Tension and Acceleration in Pulley System

[hbweb500]

I recently did a problem where two masses were connected to the same massless string in an Atwood's machine, yet they had different accelerations. This wasn't really intuitive to me. If one mass had a greater magnitude of acceleration than the other mass, then it would imply that the bit of string above the one mass also had a greater acceleration than a bit above the other mass. If tension is the same throughout and the tiny bits of string are massless, then where does the extra force come from to accelerate the one bit of string more than the other?

I know my thinking is wrong, but I'm trying to find out where. I think it might lie in my hazy understanding of tension. Also, I realize my question might be hazy as well, I will try to post a diagram of what I am asking later when I have time.

 

[hbweb500]

I had some time, anyways:

So if the right mass is much greater than the mass attached to the movable pulley, then the right mass will fall and the center pulley will be lifted. All the time, the string above the mass to the right will accelerate at the same rate as the mass, while a bit of string above where the string is attached to the ground to the left will have no acceleration.

My question is, if the string has uniform tension, then wouldn't a force diagram for the bit of string above the mass and the bit of string attached to the ground look the same? And if they do, then how is one accelerating while the other isn't?

 

[Ben Niehoff]

The acceleration of the middle pulley should be exactly half the acceleration of the mass on the right. Look at the length of string that has to move in each case.

 

[hbweb500]

Right, I understand and can get the right answer. I understand string conservation. I just don't quite get how different parts of the same string can have different accelerations when (seemingly) the same forces are acting on them.

 

[hbweb500]

I'm going to bump, and at the same time add another question that might be related to my understanding of the original problem.

Suppose two people are in a field, pulling a massless rope either way with 50 Newtons of force. They and the rope are stationary. The tension in the rope is 50 Newtons.

Now suppose one of them pulls with 60 Newtons of force. The system will accelerate. The free-body diagram, however, for a bit of rope has tension left equaling tension to the right, so the bit of rope shouldn't accelerate. This doesn't make sense to me.

 

[pallidin]

Right, I understand and can get the right answer. I understand string conservation. I just don't quite get how different parts of the same string can have different accelerations when (seemingly) the same forces are acting on them.

Think about the one crucial aspect you are missing here. Angles. This permits acceleration.

Then think about tension. When an angular situation permits acceleration, the tension does not remain the same. In fact, when the speed doubles, the force(at that point) is cut in half. If it did not we would have free energy and perpetual motion machines.

I have done a considerable amount of private work on mechanically accelerative systems, so I do understand your questions and confusions, as I had them as well.

Perhaps this can help: With the pulley arrangement you posted, imagine that you placed tension meters(like the simple ones used to weigh a fish) in-between each pulley. Ah!
Their readings would be different!

 

[Doc Al]

I'm going to bump, and at the same time add another question that might be related to my understanding of the original problem.

Suppose two people are in a field, pulling a massless rope either way with 50 Newtons of force. They and the rope are stationary. The tension in the rope is 50 Newtons.

Now suppose one of them pulls with 60 Newtons of force. The system will accelerate. The free-body diagram, however, for a bit of rope has tension left equaling tension to the right, so the bit of rope shouldn't accelerate. This doesn't make sense to me.

Since the bit of rope is massless, all you can say is that ΣF = ma = 0*a = 0, which tells you that the tension is the same throughout.

 

[Doc Al]

Perhaps this can help: With the pulley arrangement you posted, imagine that you placed tension meters(like the simple ones used to weigh a fish) in-between each pulley. Ah!
Their readings would be different!

You think so? (The rope is massless; the pulleys massless and frictionless.)

 

[pallidin]

Doc, I'm not talking about a static situation here. During pulley movement and segmental acceleration it is not possible for the tension to remain the same.

 

[Doc Al]

Doc, I'm not talking about a static situation here.

Neither am I. The pulleys accelerate, but the tension is the same throughout the rope.

 

[pallidin]

Trust me, I built a 23 stage accelerative pulley system within a two floor building that nearly killed me. Tension differentials are a CRITICAL aspect of such experiments.

 

[pallidin]

Neither am I. The pulleys accelerate, but the tension is the same throughout the rope.

So, we have free energy! Nice!

 

[pallidin]

I shouldn't be sarcastic. Sorry. But I have a hell of a lot of experience with this.

Perhaps were on the same page but mis-understanding each other.

 

[Doc Al]

Perhaps were on the same page but mis-understanding each other.

I sure hope so, since this is physics 101. I'm curious as to why you think uniform tension leads to free energy.

 

[pallidin]

I sure hope so, since this is physics 101. I'm curious as to why you think uniform tension leads to free energy.

Doc, I was being sarcastic. Again.

 

[pallidin]

First of all, there is no such thing as uniform tension. That would imply FTL.

 

[zmike]

I have the same problem understanding how is it possible for 2 objects attached by the same rope to have different accelerations. I thought the ropes and pulleys are assumed to be massless so how is it possible for the objects to have different a?

thanks

 

[pallidin]

Keep in mind also, that "tension" is a result of an applied force, it is not a force in itself.

