. + +Физический смысл: мера вращательного движения частицы относительно точки. + +2. Момент импульса относительно оси + +Если выбрана ось с направляющим вектором $hat(e)$, то момент импульса относительно оси определяется как проекция момента импульса на эту ось: +$ +L_"ось" = arrow(L)_O dot hat(e). +$ + +#line(length: 100%) + +*Q:* _*21*. Сформулируйте закон сохранения момента импульса тела. Приведите примеры проявления этого закона._ + +*A:* + +1. Закон сохранения момента импульса + +Если суммарный момент внешних сил, действующих на систему относительно выбранной оси (или точки), равен нулю, то *момент импульса этой системы сохраняется*: + +$ +frac(d arrow(L), d t) = arrow(M)_"внеш". +$ + +Если $arrow(M)_"внеш" = 0$, то +$ +arrow(L) = "const". +$ + +2. Физический смысл + +Закон аналогичен закону сохранения импульса в поступательном движении: отсутствие внешнего «крутящего воздействия» сохраняет вращательное движение. + +3. Примеры проявления + +- *Фигура вращающегося фигуриста:* при прижатии рук к телу радиус масс уменьшается → момент инерции уменьшается, угловая скорость увеличивается, чтобы сохранить $L = I omega$. +- *Космический спутник или планета:* в отсутствии внешних моментов сохраняет угловой момент при вращении вокруг оси или орбитальном движении. +- *Кошка в прыжке:* умеет поворачивать своё тело в воздухе, сохраняя суммарный момент импульса нулевым. +- *Сжатие газа в туманности:* уменьшение радиуса облака приводит к увеличению скорости вращения (образование галактик, звёзд). + +#line(length: 100%) + +*Q:* _*22*. Получите формулу для кинетической энергии тела, которое вращается вокруг неподвижной оси._ + +*A:* + +Пусть твёрдое тело вращается с угловой скоростью $omega$ вокруг неподвижной оси. Разобьём его на точки масс $m_i$ на расстояниях $r_i$ от оси. Скорость каждой точки $v_i = omega r_i$. Тогда + +$ +E_k = sum_i 1/2 m_i v_i^2 +=1/2 sum_i m_i (omega r_i)^2 +=1/2 omega^2 sum_i m_i r_i^2 +=1/2 I omega^2, +$ +где $I=sum_i m_i r_i^2$ — момент инерции относительно оси (в непрерывном случае $I = integral r^2 d m$). + +Итого: +$E_k = 1/2 I omega^2$. + + +#line(length: 100%) + +*Q:* _*23*. Получите формулу для кинетической энергии плоского движения твердого тела._ + +*A:* Плоское движение = поступательное движения центра масс $G$ + вращение вокруг $G$. +Для точки $i$: $arrow(v)_i = arrow(v)_G + arrow(omega) times arrow(r)_i'$, где $arrow(r)_i' = arrow(G i)$, $sum m_i arrow(r)_i' = 0$. + +$ +T& = 1/2 sum_i m_i v_i^2 +=1/2 sum_i m_i (v_G^2 + 2, arrow(v)_G dot (arrow(omega) times arrow(r)_i') + |arrow(omega) times arrow(r)_i'|^2) \ +&=1/2 M v_G^2 + 1/2 sum_i m_i,|arrow(omega) times arrow(r)_i'|^2 +=1/2 M v_G^2+ 1/2 I_G omega^2, +$ + +где $M=sum m_i$, $I_G = sum m_i (r_i')^2$ — момент инерции относительно центра масс. + +Итого: +$ +T=1/2 M v_G^2 + 1/2 I_G omega^2. +$ + +Эквивалентно для произвольной точки $O$: +$ +T = 1/2 M v_O^2 + 1/2 I_O omega^2 - M(arrow(v)_O dot(arrow(omega) times arrow(r)_(G O))), +$ +а при выборе $O=G$ смешанный член исчезает, что даёт формулу выше. + +#line(length: 100%) + +*Q:* _*24*. Чему равна кинетическая энергия диска (шара) массой $m$ и радиусом $R$, который катится без скольжения по горизонтальной плоскости, если его центр масс имеет скорость $V$ . Покажите, что получится такая же формула, если рассматривать качение относительно мгновенной оси вращения._ + +*A:* + +Через разложение «поступательное + вращение вокруг ЦМ» + +Без скольжения: $V = omega R$. +$ +T= 1/2 m V^2 + 1/2 I_"цм" omega^2. +$ + +- Диск (сплошной): $I_"цм" = 1/2 m R^2$ +$ +T=1/2 m V^2 + 1/2 (1/2 m R^2)(frac(V, R))^2 = 3/4 m V^2. +$ + +- Шар (сплошной): $I_"цм" = 2/5 m R^2$ +$ +T = 1/2 m V^2 + 1/2 (2/5 m R^2)(V/R)^2 = 7/(10) m V^2. +$ + +Через мгновенную ось вращения (точка контакта $P$) + +При чистом качении мгновенно $v_P = 0$ ⇒ движение — чистое вращение вокруг $P$: +$ +T=1/2 I_P omega^2, space.quad I_P = I_"цм" + m R^2. +$ + +- Диск: $I_P = 1/2 m R^2 + m R^2 = 3/2 m R^2$ +$ +T=1/2(3/2 m R^2)(V/R)^2 = 3/4 m V^2. +$ + +- Шар: $I_P = 2/5 m R^2 + m R^2 = 7/5 m R^2$ +$ +T = 1/2(7/5 m R^2)(V/R)^2 = 7/10 m V^2. +$ + +Обе методики дают одинаковые результаты, как и должно быть. + + +#line(length: 100%) + +*Q:* _*25*. Какие оси вращения называют свободными? Главными осями инерции? Приведите примеры вращения тел вокруг свободных осей. Поясните характерные особенности, которыми сопровождается вращение тел вокруг осей с наименьшим и наибольшим моментом инерции._ + +*A:* + +1. Свободные оси вращения + +*Свободные оси вращения* — такие оси, вокруг которых тело может вращаться без действия внешних моментов сил, сохраняя постоянное направление в пространстве. + +2. Главные оси инерции + +*Главные оси инерции* — оси, проходящие через центр масс тела и совпадающие с осями симметрии распределения массы. + +- В этих осях тензор инерции диагонален, и вращение вокруг такой оси является устойчивым. +- Для симметричных тел: это ось симметрии и любые взаимно перпендикулярные оси, связанные с симметрией. + +3. Примеры вращения вокруг свободных осей + +- Вращение волчка вокруг оси симметрии. +- Полёт снаряда, пущенного с вращением вокруг продольной оси (гироскопическая стабилизация). +- Вращение космического спутника вокруг оси с симметрией массы. + +4. Особенности вращения вокруг разных главных осей + +- *Вращение вокруг оси с наименьшим моментом инерции* и вокруг оси с наибольшим моментом инерции — устойчивое (тело сохраняет ориентацию). +- *Вращение вокруг промежуточной оси* (средний момент инерции) — неустойчивое: малейшее возмущение приводит к тому, что тело начинает «перекидываться» (пример — трюк с книгой или смартфоном, подброшенным в воздух). + +#line(length: 100%) + +*Q:* _*26*. Сформулируйте условия равновесия твердого тела. Перечислите виды равновесия. Какие изменения могут происходить с кинетической энергией тел при разных видах равновесия?_ + +*A:* Разберём по порядку: + +1. Условия равновесия твёрдого тела + +Для того чтобы твёрдое тело находилось в равновесии, необходимо: + +1. *Условие поступательного равновесия* +$ +sum arrow(F) = 0 +$ + (сумма всех внешних сил равна нулю). + +2. *Условие вращательного равновесия* +$ +sum arrow(M) = 0 +$ + (сумма моментов всех внешних сил относительно любой оси равна нулю). + +2. Виды равновесия + +- *Устойчивое равновесие* — при малом отклонении тело возвращается в исходное положение (пример: шар в чашке). +- *Неустойчивое равновесие* — при малом отклонении тело отклоняется ещё больше (шар на вершине горки). +- *Безразличное (нейтральное) равновесие* — при смещении тело остаётся в новом положении (шар на горизонтальной плоскости). + +3. Изменения кинетической энергии при малых отклонениях + +- *Устойчивое равновесие* — потенциальная энергия при отклонении возрастает, поэтому при возвращении в положение равновесия она преобразуется в кинетическую. Тело колеблется вокруг положения равновесия. +- *Неустойчивое равновесие* — потенциальная энергия при малом смещении уменьшается, и тело «уходит» дальше, кинетическая энергия возрастает. +- *Безразличное равновесие* — потенциальная энергия не меняется, поэтому кинетическая энергия также остаётся постоянной (при отсутствии внешних возмущений). + +#line(length: 100%) + +*Q:* _*27*. Какие системы отсчета называют неинерциальными? В чем заключается принципиальное отличие сил инерции от других сил, определяющих взаимодействие тел? Как меняются силы инерции и ньютоновские силы при переходе от одной неинерциальной системы отсчета к другой?_ + +*A:* + +1. Неинерциальные системы отсчёта + +*Неинерциальная система отсчёта (НИСО)* — это система, которая движется с ускорением относительно инерциальной системы. +В таких системах законы Ньютона не выполняются в их простом виде — для согласия с опытом приходится вводить *фиктивные (инерциальные) силы*. + +2. Отличие сил инерции от «настоящих» сил + +- *Ньютоновские силы* (гравитация, упругости, трения и т. д.) — это реальные взаимодействия между телами. +- *Силы инерции* — не связаны с каким-либо физическим взаимодействием, они появляются лишь как поправка при рассмотрении движения в НИСО. + Формула: +$ +arrow(F)_"инерц" = - m arrow(a)_0, +$ + где $arrow(a)_0$ — ускорение самой системы отсчёта. + +3. Как меняются силы при переходе между системами + +- При переходе от одной НИСО к другой изменяются *силы инерции*: они зависят от ускорения новой системы отсчёта. +- «Настоящие» ньютоновские силы, обусловленные взаимодействием тел, при этом *не меняются*. + +#line(length: 100%) + +*Q:* _*28*. Запишите формулу для силы инерции в неинерциальной системе отсчета, которая движется поступательно. Как направлена эта сила? Является ли сила инерции консервативной? Может ли она совершать работу?_ + +*A:* + +1. Формула силы инерции при поступательном движении НИСО + +Если система отсчёта движется с ускорением $arrow(a)_0$ относительно инерциальной, то для тела массы $m$ в этой НИСО вводят силу инерции: + +$ +arrow(F)_"инерц" = - m arrow(a)_0. +$ + +2. Направление силы + +- Сила инерции всегда направлена *против ускорения системы отсчёта*. +- То есть, если сама система ускоряется вправо, то сила инерции в ней действует влево. + +3. Консервативность + +- Сила инерции *не является консервативной*, потому что она не связана с каким-либо потенциальным полем и не является силой реального взаимодействия. + +4. Может ли она совершать работу? + +- Да, в уравнениях движения в НИСО сила инерции может входить в выражение для работы и энергии. +- Однако эта «работа» не есть физическая передача энергии от какого-то реального источника, а лишь математическая поправка, учитывающая ускорение системы отсчёта. + +#line(length: 100%) + +*Q:* _*29*. Чем различается аналитический вид второго закона Ньютона в инерциальной и неинерциальной системах отсчета? Выполняется ли третий закон Ньютона в НИСО?