A uniform rod of mass m and length l is suspended through a light wire of length l and torsional constant k as shown in figure. Find the time perid iof the system makes a. small oscillations in the vertical plane about the suspension point and b. angular oscillations in the horizontal plane about the centre of the rod.

Explanation

a. The oscillations take place about the horizontal lilne through the point of suspension and perpendicular to the plane of the ure. The moment of inertia of the rod about this line is
`(ml^2)/12+ml^2=13/12ml^2`
The time period `=2pi sqrt(I/(mgl))=2pi(sqrt(13ml^2)/(12mgl))`
`=2p(sqrt(13l)/(12g))`
b. The angular oscillations take place about the suspension wire. The moment of inertia about this line is `ml^2/12`. The time period is
`2pisqrt(I/k)=2pi(sqrt(ml^2)/(12k))`