# Pulley (Equal Weights)

## Main Question or Discussion Point

This is probably an easy question for someone but I am struggling with it. You have a bag of sand that weighs exactly the same as another person, and the bag of sand is on one side of a frictionless pulley and a wieghtless rope while the person is on the other side of the pulley. The person starts to climb the rope towards the pulley sometimes climbing faster and othertimes slowing down. What would you notice about the bag of sand on the other side of the pulley?

I would assume that the weight and the person would continue to stay the same distance from the pulley? What is a good reasoning for this?

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Why would you assume that? If the pulley is frictionless and the rope is massless, what are the two forces (magnitude and direction) acting on the bodies?

Keep in mind that the person only moves the rope relative to his or her own body.

Ok so that makes sence the person would not move in relation to someone standing on the ground watching but the bag of sand on the opposite side would rise until it hit the pulley... The two forces are the tension of the rope pulling up and (m)(g) pulling straight down. The tensions would equal each other which would actually be (m)(g) which results in a net force of zero and no acceleration until the body starts to move.

Is this correct?

Why would you assume that [both move equally]?
Why wouldn't you? It is obvious what happens if either mass is heavier than the other (and the rope is shortened only slowly), but if the masses are equal then the forces on each are equal, so..
Keep in mind that the person only moves the rope relative to his or her own body.
What do you mean by this?

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The energy the person exerts to raise himself up h would cause the bag of sand to move up h, since he would be exerting a force F, which would just cause the bag of sand to move up, while he remained in one place

Why wouldn't you? It is obvious what happens if either mass is heavier than the other (and the rope is shortened only slowly), but if the masses are equal then the forces on each are equal, so..
"Both moving equally" is not what the OP said. He didn't specify a starting position, so the two weights don't have to be the same distance from the pulley. The diagram I've attached might explain better what I mean.

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cowsh, would Newton's 3rd law not change your mind?
KN, the OP said "[both] continue to stay the same distance from the pulley", which implies moving equally.

That was my assumption and what I thought would have been the answer. The exact problem states:

An athlete grips a light rope that passes over a low-friction pulley attached to the ceiling of a gym. A sack of sand precisely equal in weight to the athlete is tied to the rope's other end. Both the sand and the athlete are initially at rest. The athlete climbs the rope, sometimes speeding up and slowing down as he does so. What happens to the sack of sand?

russ_watters
Mentor
Since there is only one force (the tension of the rope is the same everywhere), and the masses are the same, the accelerations are the same.

Gokul43201
Staff Emeritus
Gold Member
There's also the weight, but that's equal for both bodies too.

Alternatively, think about the total angular momentum.

Russ: Could you approve the attachment above?

Is the rope itself assume to have no mass?

If the sandbag moves up at all, then some of the rope (on the athlete's end) will end up resting on the floor, and an equivalent mass is removed from the sandbag side, by virtue of having been pulled through the pulley. If, on the other hand, the athlete's end of the rope is hauled up with him, then I suppose it's a different case.

Oh, wait, the "exact problem" calls for "a light rope", so I guess we assume it is massless.

So, as I understand it: the Athlete would be able to climb up but it would have no effect on the sandbag, which would stay in place assuming a massless rope. Am I right?

Doc Al
Mentor
So, as I understand it: the Athlete would be able to climb up but it would have no effect on the sandbag, which would stay in place assuming a massless rope. Am I right?
No. Read Russ's comment in post #9. (And Gokul's followup in #10.)

So the tension is the guys weight plus the impulse he applies pulling up. He pulls the rope and the bag comes up a bit but then when he moves up to a rest position (no impulse, the bag stops moving) ?

So the tension is the guys weight plus the impulse he applies pulling up. He pulls the rope and the bag comes up a bit but then when he moves up to a rest position (no impulse, the bag stops moving) ?
He only "moves up" while he is exerting the impulse/force as he "pulls the rope" (or in other words, while the rope is exerting an extra force on him... which is also when it exerts an extra force on the bag). Although, holding with legs, his lower body drops (and bag rises) only just slightly each time as he raises his hands (ie. the centers of mass remain equidistant from the pulley). Then he heaves himself (and the bag, though imperceptibly less than his legs) up..

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russ_watters
Mentor
That sounds the same to me, just with different wording...