_ + +*A:* + +1. Второй закон Ньютона в *инерциальной системе отсчёта (ИСО)* + +$ +m arrow(a) = sum arrow(F), +$ +где $sum arrow(F)$ — сумма всех *реальных* сил, действующих на тело. + +2. Второй закон Ньютона в *неинерциальной системе отсчёта (НИСО)* + +Если система ускоряется с $arrow(a)_0$ относительно ИСО, то вводится *сила инерции*: +$ +m arrow(a)' = sum arrow(F) + arrow(F)_"инерц", space.quad arrow(F)_"инерц" = -m arrow(a)_0, +$ +где $arrow(a)'$ — ускорение тела относительно НИСО. + +В НИСО второй закон Ньютона выполняется только с учётом добавления фиктивных сил инерции. + +3. Третий закон Ньютона в НИСО + +- *Третий закон Ньютона* (действие = противодействие) относится к реальным взаимодействиям тел и *справедлив всегда*, в том числе в НИСО. +- Но *силы инерции* не связаны с взаимодействием тел, поэтому они не имеют пары «действие–противодействие». + +#line(length: 100%) + +*Q:* _*30*. На тележке, которая движется по горизонтальной поверхности с постоянным ускорением $a = 5 "м/с"^2$, на нити длиной $l = 1 "м"$ висит груз массой $m = 1 "кг"$. Найдите силу натяжения нити и угол, который она образует с вертикалью, при условии, что груз неподвижен относительно тележки._ + +*A:* Решение в ускоренной (неинерциальной) системе тележки: + +$ +T sin theta = m a space.quad T cos theta = m g arrow.double tan theta = frac(a, g), space.quad T = m sqrt(g^2 + a^2). +$ + +Подставим $m=1, "кг"$, $a = 5, "м/с"^2$, $g = 9.81, "м/с"^2$: + +$ +theta = arctan(frac(5, 9.81)) approx 27.0 degree, space.quad +T=sqrt(9.81^2+5^2) approx 11.01 "Н". +$ + +Ответ: $theta approx 27 degree$ к вертикали, натяжение $T approx 11.0 "Н"$. + + +#line(length: 100%) + +*Q:* _*31*. Выполняются ли законы сохранения механической энергии и импульса в неинерциальных системах отсчета? Приведите соответствующие пояснения._ + +*A:* + +1. Закон сохранения механической энергии + +- В *инерциальных системах*: выполняется, если действуют только консервативные силы. +- В *неинерциальных системах*: + + - приходится вводить силы инерции; они *неконсервативны*, их работа зависит от траектории. + - поэтому строгий закон сохранения механической энергии *не выполняется*. + - Энергию можно «сохранить» только формально, если учитывать дополнительный член, связанный с энергией самой системы отсчёта. + +2. Закон сохранения импульса + +- В *инерциальных системах*: выполняется, если на систему не действуют внешние силы. +- В *неинерциальных системах*: + + - появляются силы инерции, зависящие от массы и ускорения системы. + - они не подчиняются третьему закону Ньютона и не компенсируются. + - поэтому импульс системы тел в НИСО *вообще не сохраняется*. + +3. Итог + +- *В НИСО законы сохранения механической энергии и импульса нарушаются* в их обычной форме. +- Они могут выполняться только при введении «фиктивных поправок» (работа сил инерции, импульс системы отсчёта). + +#line(length: 100%) + +*Q:* _*32*. В чем заключается отличие происхождения поля сил инерции и поля сил тяготения? В чем заключаются особенности поля сил инерции?_ + +*A:* + +1. Происхождение полей + +- *Поле сил тяготения* + + - Реальное поле, возникающее вследствие гравитационного взаимодействия тел по закону всемирного тяготения. + - Его источник — масса. + - Силы тяготения действуют независимо от выбора системы отсчёта. + +- *Поле сил инерции* + + - Мнимое (фиктивное) поле, возникающее при рассмотрении движения в неинерциальных системах отсчёта. + - Его «источник» — ускорение самой системы отсчёта. + - Силы инерции не связаны с реальными взаимодействиями. + +2. Особенности поля сил инерции + +1. *Зависимость от выбора системы отсчёта.* + В другой системе (например, инерциальной) эти силы полностью исчезают. + +2. *Неконсервативность.* + Силы инерции не выражаются через потенциальную энергию (работа зависит от пути). + +3. *Пропорциональность массе.* + $arrow(F)_text"инерц" = -m arrow(a)_0$. Все тела в данной НИСО испытывают одинаковое ускорение под действием этих сил (напоминает принцип эквивалентности). + +4. *Не выполняется третий закон Ньютона.* + Силы инерции не имеют пары «действие — противодействие». + +#line(length: 100%) + +*Q:* _*33*. В чем заключается принцип суперпозиции полей сил инерции и притяжения? Ощущает ли отличие поля сил инерции и поля сил тяготения наблюдатель, который находится в неинерциальной системе отсчета?_ + +*A:* + +1. Принцип суперпозиции + +В НИСО на тело действуют как *реальные силы* (например, гравитации), так и *фиктивные силы инерции*. +Поле инерции просто *накладывается* на поле тяготения (или другое реальное поле). + +Пример: в лифте, движущемся с ускорением $a$, эффективное поле силы тяжести для пассажира: +$ +arrow(g)_"эфф" = arrow(g) + arrow(a)_"лиф". +$ + +2. Может ли наблюдатель отличить поля? + +- Для наблюдателя внутри НИСО (например, в ускоряющемся лифте или ракете) *силы инерции и силы тяготения неразличимы*. +- Именно на этом основан *принцип эквивалентности Эйнштейна*: локально невозможно отличить действие гравитационного поля от равномерно ускоренного движения. + +3. Вывод + +- *Принцип суперпозиции:* в НИСО суммируются поле тяготения и поле инерции. +- *Различие:* внешний наблюдатель (в ИСО) видит, что силы инерции фиктивные, а внутренний наблюдатель (в НИСО) воспринимает их как реальные и не может отличить от гравитации. + +#line(length: 100%) + +*Q:* _*34*. Какие силы инерции возникают во вращающихся системах отсчета? По какой формуле рассчитывается центробежная сила инерции? Как определяется ее направление? Зависит ли направление центробежной силы от направления угловой скорости +вращения?_ + +*A:* + +1. Силы инерции во вращающихся системах отсчёта + +При переходе к вращающейся системе отсчёта возникают: + +- *Центробежная сила инерции* +- *Сила Кориолиса* +- (при неравномерном вращении — ещё и сила Эйлера). + +2. Центробежная сила инерции + +Для точки массы $m$, находящейся на расстоянии $r$ от оси вращения, в системе, вращающейся с угловой скоростью $arrow(omega)$: +$ +arrow(F)_"цб" = m omega^2 arrow(r)_perp, +$ +или в векторной форме: +$ +arrow(F)_"цб" = m arrow(omega) times (arrow(omega) times arrow(r)). +$ + +3. Направление + +- Всегда направлена *от оси вращения наружу*, вдоль радиуса. +- То есть противоположна центростремительному ускорению. + +4. Зависимость от направления угловой скорости + +- Величина $|arrow(F)_"цб"| = m omega^2 r$ зависит только от модуля $omega$. +- Направление силы *не зависит от знака/направления вращения*, а определяется только положением точки относительно оси. + +#line(length: 100%) + +*Q:* _*35*. Как изменится модуль центробежной силы инерции, если скорость вращения системы отсчета увеличить в $n$ раз? Дайте необходимые пояснения._ + +*A:* + +$ +arrow(F)_"цб" = m arrow(omega) times (arrow(omega) times arrow(r)), space.quad F_"цб" = m omega^2 r. +$ + +Если угловую скорость увеличить в $n$ раз ($omega arrow n omega$), то +$ +F'_"цб" = m(n omega)^2 r = n^2 m omega^2 r = n^2 F_"цб". +$ + +То есть модуль центробежной силы возрастает *в $n^2$ раз*. Причина — квадратичная зависимость от $omega$ (или эквивалентно от линейной скорости $v = omega r$: $F_"цб" = (m v^2)/r)$. + + +#line(length: 100%) + +*Q:* _*36*. При движении космического корабля по круговой орбите имеет место состояние невесомости. Почему оно пропадает, когда корабль входит в атмосферу Земли? На каких участках траектории космического корабля возникают перегрузки? Что называют перегрузкой?_ + +*A:* + +1. Невесомость на орбите + +Корабль и всё внутри него движутся по орбите с одинаковым ускорением свободного падения $g_"орб"$. + +- На корабль и на космонавтов действует только сила тяготения. +- Опоры (пол, кресла) не оказывают давления → *сила реакции опоры равна нулю*, и человек ощущает невесомость. + +2. Почему невесомость пропадает в атмосфере + +При входе в атмосферу возникает сила сопротивления воздуха (аэродинамическая сила), действующая на корпус корабля. + +- Корабль начинает замедляться. +- Космонавт внутри ещё движется по инерции, ударяется о стенку → возникает сила давления. +- Появляется весовое ощущение → невесомость исчезает. + +3. Где возникают перегрузки + +Перегрузки возникают там, где ускорение корабля (или его частей) отличается от ускорения свободного падения: + +- *При старте* ракеты — от действия реактивной тяги. +- *При манёврах* (изменение скорости или траектории). +- *При торможении в атмосфере* (вход в плотные слои, аэродинамическое замедление). + +4. Определение перегрузки + +*Перегрузка* — это отношение полной кажущейся силы (реакции опоры), действующей на человека или прибор, к его весу в состоянии покоя на поверхности Земли: +$ +n = frac(R, m g). +$ + +Если $n>1$, человек ощущает усиленный вес (давление кресла), если $n<1$ — невесомость или даже «отрицательный вес». + +#line(length: 100%) + +*Q:* _*37*. При каких условиях возникают кориолисовы силы инерции? Зависит ли сила Кориолиса от скорости движения тела во вращающейся системе отсчета? Приведите примеры проявления сил Кориолиса на Земле._ + +*A:* + +1. Условия возникновения + +*Сила Кориолиса* появляется в неинерциальной системе отсчёта, которая *вращается* с угловой скоростью $arrow(omega)$. +Она действует на тело, которое имеет скорость $arrow(v)'$ относительно этой вращающейся системы. + +2. Формула + +$ +arrow(F)_"кор" = -2 m (arrow(omega) times arrow(v)'). +$ + +- Она есть *только при движении тела* внутри вращающейся системы ($arrow(v)' eq.not 0$). +- Чем больше скорость $v'$, тем больше сила. +- Зависит от направления движения относительно оси вращения. + +3. Примеры проявления на Земле + +- *Отклонение снарядов, ракет, авиации*: в Северном полушарии вправо от направления движения, в Южном — влево. +- *Циркуляция атмосферы и океанов*: пассаты, циклоны и антициклоны закручиваются из-за силы Кориолиса. +- *Течения рек*: в северном полушарии течение интенсивнее размывает правый берег, в южном — левый. +- *Маятник Фуко*: плоскость колебаний медленно поворачивается из-за вращения Земли. + +#line(length: 100%) + +*Q:* _*38*. По какой формуле находится сила Кориолиса? Сформулируйте правило, по которому определяется направление силы Кориолиса. Почему сила Кориолиса отсутствует, когда тело движется параллельно оси вращения системы?