Angular Oscillation of a rod in a circle

[Tanya Sharma]

Homework Statement

A uniform rod moves in a vertical circle .Its ends are constrained to move on the track without friction.Find the angular frequency of small oscillation .

Homework Equations

The Attempt at a Solution

Suppose the rod of length L moves in a circle of radius R .
Let the equilibrium position of the rod be AB .X be the mid point .CD is the position of the rod when it displaced by an angle θ .Y is the mid point.

The mechanical energy of the rod in position CD is denoted by E .

The moment of inertia of the rod about its CM (the middle point) is I_cm
The moment of inertia of the rod about O is I .

$I_{cm} = ML^2/12$

$I=I_{cm} + Md^2$

$I=M(R^2-\frac{L^2}{6})$

$E= mgd(1-cos\theta)+(1/2)I\dot\theta^2$

Differentiating E w.r.t time ,we get

$dE/dt = mgdsin\theta\dot\theta+(1/2)I(2\dot\theta\ddot\theta)$

Since Mechanical energy remains conserved ,

Putting dE/dt=0 ,we get

$\ddot\theta = -\frac{mgdsin\theta}{I}$

Using small angle approximation , sinθ≈θ

$\ddot\theta = -\frac{mgd\theta}{I}$

Is my approach correct ?

 

[haruspex]

Looks right to me. (It's clearly the same as making a pendulum out of the rod by attaching a light bar length d rigidly, at right angles, to its centre.)