# Am I correct in saying the mass of a pendulum bob affects its damping rate

1. Jan 30, 2013

### JamieGreggary

How does the mass of a pendulum bob affect the time taken for the oscillation of a pendulum to diminish?

At first I instinctively thought that it would have no effect. However, thinking about the pendulum bob's momentum as it interacts with the air molecules, a higher massed pendulum should result in an oscillation which damps in a slower time period.

Consider:
Let M be the mass of the pendulum bob, and m be the mass of the group of particles it interacts with. Let the initial speed of the pendulum bob be vi and the speed of the air particles be ui which is approximately zero.

Using the conservation of momentum:
Initial momentum = Final momentum
Mvi+mui = Mvf+muf

As ui is effectively zero...

Mvi = Mvf+muf

Rearranging for the final velocity of the pendulum bob:
vf = (Mvi - muf)/M
vf = vi - (m/M)uf

So, as the mass increases (say approaches infinity), the ratio of m/M tends to 0, and so the final velocity of the pendulum bob approaches its initial velocity.

Therefore as the mass increases, the less change in velocity (vf-vi) the pendulum bob experiences. This means that less energy is taken out of the system, and thus the pendulum bob takes a longer time period to damp to a lower oscillation.

Summary
Now in theory this seems correct to me, but I'm not entirely sure if my logic is correct since I keep hearing that the mass should have no effect on the pendulums motion. For example, by my logic dropping two balls from a tower where one mass is greater than the other, the smaller mass should be more affected by air resistance and hit the ground after the heavier ball. <-- Surely that just isn't true?

So ultimately: Does the mass of a pendulum affect its "damping rate" (I know this isn't the correct term but I can't think of anything else at this moment in time) :P

Thank you very much

Last edited: Jan 30, 2013
2. Jan 30, 2013

### Dick

Sure. Your intuition is correct, if you have two identical pendulums being damped by friction then the pendulum with the larger mass will have a smaller damping rate than the smaller mass.

3. Jan 30, 2013

### JamieGreggary

But in terms of dropping balls from a tower it doesn't seem intuitively correct, and surely there cannot be one rule for one and one rule for another:

"For example, by my logic dropping two balls from a tower where one mass is greater than the other, the smaller mass should be more affected by air resistance and hit the ground after the heavier ball."

Unless this is actually what happens but I just naively assumed otherwise.

Thanks a lot for your help regardless, at least I feel I have some reassurance that it is the case ;D

4. Jan 30, 2013

### Dick

They are both less affected by friction, so yes, the smaller mass will hit later. There's really no inconsistancy between the two.