_ + +*A:* + +1. Формула силы Кориолиса + +$ +arrow(F)_"кор" = -2 m (arrow(omega) times arrow(v)'), +$ +где + +- $m$ — масса тела, +- $arrow(omega)$ — угловая скорость вращающейся системы отсчёта, +- $arrow(v)'$ — скорость тела относительно этой системы. + +2. Направление силы (правило) + +Направление $arrow(F)_"кор"$ определяется по правилу векторного произведения: + +- берём вектор $arrow(omega)$ (ось вращения, направление по правилу правого винта), +- берём вектор скорости $arrow(v)'$, +- строим $arrow(omega) times arrow(v)'$ по правилу правой руки, +- умножаем на $-2 m$, то есть получаем вектор, направленный в противоположную сторону. + +Иными словами: сила Кориолиса всегда перпендикулярна и к оси вращения, и к направлению движения. + +3. Почему сила Кориолиса отсутствует при движении вдоль оси вращения + +- Если $arrow(v)'$ параллелен $arrow(omega)$, то $arrow(omega) times arrow(v)' = 0$. +- Следовательно, $arrow(F)_"кор" = 0$. +- Физически: движение вдоль оси вращения не связано с изменением радиус-вектора относительно оси, поэтому вращение системы не создаёт поперечного отклонения. + +#line(length: 100%) + +*Q:* _*39*. Как направлена сила Кориолиса при движении тела: по экватору с востока на запад; с запада на восток; на экваторе вдоль меридиана?_ + +*A:* + +1. Формула + +$ +arrow(F)_"кор" = -2 m (arrow(omega) times arrow(v)'), +$ +где $arrow(omega)$ — угловая скорость вращения Земли (направлена вдоль земной оси с юга на север, то есть к Северному полюсу). + +2. Движение по экватору с *запада на восток* + +- Скорость $arrow(v)'$ направлена на восток. +- $arrow(omega)$ направлена на север. +- Векторное произведение $arrow(omega) times arrow(v)'$ — вверх (радиально наружу). +- С учётом минуса: сила Кориолиса направлена *к центру Земли (вниз)*. + +3. Движение по экватору с *востока на запад* + +- Скорость $arrow(v)'$ направлена на запад. +- $arrow(omega) times arrow(v)'$ — вниз (к центру). +- Сила Кориолиса с минусом — *радиально наружу (вверх)*. + +4. Движение на экваторе вдоль меридиана (север ↔ юг) + +- Здесь $arrow(v)'$ параллельна $arrow(omega)$. +- Тогда $arrow(omega) times arrow(v)' = 0$. +- *Сила Кориолиса отсутствует.* + +#line(length: 100%) + +*Q:* _*40*. Объясните, как на основе существования сил Кориолиса было экспериментально доказано суточное вращение Земли при помощи маятника Фуко._ + +*A:* + +1. Суть маятника Фуко + +Маятник Фуко — это длинный тяжёлый маятник, способный свободно качаться в любой вертикальной плоскости. + +- В идеале плоскость его колебаний остаётся неизменной в инерциальной системе. +- Но Земля вращается → для наблюдателя, находящегося на Земле (неинерциальная система), плоскость колебаний *кажется поворачивающейся*. + +2. Роль силы Кориолиса + +На колеблющийся груз действует сила Кориолиса: +$ +arrow(F)_"кор" = -2 m (arrow(omega) times arrow(v)). +$ + +- Она вызывает постепенное смещение траектории грузика. +- В результате плоскость колебаний маятника медленно поворачивается относительно поверхности Земли. + +3. Наблюдаемый эффект + +Скорость поворота плоскости: +$ +Omega = omega sin phi, +$ +где $omega$ — угловая скорость вращения Земли, $phi$ — широта. + +- На *полюсах* ($phi = 90 degree$) плоскость полностью поворачивается за 24 часа. +- На *экваторе* ($phi=0$) поворот отсутствует. +- В промежуточных широтах — период больше суток, но всегда конечен. + +4. Экспериментальное доказательство + +Фуко (1851 г., Пантеон, Париж) подвесил маятник длиной 67 м. + +- Плоскость колебаний маятника постепенно поворачивалась относительно здания. +- Это было наглядным доказательством того, что вращается именно Земля, а не маятник. + +#line(length: 100%) + +*Q:* _*41*. Существование кориолисовых сил является результатом того, что в северном полушарии правый берег реки всегда более крутой, чем левый; правый рельс железной дороги изнашивается сильнее, чем левый. Вместе с тем известно, что сила Кориолиса перпендикулярна вектору относительной скорости и, следовательно, не может выполнять работы. За счет какой энергии выполняется работа в рассмотренных выше случаях?_ + +*A:* + +1. Свойство силы Кориолиса + +Сила Кориолиса всегда перпендикулярна скорости тела: +$ +arrow(F)_"кор" perp arrow(v), +$ +поэтому работа этой силы равна нулю: +$ +A = integral arrow(F)_"кор" dot d arrow(r) = 0. +$ +Она может *изменять направление движения*, но не его кинетическую энергию. + +2. Тогда откуда работа в реальных процессах? + +Когда мы наблюдаем такие явления, как: + +- размыв правого берега рек в Северном полушарии, +- больший износ правого рельса, + +работа совершается *не силой Кориолиса напрямую*, а *реальными силами взаимодействия*: + +- Для рек: давление и трение потока о берега. Сила Кориолиса лишь отклоняет поток, перераспределяя направление скоростей → вода сильнее ударяет в правый берег, и его разрушает *гидродинамическая сила воды*. +- Для поездов: Кориолис вызывает микроскопическое смещение колёс относительно рельсов → возрастает давление на правый рельс, и именно *нормальная реакция рельса* выполняет работу изнашивания. + +3. Источник энергии + +Энергия берётся из *основного движения системы*: + +- в реках — из потенциальной энергии воды, стекающей вниз по уклону, +- в железной дороге — из работы двигателя локомотива, обеспечивающего поступательное движение поезда. + +#line(length: 100%) + +*Q:* _*42*. Дайте определение веса тела. Какова природа этой силы? Запишите формулу связи веса тела с силами инерции для системы, которая движется относительно Земли поступательно с некоторым ускорением._ + +*A:* + +1. Определение веса тела + +*Вес тела* — сила, с которой тело действует на опору или подвес вследствие действия на него сил притяжения и инерции. + +То есть вес — это *реакция опоры* (или натяжение подвеса), а не сама сила тяжести. + +2. Природа силы + +- Основная причина появления веса — *гравитационная сила*, действующая на тело. +- Но если система отсчёта неинерциальная (движется с ускорением), то к силе тяжести добавляются *силы инерции*, и вес изменяется. + +3. Формула для поступательно ускоряющейся системы + +Если система движется с ускорением $arrow(a)_0$ относительно Земли (например, лифт), то эффективный вес: + +$ +arrow(P) = m arrow(g) - m arrow(a)_0, +$ + +где + +- $m$ — масса тела, +- $arrow(g)$ — ускорение свободного падения, +- $arrow(a)_0$ — ускорение системы. + +4. Примеры + +- Лифт ускоряется *вверх* с ускорением $a$: +$ +P = m(g + a) space.quad arrow.double space.quad "кажущийся вес увеличивается". +$ + +- Лифт ускоряется *вниз*: +$ +P = m(g-a) space.quad arrow.double space.quad "вес уменьшается". +$ + +- При $a = g$: +$ +P = 0 space.quad arrow.double space.quad "невесомость". +$ + +#line(length: 100%) + +*Q:* _*43*. Какие причины обусловливают зависимость ускорения свободного падения от географической широты места? Как выражается эта зависимость?_ + +*A:* + +1. Причины зависимости $g$ от широты $phi$ + +1. *Вращение Земли* + + - При вращении Земли тело испытывает центробежное ускорение: +$ +a_"цб" = omega^2 R cos phi, +$ + направленное от оси вращения. + - На экваторе ($phi = 0 degree$) оно максимальное, на полюсах ($phi = 90 degree$) равно нулю. + +2. *Сплюснутость Земли (геоид)* + + - Радиус Земли на экваторе больше, чем на полюсах. + - По закону тяготения $g prop frac(1, R^2)$. + - Поэтому на экваторе $g$ меньше, а на полюсах больше. + +2. Общая зависимость + +Ускорение свободного падения на широте $phi$: +$ +g(phi) = g_0 (1 + alpha sin^2 phi) - omega^2 R cos^2 phi, +$ +где + +- $g_0$ — номинальное значение у поверхности Земли, +- $alpha$ — поправка на сплюснутость Земли ($approx 0.0053$), +- $omega$ — угловая скорость вращения Земли, +- $R$ — экваториальный радиус Земли. + +3. Итог + +- На экваторе $g approx 9.78 "м/с"^2$. +- На полюсах $g approx 9.83 "м/с"^2$. +- Разница $≈ 0.5 %$. + +#line(length: 100%) + +*Q:* _*44*. Какой формулой определяется угол между направлением к центру Земли и направлением линии отвеса? От чего зависит его величина? В каких границах он изменяется?_ + +*A:* + +1. Причина отклонения + +Линия отвеса определяется *суммарным ускорением*, действующим на тело: +$ +arrow(g)_"эфф" = arrow(g) + arrow(a)_"цб", +$ +где + +- $arrow(g)$ — ускорение гравитации (к центру Земли), +- $arrow(a)_"цб"$ — центробежное ускорение из-за вращения Земли. + +Центробежное ускорение направлено перпендикулярно оси вращения, поэтому отвес чуть отклоняется от истинного центра Земли. + +2. Формула для угла отклонения $delta$ + +Угол между направлением к центру Земли и линией отвеса: +$ +tan delta = frac(a_"цб,гор", g), +$ +где $a_"цб,гор" = omega^2 R cos phi dot sin phi$ — горизонтальная составляющая центробежного ускорения на широте $phi$. + +Приближённо: +$ +delta approx frac(omega^2 R, g) sin phi cos phi. +$ + +3. Зависимость и границы + +- Зависит от широты $phi$. +- На экваторе ($phi = 0 degree$) и на полюсах ($phi = 90 degree$) угол равен нулю (отвес точно указывает в центр Земли). +- Максимальное отклонение наблюдается при $phi=45 degree$. +- Величина отклонения очень мала: порядка *2′ (угловых минут)*, то есть меньше $0.05°$. + +#line(length: 100%) + +*Q:* _*45*. В каких границах изменяется вблизи поверхности Земли ускорение свободного падения? Какие причины обуславливают эти изменения?