Wait a minute!
(And, yeah, sorry for dragging this zombie back out after it died five months ago)
If the pulley has enough inertia to resist spinning, it will not only resist spinning “as the monkey accelerates”, but throughout the experiment. [The force exerted with each pull is not stored somehow to accumulate, is it?] We might as well clamp the rope to the ceiling (wherever it is up there) so it’s immobile, in which case the monkey’s action would fail to move the rope and we’d once again have simply the rope+monkey balancing off the rope+bucket [it doesn’t matter how high or low the monkey is situated on the rope] until the monkey reached the top. It would definitely “cause the monkey to ascend more quickly than the bucket once he starts climbing” because the bucket wouldn’t ascend at all.
I think you may have taken “resist spinning” to mean “prevent spinning”. This is not what I meant. No amount of inertia would prevent the pulley from spinning when a force is exerted on it. (Friction could though.) It’s just that it’s going to require a certain amount of force to cause the pulley to accelerate and thus it will absorb some of the force that would otherwise go to the bucket. If the monkey pulls with 5 lbs of force and the pulley absorbs 1 lb leaving 4 lbs for the bucket, the monkey is going to have greater acceleration than the bucket.
Inertia is not a resistance to motion, it is a resistance to change of motion.
Have you ever tried to move a heavy object, say, push a car? It takes a lot of umph to get the car rolling, and then you can’t stop the car. The inertia that resisted the initial push becomes rolling momentum (inertia in another word).
If your pulley wheel is massive, then the inertia resists the initial motion, so the monkey starts moving before the bucket does. But once the wheel starts moving, the bucket and monkey are moving at the same speed, but the monkey is slightly ahead of the bucket.
If the monkey doesn’t go all the way to the top but stops, then the inertia will keep the wheel turning, and the monkey will go down as the bucket keeps going up, until they balance out at the same height. There may be some wobbling back and forth rather than a quick settling at the same height. The inertia will cause overshoot, and then the bucket will be higher, then the bucket will start pulling more and the rope will reverse directions. Repeat ad nauseum until the momentum damps out. With no friction, that could take a while.
Without friction, the bucket and monkey won’t oscillate up and down. As you say, there’s only resistance to change of motion, so there’s not restoring force to make them go up and down.
I think the monkey will continue to descend, and the bucket will continue to rise (until the bucket hits the pulley, or the monkey hits the ground). But it’s possible they immediately come to rest when the monkey stops climbing, I’d have to think on this some more. With friction, they will eventually come to rest, but I don’t think that would necessarily be when they are at the same height.
Wouldn’t the weight imbalance of uneven monkey and bucket create the restoring force?
Of course that restoring force might not be enough to counter the momentum of the rotating wheel, in which case you’re right, the bucket would run up and crash at the top. Or the monkey will jump off and the whole experiment goes to pot.
The monkey and bucket of sand weigh the same, and we’re considering the rope to have negligible weight, so there would be no weight imbalance. (If you include the rope’s weight, you still don’t get a restoring force.)
I’ll point out that the core of my (exceedingly verbose) argument is resting on the fact that the original question fails to postulate a frictionless suspension-point for the rope and also fails to postulate a weightless and flexible connecting material when, in fact, it’s perfectly easy enough to use such words instead of “pulley” or “rope” in the premise.
There is no restoring force (or entity, seen or unseen). Newton 3 applies here, but in a different manner; the rope on the monkey’s side is going down because the monkey is exerting force in its attempt to go up.
The monkey and bucket will NOT balance-out at the same height since inertia will keep the pulley-wheel turning and, lacking any sentience, the wheel will not know when to stop
Furthermore, if the monkey’s initial attempt to ascend causes the pulley-wheel to move at all the monkey’s side of the system will have an additional weight of the rope that shifted to his (her) side and the bucket’s side of the system will have exactly that much less rope and therefore that much less weight. Therefore the system will be out of balance and the monkey and its extra rope will continue to increase that imbalance as long as gravity is the only force acting upon the system.
I believe the rate at which the monkey and its rope descend will NOT change (speed will be constant; acceleration = 0) because gravity is a constant which does not depend on weight or mass (q.v. Lucretius; Stevinus; de Soto) but this is an argument that we shouldn’t bother the monkey with; take it to another thread.
If we’re using the mass of the rope and the inertia of the pulley then the side gaining mass (and therefore the whole system) will have changed acceleration since it’s not only affected by gravity but also by the decreasing counterforce of the decreasing mass on the other side of the pulley.
Take an equal-arm balance - you know, balance scales like “the scales of justice”. Put the same weight on each side so they balance evenly. Then use your finger to lift one side and lower the other, then let go. You have equal weight on each side at uneven level, and you get a restoring force that drives the pans to the same height.
But if there’s enough inertia to resist the original motion, then that inertia will likely overcome that restoring force.
That’s just because those balances are made so that the pivots holding the two pans are lower than the pivot in the middle. Put them at the same height, like in a pulley, and there’s no restoring force. Put them higher, and there’d be a positive feedback, and the whole thing would topple over.
For the monkey and bucket of sand, where is the restoring force coming from? There’s no change in total energy. Monkey goes up a meter, and sand comes down a meter, and the change in potential energy is zero.
Nice. I’m gonna take a short cut and just believe you.
But that’s not what has been postulated in the original physics question.:smack:
The original question (a zillion responses ago and borrowed from some textbook for this discussion board to argue about) simply postulates
[ul]
[li]A pulley[/li][li]A rope[/li][li]A mass – which is sentient, sapient, conscious, and active (restless?)[/li][li]A mass – which is…well…none of the above.[/li][li]The two masses are equal and start out balanced on the pulley&rope system[/li][li]The active (etcetera) mass (called a monkey for easy identification) is somehow motivated to exert effort to move itself upward from its position on its side of the rope[/li][/ul]
All of my arguments rest heavily on the fact that the question specifies
[ul]
[li]a single pulley, rather than a fulcrum-point up in the air (or several)[/li][li]a rope, rather than a flexible connective filament with no (or negligible) mass[/li][/ul]
Everyone’s argument, by the way, presumes a stable, life-stustaining environment in which this experimental system is set up. After all, an unstable system (shifting gravity, earthquakes, etcetera) will throw our whole experiment off kilter anyway. And, if we’re doing this in a filled swimming pool, the monkey will probably just float to the top. If we’re doing this in a vacuum-sealed room we can wait until the oxygen runs out and the monkey will drop to the floor when it dies. Likewise if we’re starting in a sealed vacuum (only we’ll see results faster).
—G!
For Goodall’s sake!
Will someone just try this at a zoo somewhere?
