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 ________

Answer

6

Explanation

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)`