A system of two identical rods (L-shaped) of mass `m` and length `l` are resting on a peg `P` as shown in the figure. If the system is displaced in its plane by a small angle `theta`,find the period of oscillations:

Options

A. `2pisqrt((sqrt(2)l)/(3g))`
B. `2pisqrt((2sqrt(2)l)/(3g))`
C. `2pisqrt((2l)/(3g))`
D. `2pisqrt((l)/(3g))`

Answer

B

Explanation

Center of mass `2m` of a system is at a distance peg `P` is `(l)/(2sqrt(2))`

`tau = r xx F = -mg sin theta (l)/(2sqrt(2))`
`rArr Ialpha = -mgsin theta = (l)/(2sqrt(2)) (sin theta = 0` from small `theta`)
`rArr (2ml^(2))/(3)alpha = -mg theta(l)/(2sqrt(2)) rArr alpha = -(3g theta)/(2ssqrt(2)l) = -omega^(2)theta`
`rArr omega = sqrt((3g)/(2sqrt(2)l)) rArr T = (2pi)/(omega) = 2pi sqrt((2sqrt(2)l)/(3g))`