# Tension and Acceleration in Pulley System

1. Oct 27, 2008

### 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.

2. Oct 27, 2008

### hbweb500

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 wouldnt 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?

3. Oct 27, 2008

### 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.

4. Oct 27, 2008

### 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.

5. Oct 28, 2008

### 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.

6. Nov 2, 2008

### pallidin

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!

7. Nov 2, 2008

### Staff: Mentor

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.

8. Nov 2, 2008

### Staff: Mentor

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

9. Nov 2, 2008

### 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.

10. Nov 2, 2008

### Staff: Mentor

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

11. Nov 2, 2008

### 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.

12. Nov 2, 2008

### pallidin

So, we have free energy! Nice!!!!!

13. Nov 2, 2008

### 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.

14. Nov 2, 2008

### Staff: Mentor

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

15. Nov 2, 2008

### pallidin

Doc, I was being sarcastic. Again.

16. Nov 2, 2008

### pallidin

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

17. Nov 2, 2008

### 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

18. Nov 2, 2008

### pallidin

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

19. Nov 2, 2008

### pallidin

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

20. Nov 2, 2008

### pallidin

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?