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Консультация № 168542
30.05.2009, 08:48
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Однородная кольцевая цепочка массы m и длины l надета на гладкий конус, ось которого вертикальна. Угол полураствора конуса a. Конус привели во вращение относительно оси и каждый элемент цепочки движется со скоростью V. Найдите величину силы F натяжения цепочки.
Обсуждение
Неизвестный
30.05.2009, 23:50
общий
это ответ
Здравствуйте, Филиппов Алексей Павлович.
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На участок цепи с угловым размером 'df' действует центобежная сила от реакции конуса
dFg= dm*g*ctg(a)= (m/(2*pi))*df*g*ctg(a)
и от вращения
dFw= dm*v^2/r= (m/(2*pi))*df*v^2/(L/(2*pi))= m*df*v^2/L
растягивающая сила с которой 2 полукольца пытаются оторваться одно от другого
F2= INT[0;pi]((m/(2*pi))*g*ctg(a)*sin(f)*df+ (m*v^2/L)*sin(f)*df)
F2= ((m/(2*pi))*g*ctg(a)+ m*v^2*pi/L)*(-cos(pi)-(-cos(0)))
F2= 2*((m/(2*pi))*g*ctg(a)+ m*v^2*pi/L)
Так как эта сила приходится на 2 сечения цепочки, то растягивающая сила в цепочке будет в 2 раза меньше чем сила с которой удерживаются 2 полукольца
Ответ
F= F2/2= ((m/(2*pi))*g*ctg(a)+ m*v^2*pi/L)
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___
На участок цепи с угловым размером 'df' действует центобежная сила от реакции конуса
dFg= dm*g*ctg(a)= (m/(2*pi))*df*g*ctg(a)
и от вращения
dFw= dm*v^2/r= (m/(2*pi))*df*v^2/(L/(2*pi))= m*df*v^2/L
растягивающая сила с которой 2 полукольца пытаются оторваться одно от другого
F2= INT[0;pi]((m/(2*pi))*g*ctg(a)*sin(f)*df+ (m*v^2/L)*sin(f)*df)
F2= ((m/(2*pi))*g*ctg(a)+ m*v^2*pi/L)*(-cos(pi)-(-cos(0)))
F2= 2*((m/(2*pi))*g*ctg(a)+ m*v^2*pi/L)
Так как эта сила приходится на 2 сечения цепочки, то растягивающая сила в цепочке будет в 2 раза меньше чем сила с которой удерживаются 2 полукольца
Ответ
F= F2/2= ((m/(2*pi))*g*ctg(a)+ m*v^2*pi/L)
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