_ + +*A:* + +1. Диапазон значений $g$ на Земле + +Ускорение свободного падения у поверхности Земли изменяется примерно в пределах: + +$ +g approx 9,78 "м/с"^2 space.quad "экватор" div 9,83 "м/с"^2 space.quad "полюсы". +$ + +Разница ≈ 0,05 м/с² (около 0,5 %). + +2. Причины изменения $g$ + +1. *Вращение Земли* + + - На экваторе действует центробежное ускорение: + $ + a_"цб" = omega^2 R_"экв" approx 0,034 "м/с"^2, + $ + что уменьшает эффективное $g$. + - На полюсах центробежного ускорения нет → $g$ больше. + +2. *Сплюснутость Земли (геоид)* + + - Земля не идеально сферическая: радиус на экваторе больше, чем на полюсах. + - По закону тяготения $g prop 1/(R^2)$, поэтому на экваторе (где радиус больше) $g$ меньше. + +3. *Локальные геологические особенности* + + - Различия в плотности пород, наличие гор, впадин, полезных ископаемых создают небольшие локальные отклонения $g$ (в пределах тысячных долей). + +#line(length: 100%) + +*Q:* _*46*. Одно и то же тело взвесили на пружинных и рычажных весах на экваторе и полюсе. Различаются показания этих приборов? Если, да, то каким именно образом? Дайте необходимые пояснения._ + +*A:* + +1. Что измеряют разные весы + +- *Рычажные весы* (двухчашечные) сравнивают массы по уравновешиванию моментов сил тяжести. + ➝ Их показания определяются отношением масс и *не зависят* от величины ускорения свободного падения $g$. + +- *Пружинные весы* измеряют силу упругости пружины, которая уравновешивает вес тела. + ➝ Их показания пропорциональны *силе тяжести*, то есть зависят от значения $g$. + +2. Зависимость $g$ от широты + +На экваторе и на полюсе $g$ различно: + +- На *экваторе* меньше, потому что: + + 1. радиус Земли больше (дальше от центра → меньше $g$); + 2. есть центробежное ускорение из-за вращения Земли. + +- На *полюсе* $g$ больше (радиус меньше, центробежное ускорение отсутствует). + +3. Сравнение показаний + +- *Рычажные весы*: одинаковые на экваторе и на полюсе. +- *Пружинные весы*: покажут меньший «вес» на экваторе, больший — на полюсе. + +#line(length: 100%) + +*Q:* _*47*. Что называют гироскопом? Какие оси имеет гироскоп? В чем заключается гироскопический эффект? Какие явления называют прецессией? Нутацией?_ + +*A:* + +1. Что называют гироскопом + +*Гироскоп* — это быстро вращающееся твёрдое тело (обычно ротор), установленное так, что его ось вращения может свободно изменять ориентацию в пространстве. +Основное свойство: благодаря большому моменту импульса $arrow(L) = I arrow(omega)$ ось гироскопа сохраняет своё направление в пространстве при отсутствии внешних моментов. + +2. Оси гироскопа + +- *Ось вращения* (собственная ось ротора). +- *Ось симметрии подвеса* (через центр масс и опору). +- *Ось прецессии* (ось, вокруг которой поворачивается ось гироскопа под действием внешнего момента). + +3. Гироскопический эффект + +*Гироскопический эффект* — это свойство гироскопа сохранять неизменным направление своей оси в пространстве или изменять его особым образом (прецессией), несмотря на приложенные силы. +Основан на законе сохранения момента импульса. + +4. Прецессия + +*Прецессия* — это медленное вращение оси гироскопа вокруг другой оси под действием момента внешних сил, перпендикулярного вектору собственного момента импульса. +Скорость прецессии: +$ +Omega = M/L. +$ + +5. Нутация + +*Нутация* — это дополнительные колебательные движения оси гироскопа (наклонные колебания вокруг линии средней прецессии). +Они возникают при резком изменении внешнего момента или при несовершенстве подвеса. + +#line(length: 100%) + +*Q:* _*48*. Почему под действием момента внешних сил, перпендикулярного оси собственного импульса, гироскоп начинает совершать прецессию?_ + +*A:* + +1. Собственный момент импульса гироскопа + +Вращающееся тело (ротор) имеет момент импульса +$ +arrow(L) = I arrow(omega), +$ +направленный вдоль оси вращения. + +2. Действие внешнего момента + +Если к гироскопу приложен момент внешних сил $arrow(M)$, то по закону изменения момента импульса: +$ +frac(d arrow(L), d t) = arrow(M). +$ + +- Если $arrow(M)$ направлен *перпендикулярно* $arrow(L)$, то он не меняет величину $|arrow(L)|$, а изменяет только *направление* вектора $arrow(L)$. +- Вектор $arrow(L)$ начинает поворачиваться в направлении действия $arrow(M)$. + +3. Прецессия + +Поскольку ось гироскопа совпадает с направлением $arrow(L)$, её ориентация также начинает изменяться → ось описывает медленное вращение вокруг вертикали (или другой оси, вдоль которой приложен момент). Это движение называется *прецессией*. + +Скорость прецессии: +$ +arrow(Omega) = frac(arrow(M), |arrow(L)|). +$ + +4. Физический смысл + +- Момент сил не «опрокидывает» гироскоп (как это было бы для невращающегося тела), а лишь изменяет направление его оси. +- Поэтому гироскоп «устойчив» и реагирует на внешний момент боковым вращением — прецессией. + +#line(length: 100%) + +*Q:* _*49*. Приведите примеры практического применения гироскопического эффекта._ + +*A:* Гироскопический эффект — это способность вращающегося тела сохранять направление своей оси в пространстве (устойчивость при воздействии внешних моментов). Его активно используют в технике. + +Примеры применения: + +1. *Навигация и ориентация* + + - гирокомпасы на кораблях и подводных лодках (ориентация без магнитного поля); + - гироскопические приборы в авиации (гирогоризонт, авиагоризонт, указатель поворота); + - инерциальные навигационные системы ракет, спутников, космических аппаратов. + +2. *Транспорт* + + - стабилизация велосипедов, мотоциклов (вращающиеся колёса дают устойчивость); + - гиростабилизаторы в поездах и автомобилях для повышения устойчивости. + +3. *Космос* + + - ориентация искусственных спутников и космических станций (маховики-реакционные колёса, гиростабилизаторы). + +4. *Морская техника* + + - гиростабилизаторы на судах и яхтах для уменьшения качки. + +5. *Электроника и бытовая техника* + + - гиродатчики в смартфонах, дронах, геймпадах — для определения положения в пространстве. + +#pagebreak() +#align(center)[= _Колебания и волны_] + +*Q:* _*1*. Какое движение называется колебательным? Приведите примеры._ + +*A:* *Колебательное движение* — это движение, которое многократно повторяется во времени и происходит около положения устойчивого равновесия. +Основные признаки: + +- наличие равновесного положения, +- отклонение от него, +- возврат под действием сил, стремящихся вернуть систему в равновесие, +- периодичность или квазипериодичность. + +Примеры колебательных движений: + +- механические: + + - колебания маятника, + - колебания пружинного маятника (груз на пружине), + - вибрация струны; +- акустические: звуковые колебания воздуха; +- электрические: колебания заряда и тока в колебательном контуре $L C$; +- физические системы: колебания молекул в твёрдых телах (фононы), колебания в кристаллической решётке. + +#line(length: 100%) + +*Q:* _*2*. Какие колебания называются гармоническими? По каким критериям эти колебания в природе и технике выделяют в особую группу?_ + +*A:* *Гармонические колебания* — это такие колебания, при которых физическая величина (смещение, скорость, ток и т. п.) изменяется по закону синуса или косинуса: + +$ +x(t) = A cos(omega t + phi_0), +$ + +где $A$ — амплитуда, $omega$ — циклическая частота, $phi_0$ — начальная фаза. + +2. Критерии выделения в особую группу + +Гармонические колебания считаются базовыми, потому что: + +- *линейность*: они возникают в линейных системах при малых отклонениях от равновесия (например, закон Гука); +- *универсальность*: любое сложное периодическое движение можно разложить в сумму гармонических (ряд Фурье); +- *простота анализа*: имеют чёткие характеристики (амплитуда, период, частота, фаза), легко измеряемые и вычисляемые; +- *распространённость*: в природе и технике они встречаются очень часто — от маятника и колебаний атомов в кристалле до переменного тока и радиоволн. + +#line(length: 100%) + +*Q:* _*3*. Дайте определение основных величин, характеризующих гармоническое колебательное движение (амплитуды, фазы, периода, частоты, циклической частоты). Каков физический смысл этих характеристик?_ + +*A:* Возьмём гармоническое колебание + +$ +x(t) = A cos(omega t + phi_0). +$ + +Основные величины + +1. *Амплитуда $A$* + + - Максимальное по модулю отклонение колеблющейся величины от положения равновесия. + - Физический смысл: характеризует «размах» колебаний (максимальное смещение, максимальный ток и т. д.). + +2. *Фаза $phi(t) = omega t + phi_0$* + + - Аргумент функции косинуса/синуса, определяющий состояние колеблющейся системы в данный момент времени. + - Физический смысл: задаёт «положение» колебания на цикле (например, находится ли система в максимуме, на спаде, в равновесии). + +3. *Начальная фаза $phi_0$* + + - Значение фазы в момент $t = 0$. + - Физический смысл: определяет, с какого состояния система начинает колебаться. + +4. *Период $T$* + + - Время одного полного колебания. + - Физический смысл: длительность одного «цикла» движения. + - Формула: +$ +T = frac(2 pi, omega). +$ + +5. *Частота $nu$* + + - Число колебаний в единицу времени. + - Связь с периодом: +$ +nu = 1/T. +$ + - Физический смысл: показывает, сколько раз за секунду повторяется движение. + +6. *Циклическая частота $omega$* + + - Угловая скорость изменения фазы (в радианах в секунду). + - Связь с частотой: +$ +omega = 2 pi nu = frac(2 pi, T). +$ + - Физический смысл: показывает, с какой скоростью «бежит» фаза; удобно в формулах, потому что измеряется в рад/с. + +#line(length: 100%) + +*Q:* _*4*. Какими признаками определяются упругие силы? В чем различие упругих и квазиупругих сил?