You have a frictionless pulley over which is suspended a rope (weightless). One one end of the rope is a weight (W). On the other end of the rope is a Monkey of weight W.
What happens when the monkey gets bored and starts climbing?
I suppose the weight (W) is precisely equal for both? Otherwise it would be a ridiculously easy question.
If the weight is heavier than the monkey, the monkey can climb.
If it is lighter, the monkey will pull the weight up instead of climbing.
If precisely equal though, it would be a balancing act. The monkey could climb a little, but the the weight would rise a little.
Effectively, it would be the same as having a teeter-totter with the weight on one end and the monkey on the other.
The more rope the monkey climbs through, the higher the 'teeter-totter' would 'swing' each time the balance changed a little. If the monkey climbs up all the rope, it could get all the way to the pulley, only to fall all the way to the floor when the balance changes, and then go all the way back up when the balance changes again.
This is a thought experiment so we grant ideal conditions:
no friction or mass of pulley
no mass of rope
weights exactly equal
That should get some brain cells sparking. You should make questions like this a regular thing.
Well, according to me, the monkey should descend and the weight should rise so as to make sure that the vertical distance of the rope remains the same from the top on both sides.
In my opinion, the rope below the monkey can be neglected. So, as the monkey climbs up, the rope will move below thereby making the monkey descend and the weight rise. This will happen until the monkey has reached the top and the weight has reached the top as well.
This question really got me thinking and well, I'm not too sure that what I said makes sense ... so, well, I think you should remove the word "basic" from the post title.
It depends on how fast the monkey tries to accelerate up the rope.
The more acceleration, the more the monkey will remain in position with the rope moving through his hands & feet until the weight hits the pulley and the monkey then climbs up the rope to the pulley.
The less he tries to accelerate the more the wight will remain stationary and the monkey will ascend the rope will little or no movement of the rope.
Yes, but there is still one unequal factor that has to be taken into consideration...
Dennise reiterates it:
The momentum/inertia/acceleration of the monkey is what determines the motion of the system.
Nope. The rope is a 'perfect' rope and therefore there is no delay in propagation. The acceleration of the monkey, therefore, is not a factor.
Yes it is. When the monkey stops moving, whatever was moving will try to stay in motion.
The inertia is identical in both cases (the monkey and the weight).
But while the weight accelerates/decelerates uniformly, the monkey has independently moving parts... I think this would be the critical factor that gets the system off-balance.
But the question is an 'ideal' situation so we can ignore lateral (non vertical) vectors in the solution. All ideal problems involve some simplification, and this is no different. Assuming no non-vertical forces then the solve the problem..
Think about it this way: Linear momentum: if two objects of equal mass are moving at the same speed but in opposite directions, the total linear momentum is zero.
Similarly, for the monkey and the counterweight, they have same mass and same speed, but they are rotating in opposite directions relative to the axis of the pulley -- hence equal and opposite angular momentum, hence zero net angular momentum.
Pully is a first order simple machine, so is a beam balance.
So, place W on one pan and monkey on other pan. As the monkey stand up imagine what will happen? - A temporary unbalance
All these simplifying assumptions are actually making it harder to figure out...
Really, the solution is obvious if we just remove some of these impossible conditions.
But, if you insist on making everything equal, instant, frictionless, and two dimensional, then both the monkey and the weight should rise at the exact same rate.
This seems like an impossible answer because in real life, it is. It only becomes possible with all of these assumptions of simplified, perfectly efficient systems.
You would be surprised how often real-world systems behave like ideal systems. In a real world system the friction of the pulleys would usually be so tiny as to be negligible compared to the weight of things being hoisted by pulleys. Slippage would be almost nil, too, as would stretching effects on most ropes.
i would suspect that if this apparatus was actually built, it would behave ideally. A monkey isn't all that heavy, but well-bearinged pulleys would turn so easily that resistance would be negligible. i can confirm that it does work as expected using a heavy-duty pulley, a man, and a bucket full of roofing membrane - i've seen it myself.
But wouldn't the differences -- however minute -- throw off the delicate balance that would be required for both to rise at the same rate?
But once the rates are slightly different, the weight of the rope becomes a factor, since one side will have more rope than the other, which leads to even more unevenness, causing more rope to go to the heavy side, worsening the unbalance.
This slight difference would have a positive feedback effect which will eventually make one side significantly heavier than the other, so that one side falls while the other rises.
Yep, but since experimental error includes that case, the counter case (where the other side does the same) and the ideal answer, then that doesn't really mean anything...
It does mean that the 'ideal case' -- though possible -- is fantastically unlikely.
When the monkey and weight are rising at the same rate, it is essentially a balancing act.
Suppose we have a lever, pivot point exactly in the middle, weight on one side, and monkey on the other side... Now, the monkey moves, only a tiny bit, but it causes the weight side to fall just slightly, and the monkey side to rise just slightly... What will happen then? Unless some force acts to push the monkey side back down, the weight will continue to fall at an accelerating rate.
...That's why balancing things is precarious, any slight disturbance is magnified by the feedback effect until it causes the balance to go completely off.
The rope is the same thing; instead of lever arms, we have lengths of rope, and instead of a lever pivot point, we have a pulley... but they still operate the same way.
Now, how much of a disturbance is required to upset the balance?
All the weight-endowed objects in the system have inertia, but not so much that a very tiny amount of force couldn't move them a very tiny amount... and that tiny amount of movement would begin the feedback cycle.
The pulley might have too much friction for a small force to move it. But, if if has too much friction, the rope will stay put, and the monkey will simply climb up, since it is effectively locked in place.
So, in order for the ideal case to happen, the pulley must have enough friction to stop any random differences in the monkey's movements, but not so much that the monkey's weight cannot pull it.
Since monkeys don't climb with a particularly smooth motion, that leaves you with a pretty narrow range of acceptable values for pulley friction.
The weight of the rope would be very small in comparison to the masses of the monkey/weight so in the real world i think they would still rise together. There might be a slight difference but not much. The original problem specifies a weightless rope, of course.
In a real system, the balance would be somewhat precarious. If the monkey simply stayed put for a time, wind or some other small force would eventually cause the balance to be upset.
The important point, as I see it, is that the monkey's climbing action would not in itself unbalance the system, and the forces and acceleration on the monkey and the weight would be equal and opposite. If the system is initially balanced and motionless, and the monkey began climbing fairly quickly, the real system would indeed act pretty much like the ideal one for a period of time. Variations from 'ideal' behavior would tend to be random, thus could be balanced out over many experiments if desired.
One effect that would not be random in a real system is the increasing weight of the rope on the monkey's side in a real system, which would always cause the weight to reach the top before the monkey did. Bikerman carefully stated that the rope must be weightless. The other ideal conditions make the result simpler and surer, but the weightless rope is essential to the stated behavior.
A possible improvement in the original statement of the experiment would be to use a loop of rope, so that the rope would remain balanced regardless of the positions of the monkey and the weight.