A thin-walled tube of mass m and radius R has a rod of mass m and vry small cross section soldered on its inner surface. The side-view of the arrangement is as shown in the following figure. The entire arrangement is placed on a rough horizontal surface. The system is given a small angular displacement from its equilibrium position, as a result, the system performs oscillations. The time period of resulting oscillations if the tube rolls without slipping is

Options

A. `2pi sqrt((4R)/(g))`
B. `2pi sqrt((2R)/(g))`
C. `2pi sqrt((R )/(g))`
D. None of these

Answer

A

Explanation

Rolling can be considered as pure rotation about point of contact, `P`: Now it is compound pendulum oscillating about 'P'.

`:. T = 2pi sqrt((I)/(mgd))`, where `I = mR^(2) +mR^(2) = 2mR^(2)`.
`d = R//2 :. T = ?`