_ + +*A:* 1. Признаки упругих сил + +Упругая сила проявляется, когда тело деформируется, и характеризуется: + +- *Противодействием деформации* — стремится вернуть тело в исходное (равновесное) состояние. +- *Зависимостью только от величины деформации* (для идеальной упругости): +$ +F = -k Delta x +$ + (закон Гука, где $k$ — жёсткость, $Delta x$ — удлинение/сжатие). +- *Мгновенной обратимостью*: работа упругой силы полностью возвращается при устранении деформации (нет потерь энергии). + +2. Квазиупругие силы + +Это силы, которые внешне ведут себя *похоже на упругие*, но: + +- они не полностью восстанавливают исходное состояние; +- при деформации и обратном движении часть энергии рассеивается (в тепло, внутренние процессы); +- сила зависит не только от величины деформации, но и от скорости, времени, внутреннего трения. + +Пример: резина, полимеры, биологические ткани — при растяжении и отпускании графики $F(Delta x)$ не совпадают (явление гистерезиса). + +3. Различие + +- *Упругие силы*: строго подчиняются закону Гука, полностью обратимы, энергия сохраняется. +- *Квазиупругие силы*: лишь приближённо подчиняются закону Гука, сопровождаются диссипацией энергии, обратимость неполная. + +#line(length: 100%) + +*Q:* _*5*. Какие колебания называются свободными? При каких условиях свободные колебания будут незатухающими?_ + +*A:* *Свободные колебания* — это колебания системы, происходящие только за счёт её собственных внутренних сил после того, как система была выведена из положения равновесия, *без дальнейшего внешнего воздействия*. + +Примеры: колебания маятника после отклонения, груз на пружине после растяжения. + +2. Условия незатухающих свободных колебаний + +В реальности всегда есть силы сопротивления (трение, сопротивление воздуха), из-за которых колебания затухают. +*Незатухающими* свободные колебания будут только при идеализированных условиях: + +- отсутствуют диссипативные силы (трение, сопротивление среды, внутренние потери), +- действуют только *консервативные силы* (сила тяжести, упругости). + +В этом случае система бесконечно долго колеблется с постоянной амплитудой и постоянной собственной частотой. + +#line(length: 100%) + +*Q:* _*6*. На примере прямолинейных гармонических колебаний установите связь между амплитудами и фазами смещения скорости и ускорения материальной точки._ + +*A:* Возьмём прямолинейные гармонические колебания + +$ +x(t) = A cos(omega t + phi). +$ + +Амплитуды: +$ +v(t) = dot(x) = -A omega sin(omega t + phi) = A omega cos (omega t + phi + frac(pi, 2)) arrow.double A_v = omega A, +$ +$ +a(t)= dot(x) = -omega^2 x = -A omega^2 cos(omega t + phi) = A omega^2 cos (omega t + phi + pi) arrow.double A_a = omega^2 A. +$ + +Фазовые сдвиги: + +- Скорость опережает смещение на $frac(pi, 2)$ (четверть периода): + $phi_v = phi + frac(pi, 2)$. +- Ускорение в противофазе со смещением (сдвиг $pi$): + $phi_a = phi + pi$. +- Ускорение опережает скорость на $frac(pi, 2)$ (или скорость отстаёт на $frac(pi, 2))$. + +Итог по связям: +$ +A_v = omega A, space.quad A_a = omega^2 A, space.quad phi_v - phi = frac(pi, 2), space.quad phi_a - phi = pi. +$ + + +#line(length: 100%) + +*Q:* _*7*. В течении какой доли периода колебаний совпадают направления векторов смещения и скорости простого гармонического осциллятора?_ + +*A:* Полпериода. + +В SHM $x = A cos(omega t + phi)$, $v = -A omega sin(omega t + phi)$. Направления совпадают, когда $x$ и $v$ одного знака ⇔ $sin(2(omega t + phi)) < 0$, что выполняется на интервалах ($frac(pi, 2), pi$) и ($frac(3 pi, 2), 2 pi$) по фазе — суммарно $T/2$. Физически: совпадают, когда осциллятор уходит от равновесия к крайним положениям. + + +#line(length: 100%) + +*Q:* _*8*. При помощи метода векторных диаграмм найдите выражение для амплитуды: результирующего колебания, полученного в результате сложения двух колебаний одного направления и одинаковых частот. При каком условии амплитуда результирующего колебания равна сумме амплитуд составляющих колебаний? Модулю их разности?_ + +*A:* Классическое сложение фазоров. + +Пусть +$ +x_1 = A_1 cos( omega t+ phi_1), space.quad +x_2 = A_2 cos( omega t+ phi_2), space.quad +Delta phi = phi_2 - phi_1. +$ +Тогда результирующее колебание того же вида +$ +x = A cos( omega t + phi), +$ +где амплитуда и фаза задаются по векторной диаграмме: +$ +A = sqrt(A_1^2 + A_2^2 + 2A_1 A_2 cos Delta phi) +$ + +$ +tan phi= frac(A_1 sin phi_1 + A_2 sin phi_2, A_1 cos phi_1+A_2 cos phi_2). +$ + +Условия: + +- $A = A_1 + A_2$ при фазовом совпадении ($Delta phi = 0$, в фазе). +- $A = |A_1 - A_2|$ при противофазе ($Delta phi = pi$). + +#line(length: 100%) + +*Q:* _*9*. В каком случае при сложении двух колебаний будет наблюдаться процесс биений? Запишите уравнение такого процесса, изобразите график зависимости смещения от времени._ + +*A:* *Когда возникают биения:* при сложении двух колебаний одного направления с близкими, но различными частотами $omega_1 approx omega_2$ и сравнимыми амплитудами. + +*Уравнение суммы* (для простоты $A_1 = A_2 = A, phi_1 = phi_2 = 0$): +$ +x(t) = A cos omega_1 t + A cos omega_2 t = 2 A cos(frac(omega_2-omega_1, 2)t), cos(frac(omega_1 + omega_2, 2)t). +$ +Амплитуда медленно меняется по огибающей + +$ 2 A cos(frac(Delta omega, 2)t)$. +Частота биений $f_b = |f_2 - f_1|$, период биений $T_b = frac(1, |f_2 - f_1|)$. + +Для разных амплитуд: +$ +x = A_1 cos omega_1 t + A_2 cos omega_2 t, space.quad A_"огиб" (t) = sqrt(A_1^2 + A_2^2 + 2A_1A_2 cos(Delta omega, t)). +$ + +#line(length: 100%) + +*Q:* _*10*. Каков вид траектории точки, которая участвует одновременно в двух взаимно перпендикулярных гармонических колебаниях с одинаковыми частотами? При каких значениях разности фаз траекторией движения будет прямая и при каких — эллипс?_ + +*A:* Возьмём + +$ +x(t) = A cos omega t, space.quad y(t) = B cos(omega t + delta). +$ + +Вид траектории + +Исключая $t$, получаем связь +$ +(frac(x, A))^2 + (frac(y, B))^2 - 2 frac(x, A) frac(y, B) cos delta = sin^2 delta, +$ + +что описывает *эллипс* (вообще — повернутый). Угол поворота $theta$ удовлетворяет +$ +tan 2 theta = frac(2 A B cos delta, A^2 - B^2). +$ + +Когда прямая? + +- При $delta = 0$ или $delta = pi$: +$ +y=plus.minus frac(B, A) x, +$ + то есть траектория — *прямая* (проекции синфазны/противофазны). + +Когда эллипс? + +- При любом $delta eq.not.triple 0, pi mod(2 pi)$ — *эллипс*. +- Частный случай круга: если $A=B$ и $delta = plus.minus frac(pi, 2)$, то +$ +x^2+y^2=A^2, +$ + траектория — *окружность*. + + +#line(length: 100%) + +*Q:* _*11*. Что называется фигурами Лиссажу? Как по виду фигур Лиссажу можно установить соотношение частот складываемых колебаний?_ + +*A:* *Фигуры Лиссажу* — это траектории, которые описывает точка, участвующая одновременно в двух взаимно перпендикулярных гармонических колебаниях: + +$ +x(t) = A_x cos(omega_x t + phi_x), space.quad +y(t) = A_y cos(omega_y t + phi_y). +$ + +Если частоты отличаются, получаются характерные замкнутые или незамкнутые кривые на плоскости ($x,y$). + +2. Установление соотношения частот + +- Если отношение частот рациональное: +$ +frac(omega_x, omega_y) = frac(n, m), +$ + то траектория замкнута и называется *фигурой Лиссажу*. +- Числа $n$ и $m$ можно определить по виду фигуры: + + - *число петель* вдоль оси $x = n$, + - *число петель* вдоль оси $y = m$. + +Пример: при $omega_x : omega_y = 2:1$ получится фигура с двумя «петлями» по оси $x$ и одной по оси $y$. + +3. Особый случай + +Если $omega_x / omega_y$ иррационально, то кривая *не замыкается*, и траектория со временем заполняет всё ограниченное прямоугольное пространство. + +#line(length: 100%) + +*Q:* _*12*. Выведите формулы для периодов колебаний пружинного, математического и физического маятников._ + +*A:* Разберём каждый тип маятника. + +1. *Пружинный маятник* + +Тело массы $m$ на пружине жёсткости $k$. + +Уравнение движения: + +$ +m dot(dot(x)) + k x = 0. +$ + +Собственная циклическая частота: +$ +omega = sqrt(frac(k, m)). +$ + +Период: +$ +T = 2 pi sqrt(frac(m, k)). +$ + +2. *Математический маятник* + +Груз массы $m$ на невесомой нерастяжимой нити длиной $l$. +При малых углах $(sin phi approx phi)$: + +$ +dot(dot(phi)) + frac(g, l) phi = 0. +$ + +Собственная частота: +$ +omega = sqrt(frac(g, l)). +$ +Период: +$ +T = 2 pi sqrt(frac(l, g)). +$ + +3. *Физический маятник* + +Любое твёрдое тело, колеблющееся вокруг горизонтальной оси, не проходящей через центр масс. +Уравнение движения: +$ +I dot(dot(phi)) + m g l phi = 0, +$ +где $I$ — момент инерции относительно оси подвеса, $l$ — расстояние от оси до центра масс. +Собственная частота: +$ +omega = sqrt(frac(m g l, I)). +$ +Период: +$ +T = 2 pi sqrt(frac(I, m g l)). +$ + +#line(length: 100%) + +*Q:* _*13*. В чем состоит различие между математическим и физическим маятниками? В чем заключается физический смысл приведенной длины физического маятника?