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Здравствуйте! Прошу помощи в следующем вопросе: Точка участвует одновременно в двух взаимно перпендикулярных колебаниях, выражаемых уравнениями x=A1 cos ωt и y=A2 sin ωt, где A1=4 см, A2=6 см.ω1=2ω2 Найти уравнение траектории точки и построить ее, указав направление движения.
Обсуждение
Неизвестный
06.04.2011, 22:20
общий
Неизвестный
07.04.2011, 00:24
общий
Вы при переходе от y2 к y потеряли один из знаков. На мой взгляд с выражением x через y гораздо удобней работать. Также Вы на рисунке не приняли во внимание ограниченность x и y, которая следует из смысла условия задачи. Колебания будут проходить в обе стороны по полученной линии.
Неизвестный
08.04.2011, 00:53
общий
12.04.2011, 03:40
это ответ
Здравствуйте, Ольга!
x = A1cos[$969$]1t (1) y = A2sin[$969$]2t (2) [$969$]1 = 2[$969$]2 (3)
A1 = 4 см A2 = 6 см
Подставим (3) в (1) x = A1cos[$969$]1t = A1cos2[$969$]2t = A1(1 - sin2[$969$]2t) = = A1(1 - y2/A22)
При выводе были использованы формулы 1 - сos2x = sin2x sin2x +cos2x = 1 и равенство (2).
x = A1(1 - y2/A22) (4) Движение будет происходить по части параболы заданной равенством (4). Область значений косинуса(синуса) [-1;1]. Поэтому требуется отрисовать часть параболы, для которой выполнено неравенство -A1[$8804$]x[$8804$]A1 (-A2[$8804$]y[$8804$]A2). Движение будет в любой точке попеременно направлено, то вверх, то вниз. В концевых точках будет разворот.
Полученная кривая - это фигура Лисажу. Построение различных фигур Лисажу Если выставить свои параметры, то нарисует Вашу. Косинус сдвинут по фазе на 90 градусов относительно синуса.
Прежде чем править Ваш ответ, давайте поработаем над ним ещё: некорректной представляется следующая фраза:
Косинус сдвинут по фазе на 90 градусов относительно синуса.
С уважением.
Об авторе:
Facta loquuntur.
Неизвестный
08.04.2011, 11:33
общий
Адресаты:
Имелось ввиду, что
cos x = sin (x+[$8719$]/2) Для того, чтобы настроить рисование ситуации задачи, начальный сдвиг фазы для координаты х необходимо установить в положение 90[$186$].
Наберите, пожалуйста, самостоятельно окончательный вариант ответа. Фразу о сдвиге косинуса относительно синуса замените, пожалуйста, фразой о сдвиге колебаний. Потом просмотрим ответ ещё раз и, если всё будет в порядке, перенесём.
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