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JamieGreggary

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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

Using the conservation of momentum:

Initial momentum = Final momentum

As

Rearranging for the final velocity of the pendulum bob:

So, as the mass increases (say approaches infinity), the ratio of

Therefore as the mass increases, the less change in velocity (

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

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*v*and the speed of the air particles be_{i}*u*which is approximately zero._{i}Using the conservation of momentum:

Initial momentum = Final momentum

*Mv*_{i}+mu_{i}= Mv_{f}+mu_{f}As

*u*is effectively zero..._{i}*Mv*_{i}= Mv_{f}+mu_{f}Rearranging for the final velocity of the pendulum bob:

*v*_{f}= (Mv_{i}- mu_{f})/M*v*_{f}= v_{i}- (m/M)u_{f}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 (

*v*) 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._{f}-v_{i}**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

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