_ + +*A:* + +1. Математический маятник + +- Идеализированная модель. +- Материальная точка массы $m$, подвешенная на невесомой нерастяжимой нити длины $l$. +- Колебания происходят под действием силы тяжести. +- Период колебаний: +$ +T = 2 pi sqrt(l/g). +$ + +2. Физический маятник + +- Любое реальное твёрдое тело, способное колебаться вокруг горизонтальной оси, не проходящей через его центр масс. +- Тело имеет распределение масс и момент инерции (I) относительно оси подвеса. +- Период колебаний: +$ +T = 2 pi sqrt(frac(I, m g d)), +$ + где $d$ — расстояние от оси подвеса до центра масс. + +3. Приведённая длина физического маятника + +- Вводят для того, чтобы *сравнить физический маятник с эквивалентным математическим*. +- Определяется как: +$ +l_"пр" = frac(I, m d). +$ + +- Тогда формула периода приобретает вид: +$ +T = 2 pi sqrt(frac(l_"пр", g)), +$ + что полностью аналогично периоду математического маятника. + +#line(length: 100%) + +*Q:* _*14*. Почему период колебаний математического маятника не зависит от массы, а период физического маятника зависит от момента инерции._ + +*A:* + +1. Математический маятник + +- Период: +$ +T = 2 pi sqrt(l / g). +$ +- Здесь масса $m$ сокращается при выводе: сила тяжести $F = m g$ пропорциональна массе, и ускорение $a = F / m = g$ не зависит от массы. + Поэтому *период математического маятника не зависит от массы груза*. + +2. Физический маятник + +- Период: +$ +T = 2 pi sqrt(frac(I, m g d)), +$ + где $I$ — момент инерции относительно оси подвеса, $d$ — расстояние от центра масс до оси. + +- Здесь масса входит в момент инерции $I$. Для тела с распределённой массой $I = sum m_i r_i^2$. + +- Таким образом, период определяется не только массой, но и её распределением относительно оси. + +#line(length: 100%) + +*Q:* _*15*. Какие колебания называются затухающими? Как изменяется со временем амплитуда затухающих колебаний? Выразите эту зависимость аналитически и графически._ + +*A:* *Затухающие колебания* — это колебания с убывающей во времени амплитудой из-за потерь энергии (трение, сопротивление среды и т.п.). + +- При линейном вязком сопротивлении: +$ +x(t) = A_0 e^(-gamma t) cos(omega t + phi), space.quad A(t) = A_0 e^(-gamma t). +$ +- Полезные характеристики: +$ +Lambda = ln frac(x_n, x_(n+1)) = gamma T_d, space.quad zeta = frac(gamma, omega_0), space.quad omega = omega_0 sqrt(1 - zeta^2) (zeta < 1). +$ + +График с огибающей $plus.minus A(t)$ построен выше. + +#align(center)[#image("assets/8.png")] + +#line(length: 100%) + +*Q:* _*16*. Что понимают под периодом затухающих колебаний? Что такое коэффициент затухания, логарифмический декремент, добротность колебательной системы?_ + +*A:* + +1. Период затухающих колебаний + +Уравнение затухающих колебаний: +$ +x(t) = A_0 e^(-gamma t) cos(omega t + phi), +$ +где $gamma$ — коэффициент затухания, $omega$ — круговая частота затухающих колебаний. + +*Периодом* $T_d$ называют время между двумя последовательными колебаниями: +$ +T_d = frac(2 pi, omega). +$ + +2. Коэффициент затухания $gamma$ + +Характеризует скорость убывания амплитуды: +$ +A(t) = A_0 e^(-gamma t). +$ +Единицы измерения: [1/с]. + +3. Логарифмический декремент $Lambda$ + +Величина, показывающая, во сколько раз уменьшается амплитуда за один период: +$ +Lambda = ln frac(A(t), A(t + T_d)) = gamma T_d. +$ + +4. Добротность (Q) колебательной системы + +Показывает «качество» колебаний — насколько долго система сохраняет энергию. Определение: +$ +Q = frac(2 pi dot "(запасённая энергия)", "потерянная энергия за период"). +$ +Через параметры затухания: +$ +Q = frac(pi, Lambda) = frac(omega, 2 gamma). +$ + +#line(length: 100%) + +*Q:* _*17*. Получите формулы, выражающие зависимость кинетической и потенциальной энергий колебательной системы от времени. В каких случаях полная энергия колебательной системы будет постоянной? Представьте энергетические характеристики колебательной системы графически._ + +*A:* Сформулируем для гармонического осциллятора $m dot(dot(x)) x + k x=0$ (и с затуханием при необходимости). + +Формулы энергий + +Для решения $x(t) = A cos(omega t + phi)$, $omega = sqrt(k/m)$: +$ +v(t) = -A omega sin(omega t + phi). +$ +$ +E_k(t) = 1/2 m v^2 = 1/2 m A^2 omega^2 sin^2(omega t + phi), +$ +$ +E_p(t) = 1/2 k x^2 = 1/2 k A^2 cos^2(omega t + phi). +$ +$ +E(t) = E_k + E_p = 1/2 k A^2 = "const". +$ + +Если есть вязкое затухание $m dot(dot(x)) + 2 m gamma dot(x) + k x = 0$, $x= A_0 e^(-gamma t) cos(omega t + phi)$ с $omega = sqrt(omega_0^2 - gamma^2)$, $omega_0 = sqrt(k/m)$. Тогда амплитуда убывает как $e^(-gamma t)$, а полная энергия как +$ +E(t) prop A^2(t) approx E_0,e^(-2 gamma t) space.quad (gamma << omega_0). +$ + +Когда энергия постоянна + +Полная энергия постоянна *только* в консервативном случае: нет сил трения/сопротивления, параметры $m, k$ постоянны, внешних источников или откачки энергии нет. + +#align(center)[#image("assets/9.png")] +#align(center)[#image("assets/10.png")] + +#line(length: 100%) + +*Q:* _*18*. Дайте определение собственных и вынужденных колебаний системы. Запишите дифференциальное уравнение вынужденных колебаний. Какова зависимость амплитуды вынужденных колебаний, возбуждаемых силой, которая изменяется по гармоническому закону, от частоты этой силы? Представьте данную зависимость аналитически и графически - для нескольких различных коэффициентов сопротивления._ + +*A:* + +Определения + +- *Собственные колебания* — свободные колебания системы без внешнего периодического воздействия (возникают за счёт начального отклонения/скорости). +- *Вынужденные колебания* — колебания под действием внешней силы $F(t)$, обычно периодической. + + +Уравнение вынужденных колебаний (линейный осциллятор с вязким сопротивлением) + +$ +m dot(dot(x)) + 2 m gamma dot(x) + k x = F_0 cos(Omega t). +$ + +Здесь $omega_0 = sqrt(k/m)$ — собственная частота, $gamma$ — коэффициент затухания. + +Амплитуда установившихся вынужденных колебаний + +$ +x(t) = A(Omega) cos(Omega t - delta), space.quad A(Omega) = frac(F_0/m, sqrt((omega_0^2 - Omega^2)^2 + (2 gamma Omega)^2)). +$ +Пик около $Omega_"рез" approx sqrt(omega_0^2 - 2 gamma^2)$ (чем меньше $gamma$, тем выше и уже резонанс). + +График $A(Omega)$ для разных $gamma$ + +Я построил АЧХ для $gamma = 0.2, 0.8, 1.6$ при $omega_0 = 6$ (в безразмерных единицах). Вертикальная пунктирная — $Omega = omega_0$. + +#align(center)[#image("assets/11.png")] + +#line(length: 100%) + +*Q:* _*19*. В чем состоит суть явления резонанса? От чего зависит резонансная частота? Запишите формулу, определяющую резонансную частоту. Приведите примеры явления резонанса._ + +*A:* + +1. Суть явления резонанса + +*Резонанс* — это явление резкого увеличения амплитуды вынужденных колебаний системы, когда частота внешней силы $Omega$ близка к собственной частоте системы $omega_0$. +Физический смысл: внешняя сила подкачивает энергию синхронно с колебаниями, и энергия накапливается. + +2. От чего зависит резонансная частота + +Резонансная частота зависит: + +- от *собственной частоты системы* $omega_0 = sqrt(k/m)$, +- от *коэффициента затухания* $gamma$. + +3. Формула для резонансной частоты + +Максимум амплитуды $A(Omega)$ достигается при: +$ +Omega_"рез" = sqrt(omega_0^2 - 2 gamma^2). +$ + +- При малом затухании $gamma << omega_0$: +$ +Omega_"рез" approx omega_0. +$ + +4. Примеры резонанса + +- Раскачивание качелей: толчки с частотой, близкой к собственной. +- Колебания мостов под действием ветра или шагов (знаменитый случай — разрушение моста Такома Нарроуз в 1940 году). +- Механический резонанс в машинах и механизмах (опасен, т.к. амплитуды могут стать разрушительными). +- Электрический резонанс в колебательных контурах (LCR-цепи). +- Акустический резонанс — усиление звука в резонансных полостях (например, в музыкальных инструментах). + +#line(length: 100%) + +*Q:* _*20*. Какие системы: называют автоколебательными? Приведите пример такой системы. Из каких основных элементов состоит автоколебательная система? Приведите пример релаксационных колебаний и охарактеризуйте их._ + +*A:* + +1. Автоколебательные системы + +*Автоколебательная система* — это система, в которой колебания поддерживаются за счёт внутренних источников энергии, без внешнего периодического воздействия. +Энергия, теряемая на сопротивление, периодически восполняется самим устройством → колебания устойчиво поддерживаются. + +2. Примеры автоколебательных систем + +- Маятниковые часы (маятник получает импульс от механизма). +- Генераторы электрических колебаний (ламповые, транзисторные). +- Свисток, органная труба (подпитка колебаний потоком воздуха). +- Биологические ритмы (например, сердечные сокращения, дыхание). + +3. Основные элементы автоколебательной системы + +1. *Колебательная система* (например, механический маятник, LC-контур). +2. *Источник энергии* (механическая пружина, батарея, поток воздуха). +3. *Устройство обратной связи*, которое подводит энергию к системе в нужной фазе, компенсируя потери. + +4. Релаксационные колебания + +- Это особый тип автоколебаний, при которых процесс имеет *медленный накопительный этап* и *быстрый скачок*. +- Обычно форма сигнала далека от синусоиды (пилообразная, импульсная). + +*Примеры:* + +- мигание неоновой лампы; +- работа генератора импульсов на электронной лампе или транзисторе; +- сердечные ритмы, дыхательные циклы. + +*Характеристика:* период релаксационных колебаний определяется временем медленного накопления энергии и моментом её быстрой разрядки. + +#line(length: 100%) + +*Q:* _*21*. Что называют волновым движением? Какие силы должны действовать между частицами среды, чтобы в ней могли распространяться колебания?_ + +*A:* *Волновое движение* — это процесс распространения колебаний в пространстве, при котором переносится энергия (и импульс), но вещество среды при этом не переносится. +Каждая частица среды лишь совершает колебания около своего положения равновесия, а возмущение передаётся соседним частицам. + +2. Условия для распространения колебаний + +Чтобы волна могла распространяться в среде, между её частицами должны действовать силы, которые: + +- *стремятся вернуть частицу в равновесие* (силы упругости, давления, тяжести и др.); +- *связывают движение соседних частиц* (иначе колебание не будет передаваться). + +То есть необходимы *силы взаимодействия* между частицами среды (упругие силы, силы давления, электромагнитные взаимодействия и т.