Acceleration of two masses and a pulley

[Hannisch]

Homework Statement

P is a pulley with neglectable mass which is being affected by an external force F. Determine the accelerations of P, m and M.

I've got a picture, but it goes something like this:

There's a pulley and from it (on each side) there are two masses hanging. On the left is m and on the right is M. The force F is from the centre of the pulley and is pointing straight up. (I hope you understand what I mean.)

And it's also assumed that the rope is massless.

Homework Equations

F = ma

The Attempt at a Solution

There are only forces in the y-direction and I'm using up as the positive axis.

m: T - mg = ma_m

M: T - Mg = Ma_M

And I know that since the pulley is massless (or can be considered as such) the forces on it need to balance out or it'd be accelerating with infinite acceleration. The forces acting on it are F and 2T, tension from each of the sides.

That means that T = ½F and:

½F - mg = ma_m

a_m = ½(F/m) - g

and doing the same for M I get:

a_M = ½(F/M) - g

This is stated as correct in my book. My problem is the acceleration of the pulley - I think I'm not seeing something.

Because for some reason (this I can see from the answer that is supposed to be correct) a_P = ½a_m + ½a_M

Why exactly is that? Is it because the tension from the rope is half that of the force? I'm a bit confused and I don't want to guess a reason, because that's not going to be so good if I'm not right..

Ah, I'll be thankful for any help :)

- Hanna

 

[Lok]

If the 2 weights are different are they moving?

 

[Lok]

That means that T = ½F and:

½F - mg = ma_m

a_m = ½(F/m) - g

and doing the same for M I get:

a_M = ½(F/M) - g

T = F1+F2 =F1/2+F1/2=F for a start. But is the tension necesarry?

 

[Hannisch]

I guess they're different (it doesn't say that they're the same and the question clearly differentiates them), so I'd say they probably are moving relative to each other.

Otherwise their acceleration would be the same as that of the pulley, right?

 

[Hannisch]

Why wouldn't the tension be necessary? It's the only force affecting the masses up and the only force affecting the pulley down.

And I don't agree that T = F. I definitely don't, since there's tension from both sides of the pulley and that means that 2T = F.

 

[torquil]

Hint: I think you will have to use the fact that the rope has a constant length, and the pulley will keep it tight all through the movement. So the pulley has to move at some acceleration related to the accelerations of the masses.

Just to clarify: I agree with everything you have done in the first post, but I think you need to use what I wrote above to derive the movement of the pulley.

Good luck!

Torquil

 

[Hannisch]

Torquil: That makes a lot of sense. Since the tension is affecting the pulley equally on both sides, I guess that means it would have the half the acceleration of m and half that of M. Thanks!

 

[Lok]

Why wouldn't the tension be necessary? It's the only force affecting the masses up and the only force affecting the pulley down.

And I don't agree that T = F. I definitely don't, since there's tension from both sides of the pulley and that means that 2T = F.

My approach works just as well, as I see now.

Anyway if the 2 masses are equal then aM=am=aP=F/(m+M)

But as the masses are different then the smaller one will be accelerated faster then the pulley and the other slower. The pulley will be accelerated faster then in the previous case.