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erobz

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It didn't dawn on me until you were talking about a sphere in the other post. Aren't we really just conserving angular momentum?jbriggs444 said:The top of the belt may or may not accelerate with ##F=ma## (or ##F=\frac{dp}{dt}##). You get an effective mass and an effective momentum regardless.

In particular, for a free-wheeling belt (no motor), the effective mass will be the mass of the belt plus a mass contribution from the angular momentum of the rollers if they are not ideal. If you are being picky, the belt material that rolls around the end has an increased effective mass due to rotation as well. For a thin, continuously flexible belt the rotational contribution from the belt material at the ends is negligible. For a belt constructed of thick plates (like a Caterpillar tractor), it may not be negligible.

If it does not accelerate at all, that's an effective infinite mass and infinite momentum. You may need to exercise due care cancelling infinities in the math.

If it does accelerate, that's a finite effective mass.

You can use this same analytical trick when considering the motion of two weights hung by a string over a pulley. You have the net force of gravity ##g(m_1 - m_2)## and the total mass of the pair ##m_1 + m_2##. The effective mass in this case is equal to the total mass. Assuming an ideal pulley and ideal string.

If you are adventurous, you can use the trick with more complex pulley arrangements, multiplying or dividing the effective mass of each weight based on the pulley setup.

Edit: Let's try explaining things another way

Pretend that we have a long flat plate of mass ##M## moving at some speed ##v##. As long as we do not run out of plate, the relevant behavior of this long flat plate will be identical to the top surface of a conveyor belt of total mass ##M## whose surfaces are both moving at speed ##v##.

Since the relevant behavior of the two is identical, we can reason about the behavior of the flat plate even though the reality is a looping belt. Yes, that does mean that when we speak about the "momentum of the belt", we are jumping scenarios and talking about the equivalent situation rather than the literal physical truth.

Maybe I’m behind the 8 ball in this…

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