п.), которые обеспечивают передачу энергии колебаний от одной частицы к другой. + +#line(length: 100%) + +*Q:* _*22*. Какие волны называются продольными? поперечными? Объясните механизм распространения деформаций в каждом из этих случаев. Какие волны могут распространяться в твердых телах, жидкостях и газах?_ + +*A:* + +1. Продольные волны + +- *Определение:* частицы среды колеблются *вдоль направления распространения волны*. +- *Механизм:* в среде возникают *периоды сжатия и разрежения*. Сжатые участки передают давление на соседние, и волна распространяется дальше. +- *Примеры:* звуковые волны в воздухе, газах и жидкостях. + +2. Поперечные волны + +- *Определение:* частицы среды колеблются *перпендикулярно направлению распространения волны*. +- *Механизм:* возникает чередование участков *сдвиговых деформаций*; упругие силы, стремящиеся вернуть частицы в исходное положение, передают возмущение соседям. +- *Примеры:* волны на поверхности воды, электромагнитные волны, колебания в струне. + +3. Где какие волны распространяются + +- *В твёрдых телах:* возможны *оба типа* (продольные и поперечные), так как твёрдое тело сопротивляется как сжатию, так и сдвигу. +- *В жидкостях и газах:* возможны *только продольные* волны, потому что они не обладают упругостью сдвига (не могут поддерживать поперечные деформации). + +#line(length: 100%) + +*Q:* _*23*. Что можно сказать о переносе энергии упругой деформации и переносе массы при распространении механической волны?_ + +*A:* + +1. Перенос энергии + +- При распространении механической волны *переносится энергия упругой деформации и кинетическая энергия колебаний частиц*. +- Частицы среды совершают колебания около положения равновесия, но их взаимодействие передаёт энергию соседям → возмущение распространяется. +- Таким образом, *энергия волны перемещается вместе с волной*. + +2. Перенос массы + +- Масса среды *не переносится* вместе с волной. +- Каждая частица лишь колеблется около своей равновесной точки, выполняя движение вперёд-назад (продольные) или вверх-вниз (поперечные), но *не смещается на большие расстояния*. +- В среднем перемещение массы отсутствует — переносится только энергия. + +#line(length: 100%) + +*Q:* _*24*. Объясните качественно зависимость скорости упругих волн от модуля Юнга (модуля сдвига) и плотности среды._ + +*A:* + +1. Что определяет скорость волны? + +Скорость распространения механической волны зависит от двух факторов: + +- *упругость среды* (способность сопротивляться деформации), +- *инертность среды* (масса/плотность частиц, которые нужно разгонять). + +2. Продольные волны (звук в твёрдом теле, жидкостях, газах) + +Для стержня или твёрдого тела: +$ +v = sqrt(frac(E, rho)), +$ + +где $E$ — модуль Юнга (характеризует упругость при растяжении/сжатии), $rho$ — плотность. + +- Чем больше $E$, тем быстрее волна (среда «жёстче», быстрее восстанавливает форму). +- Чем больше $rho$, тем медленнее волна (частицы тяжелее, труднее разогнать). + +3. Поперечные волны (в твёрдых телах) + +$ +v = sqrt(frac(G, rho)), +$ +где $G$ — модуль сдвига. + +- Чем больше $G$, тем выше скорость поперечной волны. +- Чем больше плотность $rho$, тем меньше скорость. + +4. Качественный вывод + +- *Жёсткие, но лёгкие материалы* (например, сталь по сравнению с резиной) → высокая скорость волн. +- *Мягкие или плотные материалы* → низкая скорость. + + +#line(length: 100%) + +*Q:* _*25*. В чем состоит различие между гармоническими колебаниями и волновым процессом? Запишите уравнение плоской монохроматической волны, дайте определение физических характеристик волны. Какова связь между этими величинами?_ + +*A:* + +1. Гармонические колебания vs волновой процесс + +- *Гармонические колебания* — это периодические изменения физической величины (смещения, скорости, давления и т. п.) во времени около положения равновесия в *одной точке*. +$ +x(t) = A cos(omega t + phi). +$ + +- *Волновой процесс* — это распространение колебаний в пространстве: возмущение от одной точки передаётся другим → возникает пространственно-временной процесс. +$ +xi(x,t) = A cos(omega t - k x + phi). +$ + +2. Уравнение плоской монохроматической волны + +$ +xi(x,t) = A cos(omega t - k x + phi), +$ +где: + +- $A$ — амплитуда колебаний, +- $omega = 2 pi f$ — циклическая частота, +- $f$ — частота, +- $T = frac(1, f)$ — период, +- $k = frac(2 pi, lambda)$ — волновое число, +- $lambda$ — длина волны, +- $v$ — фазовая скорость волны. + +3. Связь характеристик волны + +Основное соотношение: +$ +v = frac(omega, k) = lambda f. +$ + +#line(length: 100%) + +*Q:* _*26*. Нарисуйте график зависимости координат точек среды, в которой распространяется бегущая волна, от расстояния до источника волны. На каком расстоянии друг от друга находятся соседние точки среды, которые колеблются в одинаковых фазах?_ + +*A:* График профиля бегущей волны $xi(x, t_0)$ готов. +Соседние точки, колеблющиеся в одинаковых фазах, разделены расстоянием *одна длина волны* $lambda$ (и вообще $n lambda$, $n in Z$). + +#align(center)[#image("assets/12.png")] + +#line(length: 100%) + +*Q:* _*27*. Что называется фронтом волны? Чем отличается фронт волны от волновой поверхности? Какие волны называются плоскими, сферическими? Приведите примеры._ + +*A:* + +1. Фронт волны + +*Фронт волны* — геометрическое место точек, до которых дошло возмущение в данный момент времени. + +- Иначе говоря, это граница раздела между «возмущённой» и «невозмущённой» частями среды. +- Обычно фронт изображают как поверхность, движущуюся со скоростью распространения волны. + +2. Волновая поверхность + +*Волновая поверхность* — геометрическое место точек, которые колеблются в *одной и той же фазе*. + +- В отличие от фронта, это не граница области, а совокупность «фазово-синхронных» точек. + +Фронт волны всегда является *частным случаем волновой поверхности*, которая соответствует определённой фазе (например, началу возмущения). + +3. Типы волн + +- *Плоские волны* — волновые поверхности представляют собой параллельные плоскости. + + - Пример: электромагнитная волна, распространяющаяся от антенны вдоль определённого направления; акустическая волна в узкой трубе. + +- *Сферические волны* — волновые поверхности представляют собой концентрические сферы с центром в источнике. + + - Пример: звук от точечного источника (хлопок, взрыв); волны на воде от падающей капли. + +#line(length: 100%) + +*Q:* _*28*. Что называют фазовой и групповой скоростями? Чем вызвано их различие? Могут ли эти скорости совпадать? Что называют дисперсией волн?_ + +*A:* + +1. Фазовая скорость + +$ +v_"ф" = frac(omega, k) +$ +— скорость распространения *отдельной фазы* (например, гребня, впадины) гармонической волны. + +2. Групповая скорость + +$ +v_"гр" = frac(d omega, d k) +$ +— скорость распространения огибающей суперпозиции волн (волнового пакета), т. е. скорость переноса энергии и информации. + +3. Причина различия + +- Различие фазовой и групповой скоростей возникает, если *частота зависит от волнового числа* $omega = omega(k)$ не линейно. +- Это связано с тем, что разные гармонические компоненты распространяются с разными скоростями. + +4. Когда они совпадают? + +- Если среда *недисперсионная*, т. е. $omega prop k$ (линейная зависимость). +- Пример: электромагнитные волны в вакууме — там $v_"ф" = v_"гр" = c$. + +5. Дисперсия волн + +*Дисперсия* — это зависимость фазовой скорости (или частоты) от длины волны (или волнового числа): +$ +v_"ф" = v_"ф" lambda space.quad "или" space.quad omega = omega(k). +$ + +- В дисперсионных средах $v_"ф" eq.not v_"гр"$. +- Примеры: + + - свет в стекле (разные цвета распространяются с разными скоростями), + - поверхностные гравитационные волны на воде, + - волны в волноводах. + +#line(length: 100%) + +*Q:* _*29*. Как образуются стоячие волны? Перечислите свойства, которые отличают стоячую волну от бегущей. Что называют узлами и пучностями смещений в стоячей волне?_ + +*A:* + +1. Образование стоячих волн + +Стоячая волна возникает при *наложении (интерференции) двух встречных волн* одинаковой частоты, амплитуды и скорости. +Чаще всего — это падающая и отражённая волна в ограниченной системе (струна, труба, столб воздуха). + +Математически: +$ +xi_1 = A cos(omega t - k x), space.quad xi_2 = A cos(omega t + k x). +$ +Сумма: +$ +xi(x,t) = 2 A cos(k x) cos(omega t). +$ + +2. Отличия стоячей волны от бегущей + +- *Нет переноса энергии вдоль среды* (энергия «колеблется» локально). +- *Амплитуда колебаний зависит от координаты*: в одних точках смещения всегда нулевые, в других — максимальные. +- *Нет движения волнового фронта*, как в бегущей волне; остаётся лишь чередование узлов и пучностей. + +3. Узлы и пучности + +- *Узлы* — точки, где $cos(k x)=0$ → смещение всегда равно нулю. +- *Пучности* — точки, где $|cos(k x)| = 1$ → амплитуда максимальна. + +Расстояние между соседними узлами или пучностями равно $frac(lambda, 2)$. + +#line(length: 100%) + +*Q:* _*30*. В чем заключается суть эффекта Доплера? Запишите формулу частоты воспринимаемого звука для случая неподвижного источника и движущегося наблюдателя; неподвижного наблюдателя и движущегося источника. Приведите примеры проявления эффекта Доплера для механических волн._ + +*A:* + +1. Суть эффекта Доплера + +*Эффект Доплера* — это изменение частоты и длины волны, воспринимаемой наблюдателем, если источник и наблюдатель движутся относительно друг друга. + +- Если они сближаются → воспринимаемая частота выше. +- Если удаляются → воспринимаемая частота ниже. + +2. Формулы для звука + +Пусть: + +- $f$ — частота источника, +- $v$ — скорость звука в среде, +- $v_"набл"$ — скорость наблюдателя (считается положительной при движении к источнику), +- $v_"ист"$ — скорость источника (положительная при движении к наблюдателю). + +(а) Неподвижный источник, движется наблюдатель + +$ +f' = f (1 plus.minus frac(v_"набл", v)), +$ +где «+» — при движении навстречу источнику, «–» — при удалении. + +(б) Неподвижный наблюдатель, движется источник + +$ +f' = f dot frac(v, v minus.plus v_"ист"), +$ +где «–» в знаменателе — источник движется к наблюдателю, «+» — от него. + +3. Примеры проявления эффекта Доплера для механических волн + +- Изменение тона сирены проезжающей машины или поезда. +- Изменение звука скорой помощи при приближении и удалении. +- Изменение частоты звука пропеллера или винта самолёта при пролёте мимо. + +#pagebreak() + +#align(center)[= _Основы специальной теории относительности (СТО)_] + +*Q:* _*1.