## How does a simple pendulum conserve angular momentum?

I know this is probably a dumb question, but it's a long time since I did basic mechanics.

Consider a simple pendulum, consisting of a weight suspended a hook on the ceiling by a string. Say the pendulum is moving in a plane, so that its motion can be described by one coordinate theta as a function of time (theta(t)).

The angular momentum of the pendulum changes with time, along a sinusoidal curve. How is conservation of angular momentum satisfied? Is it something to do with the angular momentum of the room to whose ceiling the pendulum is fixed (and ultimately, the Earth, if the room is on solid ground on Earth),

Are the fluctuations in angular momentum of the pendulum matched by opposite fluctuations in the angular momentum of the Earth?

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hi andrew!

just as ordinary (linear) momentum is conserved in any direction in which there is no external force,

so angular momentum is conserved about any point about which (and along any axis parallel to which) there is no external torque

for a pendulum, the only direction in which there's no external torque (about the pivot) is the vertical direction (because that's parallel to the weight),

so only the vertical angular momentum will be conserved …

in your example, that's zero since you specified that it moves in a plane
 Quote by andrewkirk Are the fluctuations in angular momentum of the pendulum matched by opposite fluctuations in the angular momentum of the Earth?
both the external forces (the weight, and the reaction force at the pivot) come from the Earth, so yes

 Thank you Tiny Tim.

 Tags angular momentum, pendulum