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

[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 v_i and the speed of the air particles be u_i which is approximately zero.

Using the conservation of momentum:
Initial momentum = Final momentum
Mv_i+mu_i = Mv_f+mu_f

As u_i is effectively zero...

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_f-v_i) 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

 

[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.

 

[JamieGreggary]

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.

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

 

[Dick]

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

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

 

[Nickie]

for your question and for sharing your thoughts on the topic. I appreciate your curiosity and your willingness to think critically about the concept.

Firstly, to answer your main question, yes, the mass of a pendulum bob does affect its damping rate. This is because the mass of the pendulum bob affects the momentum of the system, which in turn affects how much energy is transferred to the air molecules during each oscillation. A higher mass pendulum bob will have a higher momentum, meaning it will transfer more energy to the air molecules, resulting in a slower damping rate.

Your reasoning and calculations are correct. However, I can understand your hesitation in accepting this concept, as it may seem counterintuitive based on everyday experiences. For example, as you mentioned, dropping two balls of different masses from a tower may lead one to believe that the heavier ball would reach the ground first due to its larger mass. But in that scenario, gravity is the dominant force, and the difference in mass is not significant enough to affect the acceleration due to gravity.

In the case of a pendulum, the dominant force is air resistance, which is affected by the mass of the pendulum bob. So, while mass may not have a significant effect on the motion of objects in free fall, it can have a significant impact on the motion of objects interacting with other forces, such as air resistance.

I hope this helps clarify your doubts and provides a better understanding of how the mass of a pendulum bob affects its damping rate. Keep questioning and exploring, that's the essence of science!