Question About Conservation of Momentum

[Master Wayne]

This has been bugging me for a while. Imagine a pendulum swinging. Since it has mass and velocity, it has momentum. After a while, though, it will swing with less and less amplitude until it has stopped. Where did the momentum go? Thanks a lot for your help!

 

[Dale]

A pendulum is not an isolated system. Its momentum is not conserved.

 

[Master Wayne]

A pendulum is not an isolated system. Its momentum is not conserved.

I understand. So where did the momentum go? Which masses have now the velocity? Outside of the pendulum, that is.

 

[Dale]

I understand. So where did the momentum go? Which masses have now the velocity? Outside of the pendulum, that is.

The earth. The resulting change in the Earth's velocity is incredibly small, but that is where the momentum went.

 

[Master Wayne]

The earth. The resulting change in the Earth's velocity is incredibly small, but that is where the momentum went.

That's quite interesting. But how does that transfer of momentum happen in terms of the forces involved?

 

[HallsofIvy]

The reason the pendulum is NOT a 'closed system' is that you have friction both at the fulcrum and with the air. As the pendulum moves to the right, say, the friction between it and the support applies a force to the left on the support. If the support is rigidly attached to, say, a table which in turn is attached to a floor which ... is attached to the earth, that force is transmitted to the earth. The air resistance, on the other hand, reduces the momentum of the pendulum by transmitting momentum to individual molecules in the air.

 

[Dale]

That's quite interesting. But how does that transfer of momentum happen in terms of the forces involved?

Every force that the Earth exerts on the pendulum has an equal and opposite force that the pendulum exerts on the earth. Therefore, whatever momentum the pendulum loses is gained by the earth. This is a basic feature of Newtonian mechanics.

PS in case it is not clear, I am including the air and the support structure as part of the earth, not just dirt and rocks etc.

 

[Philip Wood]

And, of course, it's not just as the oscillations die down that the pendulum loses its momentum. Every half-swing the pendulum loses its momentum, and in the next half cycle it acquires (and then loses) momentum in the opposite direction.

 

[Master Wayne]

I believe I understand this now. Thanks a lot for your help!