Solving Atwood's Machine Problem

[Leilei]

Atwood's Machine Problem -- HELP

Okay, I had to do a problem concerning the Atwoods machine over the weekend. I'm not doing very well in Physics right now, and I don't have any friends in that class, so I thankfully found this place. I have already tried thinking about these problems...

1. If two objects of equal mass are suspended from either end of a string passing over a light pulley (an Atwood's machine), what lind of motion do you expect to occur? Why?

I said that if the two objects of equal masss are suspended over a light pulley, the objects will move only until they are both at wqual distances from the ground. Then they will not move at all.

But the part I'm stuck on is -- why? Why does this occur? How can I explain this correctly?

2. For an Atwood's machine, how would you expect the acceleration to change if you:

- Move mass from one side to the other, keeping the total mass constant?

I'm not sure if this is correct: I said, the acceleration would increase. The mass of one side is gerater than the mass of the other, meaning the heaver side has a greater weight force (mg) acting upon it equating to a greater accelearation.

- How about if you gradually increase the mass of both sides, keeping the difference in mass constant?
I said the acceleration will remain the same because ratio of masses are equal. I am not sure if this is correct either.

3. Why do the two masses have the same acceleration?
I said the two masses have the same acceleration because the tension throughout the string is equal. The combined mass pulls each other at the same rate becauseb oth have the same foreces acting upon them.

I'm not sure at all if any of my answers are correct. What I need the most help on is #1. If somebody could help me, I would appreciate it very much.

 

[HallsofIvy]

1. If two objects of equal mass are suspended from either end of a string passing over a light pulley (an Atwood's machine), what lind of motion do you expect to occur? Why?

Are you assuming a massless string? If not you will have to consider the weight of the different lengths of string on both sides.
If the string is massless, then calculate the downward force on both sides due only to the objects and see what happens.


For 2, you are basically correct: shifting mass from one side to the other will increase the force on one side over the other.

For 3, where you increase (or decrease) mass on both sides equally, yes, the acceleration will not change. Have you given any thought to exactly what that acceleration will be?
How about trying an "extreme" case: both masses are very, very small?

 

[Leilei]

Yes, we are assuming the mass of the weight is negligible.

But I'm confused as to how I explain why there would be no acceleration (Question 1)

For question 2, if the masses were small... wouldn't the acceleration be less than? And wouldn't the accleration be gravity?

 

[Dissident Dan]

Originally posted by Leilei
1. If two objects of equal mass are suspended from either end of a string passing over a light pulley (an Atwood's machine), what lind of motion do you expect to occur? Why?

I said that if the two objects of equal masss are suspended over a light pulley, the objects will move only until they are both at wqual distances from the ground. Then they will not move at all.

Always be mindful of the equations:
F = ma
F = m₁g - m₂g = g(m₁ - m₂)

The difference from the ground won't matter. We assume gravity to be constant. Mass is constant, too. You described a change in motion and a change in acceleration (moving until they are both at equal height), but mass and gravity are both constant, so how can you have a change in acceleration?

I don't want to give you the answer, but that should help you out.

2. For an Atwood's machine, how would you expect the acceleration to change if you:

- Move mass from one side to the other, keeping the total mass constant?

I'm not sure if this is correct: I said, the acceleration would increase. The mass of one side is gerater than the mass of the other, meaning the heaver side has a greater weight force (mg) acting upon it equating to a greater accelearation.

If you are increasing an inequality in mass, then that is correct. But what happens if you move mass from the more massive side (if the sides don't start off equal) to the less massive side?

- How about if you gradually increase the mass of both sides, keeping the difference in mass constant?
I said the acceleration will remain the same because ratio of masses are equal. I am not sure if this is correct either.

Be sure to understand the difference between keeping a constant difference and a constant ration. Difference is an additive/subtractive relationship, and a ratio is a multiplicatin/divisive relationship. When you keep a constant difference and change the total amount of mass, you cannot keep a constant ratio.

3. Why do the two masses have the same acceleration?
I said the two masses have the same acceleration because the tension throughout the string is equal. The combined mass pulls each other at the same rate becauseb oth have the same foreces acting upon them.

That is correct. Another way to explain it is to say that the string is of constant length (an assumption), so if one end accelerates, the other end must accelerate at the same rate for the string to keep a constant length.

 

[Leilei]

Originally posted by Dissident Dan
Always be mindful of the equations:
F = ma
F = m₁g - m₂g = g(m₁ - m₂)

The difference from the ground won't matter. We assume gravity to be constant. Mass is constant, too. You described a change in motion and a change in acceleration (moving until they are both at equal height), but mass and gravity are both constant, so how can you have a change in acceleration?

But let's say you pull the string over the pulley, each side with objects of equal mass. And let's assume you made it so one side was lower to the ground then the other one. When you let go, wouldn't the objects move so they were at equal distance from the ground, but then stop moving once it was done? I can't explain it very well, sorry.

 

[Dissident Dan]

If both sides have the same mass, then they're pulling with the same force, right? If they are pulling with the same force, then the net force is 0. If the net force is 0, then you have no acceleration. If you have no acceleration, and everything was initially at rest, then it will remain at rest.