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Консультация № 174469
23.11.2009, 10:33
35.00 руб.
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Добрый день, Уважаемые Эксперты. Помогите пожалуйста решить задачу по физике. Спасибо Вам.
Однородный диск радиуса R = 30 см колеблется около горизон-
тальной оси, проходящей через одну из образующих цилиндрической по-
верхности диска. Определить период колебаний диска.
Однородный диск радиуса R = 30 см колеблется около горизон-
тальной оси, проходящей через одну из образующих цилиндрической по-
верхности диска. Определить период колебаний диска.
Обсуждение
23.11.2009, 12:32
общий
это ответ
Здравствуйте, Рыскалев Максим Юрьевич.
Момент инерции диска относительно оси, проходящей через его центр
J0=0R[$8747$](m/(пR2))*2пr*r2 dr=(2m/R2)0R[$8747$]r3 dr=(2m/R2)*R4/4=mR2/2
Согласно теореме Гюйгенса-Штейнера момент инерции тела относительно некой оси равен J=J0+md2, где d - расстояние от данной оси до центра массы тела.
В данном случае d=R
J=mR2/2+mR2=(3/2)mR2
при отклонении от положения равновесия на угол [$966$] на диск действует момент силы тяжести M=mgR*sin[$966$]
Угловое ускорение ε=M/J=2g*sin[$966$]/3R
сделав приближение sin[$966$]≈[$966$], получаем дифференциальное уравнение ε=d2[$966$]/dt2=2g*[$966$]/3R
его решение имеет вид [$966$]=[$966$]m*cos([$969$]t), где угловая частота [$969$]=[$8730$](ε/φ)=[$8730$](2g/3R)
Период равен T=2п/[$969$]=2п[$8730$](3R/2g)=2п[$8730$](3*0,3м/(2*9,8м/с[sup]2[/sup]))=1,35 с
Момент инерции диска относительно оси, проходящей через его центр
J0=0R[$8747$](m/(пR2))*2пr*r2 dr=(2m/R2)0R[$8747$]r3 dr=(2m/R2)*R4/4=mR2/2
Согласно теореме Гюйгенса-Штейнера момент инерции тела относительно некой оси равен J=J0+md2, где d - расстояние от данной оси до центра массы тела.
В данном случае d=R
J=mR2/2+mR2=(3/2)mR2
при отклонении от положения равновесия на угол [$966$] на диск действует момент силы тяжести M=mgR*sin[$966$]
Угловое ускорение ε=M/J=2g*sin[$966$]/3R
сделав приближение sin[$966$]≈[$966$], получаем дифференциальное уравнение ε=d2[$966$]/dt2=2g*[$966$]/3R
его решение имеет вид [$966$]=[$966$]m*cos([$969$]t), где угловая частота [$969$]=[$8730$](ε/φ)=[$8730$](2g/3R)
Период равен T=2п/[$969$]=2п[$8730$](3R/2g)=2п[$8730$](3*0,3м/(2*9,8м/с[sup]2[/sup]))=1,35 с
5
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