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

• JamieGreggary
In summary, the mass of a pendulum bob does affect the time taken for its oscillation to diminish. This is because a higher massed pendulum will experience less change in velocity and thus less energy will be taken out of the system, resulting in a longer damping period. However, this concept may seem counterintuitive when compared to the scenario of dropping two balls from a tower, where the smaller mass would be more affected by air resistance and hit the ground later. Ultimately, the mass of a pendulum does have an effect on its damping rate and it is consistent with other examples of mass and air resistance.
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:
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.

Dick said:
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

JamieGreggary said:
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.

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!

## What is a pendulum bob?

A pendulum bob is a weight attached to the end of a pendulum that swings back and forth.

## What is damping rate?

Damping rate is a measure of how quickly a pendulum's oscillations decrease in amplitude due to external factors such as air resistance or friction.

## How does the mass of a pendulum bob affect its damping rate?

The mass of a pendulum bob does affect its damping rate. A heavier bob will experience more air resistance and friction, causing its oscillations to decrease at a faster rate compared to a lighter bob.

## Is the relationship between mass and damping rate linear?

No, the relationship between mass and damping rate is not linear. It follows a logarithmic pattern, meaning that as the mass of the bob increases, the damping rate will increase at a decreasing rate.

## What other factors can affect the damping rate of a pendulum bob?

Aside from mass, other factors that can affect the damping rate of a pendulum bob include the length of the pendulum, the material and shape of the bob, and the amount of air resistance and friction present in the environment.

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