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Консультация № 170902
29.07.2009, 19:05
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Здравствуйте,уважаемые эксперты!помогите пожалуйста решить задачу:
на гладкий горизонтальный стол поставили вертикально гантельку,состоящую из невесомого жесткого стержня с двумя маленькими шариками на концах,массы которых 2m и m.верхнему шарику массы 2m ударом сообщают скорость v в горизонтальном направлении.при какой длине гантельки L нижний шарик сразу оторвется от стола?
заранее благодарен
на гладкий горизонтальный стол поставили вертикально гантельку,состоящую из невесомого жесткого стержня с двумя маленькими шариками на концах,массы которых 2m и m.верхнему шарику массы 2m ударом сообщают скорость v в горизонтальном направлении.при какой длине гантельки L нижний шарик сразу оторвется от стола?
заранее благодарен
Обсуждение
Неизвестный
31.07.2009, 19:04
общий
Сообщение верхнему шарику горизонтальной скорости v вызовет вращение гантельки вокруг центра масс и поступательное движение самого центра масс.
Горизонтальная скорость, которую приобретёт находящийся на высоте (2/3)L центра масс системы, V = (2mv)/(3m) = (2/3)v.
Тогда в системе координат, связанной с центром масс, абсолютная величина скорости верхнего шарика v2m = v/3, а абсолютная величина скорости нижнего шарика vm = 2v2m = (2/3)v.
Нормальное ускорение нижнего шарика для вращения вокруг центра масс an = (vm)2/(2L/3) = (2/3)(v2/L)
Чтобы нижний шарик сразу оторвался от стола, нужно, чтобы его нормальное ускорение было больше g.
Отсюда L < (2/3)(v2/g).
Горизонтальная скорость, которую приобретёт находящийся на высоте (2/3)L центра масс системы, V = (2mv)/(3m) = (2/3)v.
Тогда в системе координат, связанной с центром масс, абсолютная величина скорости верхнего шарика v2m = v/3, а абсолютная величина скорости нижнего шарика vm = 2v2m = (2/3)v.
Нормальное ускорение нижнего шарика для вращения вокруг центра масс an = (vm)2/(2L/3) = (2/3)(v2/L)
Чтобы нижний шарик сразу оторвался от стола, нужно, чтобы его нормальное ускорение было больше g.
Отсюда L < (2/3)(v2/g).
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