 

[pallidin]

zmike, your questions are interesting and important. Please hold as I consider...

 

[pallidin]

I have the same problem understanding how is it possible for 2 objects attached by the same rope to have different accelerations. I thought the ropes and pulleys are assumed to be massless so how is it possible for the objects to have different a?

thanks

As the acceleration increases, the force drops proportionally(in your closed system scenario).

I know this is difficult, but does it make any sense to you?

 

[russ_watters]

First of all, there is no such thing as uniform tension. That would imply FTL.

I don't know where you are getting these things you are saying from, but it implies nothing of the sort. In an ideal case, where a string is assumed to be massless, the tension must be constant because there is only one force pair acting on the string. I don't even see how you could think this has anyting to do with FTL travel.

In your real-world example, you probably took into account the mass of the string or friction in the pulleys or both. These things will create different tensions in different parts of a string.

 

[russ_watters]

I have the same problem understanding how is it possible for 2 objects attached by the same rope to have different accelerations. I thought the ropes and pulleys are assumed to be massless so how is it possible for the objects to have different a?

thanks

It's simply a matter of the pulleys causing the distances to be different. It's like gears: a motor drives a small gear which drives a big gear. The little gear spins faster. Same idea exactly. This is what mechanical advantage is. Pulleys and gears multiply and divide force and distance.

 

[Doc Al]

I have the same problem understanding how is it possible for 2 objects attached by the same rope to have different accelerations.

It depends on how the objects are connected via the rope. Using pulleys, for example, you can have the same rope pull twice on one object and once on another. That's what's going on in the pulley arrangement shown in post #2.

I thought the ropes and pulleys are assumed to be massless so how is it possible for the objects to have different a?

Note that the objects are not simply tied to a rope and dragged along. By having the rope double (or triple) fold in parts, using pulleys, you can get more force and less distance, or vice versa.

 

[pallidin]

I don't know where you are getting these things you are saying from, but it implies nothing of the sort. In an ideal case, where a string is assumed to be massless, the tension must be constant because there is only one force pair acting on the string. I don't even see how you could think this has anyting to do with FTL travel.

In your real-world example, you probably took into account the mass of the string or friction in the pulleys or both. These things will create different tensions in different parts of a string.

Indeed, there is no such thing as a massless string or non-frictional pulley, so hypothetical mussing on those aspects have no-real world impact. In fact, they have no application at all.

 

[pallidin]

In closing, I believe one must be careful in attributing impossible conditions to reality.

Perhaps mathematically or conceptually entertaining; if that is the goal, OK.

But if there is an attempt to flip science to a pseudo-reality I have a problem.

 

[Doc Al]

Indeed, there is no such thing as a massless string or non-frictional pulley, so hypothetical mussing on those aspects have no-real world impact. In fact, they have no application at all.

One must learn to walk before running. First deal with the simple case, then add the complications of mass and friction.

To say such approximations have "no application at all" is ludicrous.

 

[arildno]

Indeed, there is no such thing as a massless string or non-frictional pulley, so hypothetical mussing on those aspects have no-real world impact. In fact, they have no application at all.

Totally untrue.

First and foremost, these are excellent exercises in building up METHODICAL THINKING.

This is by no means something we are born with, nor is it anything easy to achieve.
The build-up must be done step-by-step, by gradually adding more (realistic) detail.

To think about "manageable problems" will develop the faculty that later on can be used for "realistic problems".

 

[zmike]

Could you recommend me a textbook or some similar text (at the undergrad level) that would adequately explain this in detail?

The only mention of "pulleys affecting tension" in my current textbook is 3 sentences long...

thanks

 

[aniketp]

try halliday, resnick and walker.

 

[rcgldr]

Right, I understand and can get the right answer. I understand string conservation. I just don't quite get how different parts of the same string can have different accelerations when (seemingly) the same forces are acting on them.

Because the left end of the string is not moving at all, while the right end is accelerating. The left pulley doesn't move at all, while the right pulley rotates and angularly accelerates twice as fast as the middle pulley, because the middle pulley is acceleration upwards at 1/2 the rate of the weight on the right end downwards acceleration.

In this idealized case, the tension is the same everywhere in the string. Since there are two lengths of string attached to the middle pulley, the upwards force on the middle pulley is doubled, and the rate of movement is halved, compared to the right most weight (since the left end of the string is fixed). It's a similar principle to gearing, doubling the torque by halving the angular velocity.

 

[Phrak]

First of all, there is no such thing as uniform tension. That would imply FTL.

I see where you're comming from. A change in tension is transmitted through a cable at the speed of sound of the cable material. But that's not the issue in large, multiple pulley systems. Ropes and cables act like stretching spring elements and the pulleys have friction. As the cable is wound on a winch the first cable elements take the burden of the load plus all the pulley frictions, combined. The last cable element is only burdened by the load.