Question

Derive time period and angular frequency of a compound pendulum.

Answer 1

• Compound pendulum is a rigid body of any shape, capable of oscillating about a horizontal axis passing through it.

• Figure below shows vertical section of rigid body capable of oscillating about the point A.

• 

• Distance l between point A and the centre of gravity G is called length of the pendulum.

• When this compound pendulum is given a small angulr displacement θ and is then released it begins to oscillate about point A.

• At angular displacement θ its center of gravity now takes new position at G'.

• Weight of the body and its reaction at the support constitute a reactive couple or torque given by

•      τ=-mg G'B
      =-mglsinθ          (24)

• Equation 24 gives restoring couple which tends to bring displaced body to its original position.

• If α is the angular acceleration produced in this body by the couple and I is the moment of inertia of body about horizontal axis through A then the couple is

•      Iα=-mglsinθ

• if θ is very small then we can replace sinθ≅θ, so that

•      α=-(mgl/I)θ          (25)

• From above equation (25) we se that pendulum is executing Simple Harmonic Motion with time period

•