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Консультация № 196515
30.09.2019, 17:05
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Математический маятник длиной l = 0,8 м в начальный момент отклонили на угол θ = 10° и без толчка отпустили. Определить, во сколько раз увеличится максимальный угол отклонения маятника θmax, если в начальный момент времени его нижнему концу дополнительно сообщить скорость v = 0,5 v/c, направленную к положению равновесия.
Математический маятник длиной l = 0,8 м в начальный момент отклонили на угол θ = 10° и без толчка отпустили. Определить, во сколько раз увеличится максимальный угол отклонения маятника θmax, если в начальный момент времени его нижнему концу дополнительно сообщить скорость v = 0,5 v/c, направленную к положению равновесия.
Обсуждение
04.10.2019, 14:54
общий
это ответ
Здравствуйте, dar777!
Дано: l=0,8 м -- длина математического маятника; [$629$]0=10[$186$]=[$960$]/18 рад -- начальный угол отклонения маятника; (v0)1=0 -- начальная окружная скорость маятника в первом случае; (v0)2=0,5 м/с -- начальная окружная скорость маятника во втором случае.
Определить: ([$629$]max)2/([$629$]max)1 -- отношение максимальных углов отклонения маятника во втором и в первом случаях.
Решение
Если предположить, что колебания математического маятника являются гармоническими, то, согласно [1, с. 131], максимальный угол отклонения маятника, равный амплитудному значению угла отклонения, составляет
где [$629$]'0 -- угловая скорость маятника в начальный момент времени; [$969$]0=[$8730$](g/l) -- циклическая частота собственных колебаний маятника. Значит, согласно [1, с. 20], в первом случае угловая скорость маятника в начальный момент времени составляет ([$629$]'0)1=(v0)1/l=0, а во втором случае -- ([$629$]'0)2=(v0)2/l=0,5/0,8=0,625 (рад/с); в первом случае, в силу закона сохранения механической энергии [1, с. 72], ([$629$]max)1=([$629$]0)1=10[$186$], а во втором случае -- ([$629$]max)2=[$8730$](([$960$]/18)2+(0,625/([$8730$](9,81/0,8)))2)[$8776$]0,182 (рад). Тогда
Ответ: в 1,04 раза.
Литература
1. Яворский Б. М., Детлаф А. А. Справочник по физике для инженеров и студентов вузов. -- М.: ООО "Издательство Оникс", 2007. -- 1056 с.
Дано: l=0,8 м -- длина математического маятника; [$629$]0=10[$186$]=[$960$]/18 рад -- начальный угол отклонения маятника; (v0)1=0 -- начальная окружная скорость маятника в первом случае; (v0)2=0,5 м/с -- начальная окружная скорость маятника во втором случае.
Определить: ([$629$]max)2/([$629$]max)1 -- отношение максимальных углов отклонения маятника во втором и в первом случаях.
Решение
Если предположить, что колебания математического маятника являются гармоническими, то, согласно [1, с. 131], максимальный угол отклонения маятника, равный амплитудному значению угла отклонения, составляет
[$629$]max=[$8730$]([$629$]02+([$629$]'0/[$969$]0)2),
где [$629$]'0 -- угловая скорость маятника в начальный момент времени; [$969$]0=[$8730$](g/l) -- циклическая частота собственных колебаний маятника. Значит, согласно [1, с. 20], в первом случае угловая скорость маятника в начальный момент времени составляет ([$629$]'0)1=(v0)1/l=0, а во втором случае -- ([$629$]'0)2=(v0)2/l=0,5/0,8=0,625 (рад/с); в первом случае, в силу закона сохранения механической энергии [1, с. 72], ([$629$]max)1=([$629$]0)1=10[$186$], а во втором случае -- ([$629$]max)2=[$8730$](([$960$]/18)2+(0,625/([$8730$](9,81/0,8)))2)[$8776$]0,182 (рад). Тогда
([$629$]max)2/([$629$]max)1=0,182/([$960$]/18)[$8776$]1,04.
Ответ: в 1,04 раза.
Литература
1. Яворский Б. М., Детлаф А. А. Справочник по физике для инженеров и студентов вузов. -- М.: ООО "Издательство Оникс", 2007. -- 1056 с.
5
Это самое лучшее решение!
Об авторе:
Facta loquuntur.
Facta loquuntur.
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