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. `2pisqrt((4R)/(g))`
B. `2pisqrt((2R)/(g))`
C. `2pisqrt((R )/(g))`
D. none of these

Answer

B

Explanation

Rolling can be considered as pure rotation about point of contact. We can consider this system as a compound pudulum oscillating about P.
Time period `T=2pisqrt((I_P)/(mgd))`
`I_P=`moment of inertia about rotation axis `=I_0+mR^2=mR^2+mR^2=2mR^2`
The centre of mass of tube and rod will be at a height `(R )/(2)` from P, hence `d=(R )/(2)`
`impliesT=2pisqrt((2mR^2)/((mgR)/(2)))=2pisqrt((4R)/(g))`