* В чем различие ньютоновских представлений о пространстве и времени в классической механике и представлений об этих формах существования материи в специальной теории относительности?_ + +*A:* *1. Ньютоновские представления (классическая механика)* + +- *Пространство* — абсолютное, существует само по себе, одинаково для всех наблюдателей. +- *Время* — абсолютное, течёт одинаково и независимо от движения наблюдателя или тел. +- Пространство и время рассматриваются *раздельно*. +- Преобразования Галилея: + +$ +x' = x - v t, space.quad y' = y, space.quad z' = z, space.quad t' = t. +$ + +*2. Представления в специальной теории относительности (СТО)* + +- *Пространство и время* — взаимосвязанные, образуют единое *пространственно-временное континуум* (4-мерное пространство Минковского). +- *Время не абсолютно*: интервалы времени зависят от движения наблюдателя (эффект замедления времени). +- *Пространство не абсолютно*: длины зависят от движения (эффект сокращения длин). +- Законы природы инвариантны относительно *преобразований Лоренца*, а не Галилея: + +$ +x' = gamma (x - v t), space.quad t' = gamma (t - frac(v x, c^2)), space.quad gamma = frac(1, sqrt(1 - (v^2)/(c^2))). +$ + +*3. Сравнение* + +#table(columns: 3)[*Характеристика*][*Классическая механика (Ньютон)*][*СТО (Эйнштейн)*][Пространство][Абсолютное, одинаковое для всех][Относительно, зависит от системы отсчёта][Время][Абсолютное, одинаково для всех][Относительно, зависит от движения][Связь][Независимы друг от друга][Единый континуум «пространство–время»][Преобразования][Галилея][Лоренца][Предельная скорость][Не ограничена][Существует предел — скорость света $c$] + +#line(length: 100%) + +*Q:* _*2.* Сформулируйте первый и второй постулаты Эйнштейна. Как эти постулаты подтверждаются преобразованиями Лоренца? Как связаны между собой эти преобразования с преобразованиями Галилея?_ + +*A:* *1. Постулаты Эйнштейна (специальная теория относительности)* + +1. *Принцип относительности* + Законы физики одинаковы во всех инерциальных системах отсчёта. + → Ни одна ИСО не является «привилегированной». + +2. *Постулат постоянства скорости света* + Скорость света в вакууме одинакова для всех инерциальных систем отсчёта и не зависит от скорости движения источника или наблюдателя: + +$ +c = "const". +$ + +*2. Как эти постулаты связаны с преобразованиями Лоренца* + +- *Принцип относительности* → преобразования координат должны обеспечивать одинаковый вид физических законов (например, уравнений Максвелла) во всех ИСО. +- *Постоянство скорости света* → в любых ИСО световой фронт подчиняется уравнению: + +$ +x^2 + y^2 + z^2 = c^2 t^2. +$ + +Чтобы это условие сохранялось, преобразования между системами должны быть *именно преобразованиями Лоренца*: + +$ +x' &= gamma(x - v t), \ +y' &= y, \ +z' &= z, \ +t' &= gamma (t - frac(v x, c^2)), space.quad gamma = frac(1, sqrt(1 - (v^2)/(c^2))). +$ + +*3. Связь преобразований Лоренца и Галилея* + +- При малых скоростях $v << c$: + +$ +gamma approx 1, space.quad t' approx t, +$ + +и преобразования Лоренца переходят в *преобразования Галилея*: + +$ +x' = x - v t, space.quad y' = y, space.quad z' = z, space.quad t' = t. +$ + +#line(length: 100%) + +*Q:* _*3*. Времениподобным или пространственноподобным является интервал между стартом и возвращением космического корабля?_ + +*A:* *1. Интервал в СТО* + +Интервал между событиями в пространстве-времени: + +$ +s^2 = c^2 Delta t^2 - Delta r^2, +$ + +где $Delta t$ — разность времен, $Delta r$ — пространственное расстояние между событиями. + +- Если $s^2 > 0$ → интервал *времениподобный* (события можно связать причинно). +- Если $s^2 < 0$ → интервал *пространственноподобный* (события нельзя связать сигналом со скоростью ≤ c). +- Если $s^2 = 0$ → интервал *светоподобный*. + +*2. События: старт и возвращение корабля* + +- Пространственные координаты *совпадают* (старт и возвращение происходят в одной точке, например, на Земле): + +$ +Delta r = 0. +$ +- Временной интервал $Delta t > 0$. + +Тогда: + +$ +s^2 = c^2 Delta t^2 > 0. +$ + +#line(length: 100%) + +*Q:* _*4*. Предположим, что скорость света стала бесконечно большой. Что произошло бы при этом с предсказаниями теории относительности по поводу сокращения длины и замедления времени?_ + +*A:* *1. Напомним эффекты СТО* + +- *Замедление времени:* + +$ +Delta t' = gamma Delta t, space.quad gamma = frac(1, sqrt(1 - (v^2)/(c^2))) +$ + +- *Сокращение длины:* + +$ +L' = frac(L, gamma). +$ + +*2. Пусть $c arrow infinity$* + +- Тогда $frac(v^2, c^2) arrow 0$. +- Следовательно: + +$ +gamma arrow 1. +$ + +*3. Последствия* + +- *Замедление времени исчезает:* + $Delta t' = Delta t$. Время становится абсолютным. +- *Сокращение длины исчезает:* + $L' = L$. Пространство становится абсолютным. + +#line(length: 100%) + +*Q:* _*5*. Какие вам известны величины, сохраняющиеся при переходе от одной инерциальной системы отсчета к другой?_ + +*A:* *Сохраняющиеся величины при переходе от одной ИСО к другой* + +При преобразованиях Лоренца (или Галилея в классическом пределе) *формы законов физики* и некоторые физические величины остаются неизменными. + +1. *В классической механике (преобразования Галилея):* + +- *Время*: $t' = t$. +- *Масса*: постоянна и не зависит от скорости. +- *Расстояния между точками в один и тот же момент времени* (в пределах абсолютного пространства). +- *Импульс и энергия* сохраняются в замкнутых системах при любых переходах. + +2. *В специальной теории относительности (преобразования Лоренца):* + +- *Интервал в пространстве-времени*: + +$ +s^2 = c^2 t^2 - x^2 - y^2 - z^2 +$ +- *Скорость света* $c$. +- *Собственное время* между событиями на мировой линии. +- *Собственная длина* (в системе, связанной с телом). +- *Масса покоя* частицы. +- *Законы сохранения энергии, импульса и заряда*. + +#line(length: 100%) + +*Q:* _*6*. Две частицы удаляются в противоположные стороны от некоторого наблюдателя со скоростью $v = 3/5 "с"$ каждая. Каков модуль их относительной скорости?_ + +*A:* Ответ: $u_"отн" = frac(15, 17)c approx 0.882 "c".$ + +Пояснение: скорости вдоль одной прямой составляются по СТО: + +$ +u_"отн" = frac(u + v, 1 + frac(u v, c^2)). +$ + +Здесь $u = v = frac(3, 5)c$ (в противоположные стороны относительно наблюдателя), поэтому + +$ +u_"отн" = frac(3/5 "c" + 3/5 "c", 1 + (3/5 "c")^2/("c"^2)) = frac(6/5 "c", 1 + 9/25) = frac(6/5 "c", 34/25) = frac(15, 17)"c" approx 0.882"c". +$ + +(Классическое сложение дало бы $>c$, что неверно.) + + +#line(length: 100%) + +*Q:* _*7*. Полная энергия релятивистской частицы в 𝑛 раз превышает ее энергию покоя. Каково отношение модуля импульса частицы к произведению ее массы на скорость света $p/(m c)$._ + +*A:* + +$ +frac(p, m c) = sqrt(n^2 - 1). +$ + +Обоснование: $E = n m c^2$, а $E^2 = (p c)^2 + (m c^2)^2$ ⇒ $(p c)^2 = (n^2 - 1)(m c^2)^2$ ⇒ $p=m c sqrt(n^2 - 1)$. + + +#line(length: 100%) + +*Q:* _*8*. Чему равна производная энергии релятивистской частицы по ее импульсу $frac(diff E, diff p)$?_ + +*A:* $frac(d E, d p) = frac(p c^2, E) = v.$ + +Вывод: $E = sqrt((p c)^2 + (m c^2)^2) arrow.double frac(d E, d p) = frac(p c^2, sqrt((p c)^2 + (m c^2)^2))$. С учётом $v = frac(p c^2, E)$ получаем $frac(d E, d p) = v$. + + +#line(length: 100%) + +*Q:* _*9*. К телу массой $m$ в течение бесконечного периода времени приложена постоянная сила. Как изменяются со временем скорость и импульс этого тела?_ + +*A:* Зависит от рамки (классика vs СТО). + +Классическая механика (Ньютон) + +$ +arrow(F)="const" arrow.double +arrow(p)(t)= arrow(p)_0 + arrow(F)t, space.quad +arrow(v)(t)= arrow(v)_0 + frac(arrow(F), m)t. +$ + +И импульс, и скорость растут линейно (без ограничения). + +Специальная теория относительности + +$ +frac(d arrow(p), d t)=arrow(F) arrow.double arrow(p)(t) = arrow(p)_0 + arrow(F)t +$ + +$ +arrow(v)(t) = frac(arrow(p) c^2, E) = frac(arrow(p)(t)c, sqrt(arrow(p)(t)^2 + (m c)^2)). +$ + +Импульс растёт линейно, а скорость *монотонно приближается к $c$* и никогда его не достигает. + + +#line(length: 100%) + +*Q:* _*10*. Релятивистская частица движется со скоростью, при которой ее кинетическая энергия равна энергии покоя. Какую долю составляет ее скорость от скорости света в вакууме?_ + +*A:* Когда $K = E_0$, то $gamma - 1 = 1 arrow.double gamma = 2$. + +$ +gamma = frac(1, sqrt(1 - beta^2))=2 arrow.double 1 - beta^2 = 1/4 arrow.double beta = sqrt(3/4) = frac(sqrt(3), 2) approx 0.866. +$ + +Ответ: $v approx 0.866 "c"$. + + + diff --git a/theory.pdf b/theory.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3c52b759d44177dbbf95df23b6da79c3c3adbb0c GIT binary patch literal 285340 zcmbTcV|1s@vNs$%nb@|S{A1g;ZF^$dwmF&DwmGqF+j-{Rd!KdAyViN156|_fx~sYi z-Sw+l*J?5a5ivR@dNz2nz5SU#@Gxvltb~k&_C{9lJUk3CmUgCwPL^J#CWK53GK9<= zY%Hvd4Dy6*OaKN^LUsTvCxAhYkcEjsz>^&JNJ!yx5VN## zF?C`PvoUlr{r3_ENfB)xHZev~QBfgbK_*sqHUNtVBZn|E6Ctfx#*eF&4KVX%KxnFStnkSkS1X7hxh5zR2(v zP+~N#DIvQvXZ*+aF$E|J3FsfvT0ID{d5LC%gA;7z13w6eF>WAW2p~vtxIB26zv2Cl zg8u~C*~QSw#ly+e3?7CN9)^reR9*}o=6?qN|Aqd0U;yU-J}^;201Ny7dLD!THs=3| z*xx08FA9K}kwKY|ot5$b6A1$Y12Y4Ic6ee6>4s1+J%#8d%ziAeo 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