InterviewSolution
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InterviewSolution
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A seimicircular uniform wore of radius `R = pi` can rotate frelly about x-axis. The cnetre of curvature is at origin as shown. The acceleration due to gravity is given by `vec(g)=-x^(2)hatj` units. For small oscillations of the wire its time period is given by `pisqrt(x)`. the value of `x` is ________ |
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Answer» Correct Answer - 6 Let `R d theta `be an element of wire located an angular position of `theta `wrt-ve axis When slightly rotated by angle `beta` torque due to gravity is `vec(tau) = int d vec(tau) =int dm vec(g) xx vec(r)` `=- int [R dtheta(M)/(piR)]x^(2) (R cos theta). sin beta` `rArr tau = int R ((Md theta)/(pi)x^(2)R cos theta) beta`, for small `beta` as `x = R sin theta tau = beta overset((pi)/(2))underset((-pi)/(2))int (MR cos theta)/(pi)R^(2) sin^(2) theta d theta rarr (1)` But `tau= I alpha =(MR^(2))/(2) alpha rarr (2)` where `alpha = (d^(2)beta)/(dt^(2))` `alpha =- omega^(2) beta rArr T = (2pi)/(omega)` |
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