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Консультация № 198326
22.04.2020, 10:53
0.00 руб.
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Здравствуйте! У меня возникли сложности с таким вопросом:
Найти координаты центра масс фигуры, ограниченной осями координат и дугой астроиды, расположенной в первом квадранте. (ответ: xc = yc = 256a/315pi)
Найти координаты центра масс фигуры, ограниченной осями координат и дугой астроиды, расположенной в первом квадранте. (ответ: xc = yc = 256a/315pi)
Обсуждение
27.04.2020, 12:24
общий
Адресаты:
Вариант с dl был бы правильным для кривой, а у Вас - фигура, ограниченная кривой, для ней используется элемент dS, и интеграл считается по-другому. Я попробую решить вашу задачу в ближайшие день-два.
29.04.2020, 07:31
общий
это ответ
Здравствуйте, petrencko0607!
В общем случае координаты центра масс плоской фигуры D с плотностью [$961$](x, y) определяются выражениями
где интеграл в знаменателе - масса фигуры. Если плотность фигуры постоянна и равна [$961$] = 1, выражения для центра масс принимают вид
где интеграл в знаменателе - площадь фигуры. Если координаты заданы параметрически в виде x = x(a, t), y = y(a, t), переход к интегрированию по переменным a, t производится по формуле
где |J| - модуль якобиана, вычисляемого как определитель
В данном случае якобиан будет иметь вид
следовательно, выражения для координат центра масс примут вид
Вычислим соответствующие интегралы:
Тогда
В общем случае координаты центра масс плоской фигуры D с плотностью [$961$](x, y) определяются выражениями
где интеграл в знаменателе - масса фигуры. Если плотность фигуры постоянна и равна [$961$] = 1, выражения для центра масс принимают вид
где интеграл в знаменателе - площадь фигуры. Если координаты заданы параметрически в виде x = x(a, t), y = y(a, t), переход к интегрированию по переменным a, t производится по формуле
где |J| - модуль якобиана, вычисляемого как определитель
В данном случае якобиан будет иметь вид
следовательно, выражения для координат центра масс примут вид
Вычислим соответствующие интегралы:
Тогда
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