# Compound pendulum experiment to find the acceleration due to gravity

• VVS2000
In summary, the graph between time period and distance from the point of oscillation for a pendulum has a steep increase and decrease because of the change in gravitational potential energy and the net torque acting on the pendulum. As the pendulum gets longer, the time period also increases, and when the bar is attached at its middle at the center of gravity, there is no reason for oscillation, causing the time period to become infinite. The minimum in between is due to the balance between torque and rotational inertia, with the torque increasing linearly and the inertia increasing parabolically as the suspension point moves away from the center of gravity.

#### VVS2000

In this experiment, I still can't figure out why the graph between time period and distance from point of oscillation is like that. Why does it first decrease and increase so steeply? I got the second part because it goes near the centre of gravity and time period becomes almost infinite there but why does it first decrease and then increase?

As a pendulum gets longer the time period gets longer, so it is clear why T should be going up at the ends. When the bar is attached at its middle at the center of gravity, there is no change in gravitational potential energy as the bar swings, so there is no reason to oscillate, so T goes to infinity. Near there, with very little net torque to make it oscillate T is very large, so it is clear why it should go up in the middle. There must be a minimum in between.

Another way to look at it is comparing the torque to the rotational inertia. The angular acceleration is inversely related to the period, and angular acceleration is torque / inertia. So you can see how the period changes by examining torque / inertia. Picture the bar tilted from vertical by some particular angle. The torque increases linearly as you move the suspension point away from the center of gravity. However, the inertia does not change linearly. If L is the total length of the rod and x is how far the suspension point is from the center, then the moment of inertia is

## \frac 1 3 \frac M L (\frac {L^3} 4 + 3 L x^2)##

Which is parabolic in x.

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