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Уважаемые эксперты! Пожалуйста, ответьте на вопрос: Груз 1, подвешенный на нерастяжимой нити, намотанной на барабан 2, опускается по закону x=x(t). По заданному уровнению прямолинейного поступательного движения груза 1-x=x(t) опредилить скорость и ускорение точки М механизма в момент времени, когда груз пройдет расстояние S.
На рисунке, который приведен в условии задачи видно, что нить закреплена на барабане 3. Это противоречит тексту условия, в котором сказано, что
Кроме того, на рисунке указаны радиусы шкива 3. Резонно предположить, что в условии должно быть так:
Тем не менее, рассмотрим два случая: 1) нить намотана на барабан 2; 2) нить намотана на барабан 3.
Если нить намотана на барабан 2, то записав уравнение движения груза
и дважды продифференцировав его, получим
Выражение (2) показывает, что скорость груза с течением времени изменяется линейно, а выражение (3) - что ускорение груза равно Учитывая выражение (1) и общую формулу для пути при равноускоренном движении
можно придти к выводу, что т. е. в некоторой системе координат начальное положение груза, отсчитываемое по оси перемещения, характеризуется величиной 30 см, начальная скорость равна нулю а ускорение равно 16 см/с2 (в отношении ускорения к этому же приводит и формула (3)).
Согласно условию, груз за время переместился на расстояние следовательно, его текущая координата составляла Подставим это значение в выражение (1) и получим
Из выражения (2) найдём скорость груза в этот момент времени:
Ускорение груза, понятно,
Точки обода барабана 2 движутся одинаково. Их линейная скорость в рассматриваемый момент времени равна скорости груза, а касательное ускорение равно ускорению груза. Нормальное ускорение равно отношению квадрата линейной скорости к радиусу барабана 2. Поэтому
Вектор касательного ускорения точки M направлен перпендикулярно к радиусу-вектору этой точки в ту же сторону, что и вектор её линейной скорости, а вектор нормального ускорения - вдоль радиуса-вектора к оси вращения барабана 2. Полное ускорение точки M составляет
Если нить намотана на барабан 3, то линейная скорость любой точки внутреннего обода этого барабана равна скорости груза в рассматриваемый момент времени: Угловая скорость этой точки составляет С этой же угловой скоростью движется и любая точка внешнего обода барабана 3, поэтому
линейная скорость любой точки внешнего обода барабана 3 составляет
Обод барабана 2 контактирует (будем считать, что без скольжения) с внешним ободом барабана 3. Поэтому линейные скорости точек обода барабана 2 (в том числе, точки M) равны линейным скоростям точек внешнего обода барабана 3:
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