For procedures, see your mechanical laboratory manual.

Title: Compound Pendulum - Conditions for the minimum period


Aim

To suspend a given rigid body from several points and to study the variation of the period with the distance of the suspension point from the center of mass.

Theory



A simple pendulum consists of a small body called a “bob” (usually a sphere) attached to the end of a string the length of which is great compared with the dimensions of the bob and the mass of which is negligible in comparison with that of the bob. Under these conditions, the mass of the bob may be regarded as concentrated at its center of gravity, and the length of the pendulum is the distance of this point from the axis of suspension. When the dimensions of the suspended body are not negligible in comparison with the distance from the axis of suspension to the center of gravity, the pendulum is called a compound, or physical, pendulum. A rigid body mounted upon a horizontal axis to vibrate under the force of gravity is a compound pendulum. 

A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot.


In Fig.1 a body of irregular shape is pivoted about a horizontal frictionless axis through P and is displaced from its equilibrium position by an angle θ. In the equilibrium position, the center of gravity G of the body is vertically below P. The distance GP is l and the mass of the body is m. The restoring torque for an angular displacement θ is 
For small amplitudes (θ ≈ 0),
where 'I' is the moment of inertia of the body through the axis P. Eq. (2) represents a simple harmonic motion and hence the time period of oscillation is given by 
Now 
where 
is the moment of inertia of the body about an axis parallel with the axis of oscillation and passing through the center of gravity G.
where K is the radius of gyration about the axis passing through G. Thus, 
The time period of a simple pendulum of length L is given by 
Comparing with Eq. (5) we get
This is the length of “equivalent simple pendulum”. If all the mass of the body were concentrated at a point O (See Fig.1) such that
we would have a simple pendulum with the same time period. Point O is called the ‘Centre of Oscillation’. Now from Eq. (7) 

References

  1. Moment of inertia, wikipedia.org.
  2. Pendulum, wikipedia.org.
  3. The compound pendulum, The University of Texas.
  4. Nonlinear Physics with Mathematica for Scientists and Engineers pp (575-576).
  6. The Compound Pendulum, Physics 233, Simon Fraser University.
  7. Fundamentals of Physics: Resnick & Halliday.
  8. Practical physics: R.K. Shukla, Anchal Srivatsava, New Age International (P) Ltd, New Delhi.
  9. Eric J. Irons, American Journal of Physics, Vol. 15, Issue 5, pp.426 (1947).