Multiple pulleys is driving me crazy

[Alora]

Could somebody please help me with this problem? I am very confused with calculations for multiple pulleys, especially those with movable ones...

http://www.geocities.com/cdy12/pulleys.jpg

I am supposed to find the tension forces T1 and T2, as well as the acceleration of both masses.

I can do it if the whole thing is stationary, but as soon as the masses start accelerating I just don't know what to do. Please help!

 

[arildno]

Welcome to PF!
Try to set up Newton's 2.law of motion for the two masses and the (massless) moveable pulley.
In addition, since the rope lengths are constant, relate the various accelerations to each other.

 

[Doc Al]

You may find this thread helpful for a discussion of how to determine acceleration constraints in a multiple pulley problem: https://www.physicsforums.com/showthread.php?t=38121

(It's practically the same problem! )

 

[Alora]

Thanks arildno :)

Using Newton's second law for the two masses and the movable pulley I now have 3 equations but with 4 variables (T1, T2, a1, a2) and I need to find a relationship between the two accelerations.

But something's still not right ...

Here is my reasoning:

Make the ropes R1 and R2 (with tensions T1 and T2 respectively).
Call the 10kg mass m1 and the 60kg mass m2.

As m2 moves down it pulls R1 and R2 equal distances, so if there was no R2 then m1 and m2 should have the same acceleration.

However, m1 is also attached to a movable pulley, which is then attatched to R2, so m1 is really being pulled up by R1 as well as R2.

Due to gravity on m2, both ropes are being pulled equal distances, so the magnitude of acceleration of m1 is twice that of m2.

It sounds good in my head, but the answer says that acceleration of m1 is THREE times that of m2.

What am I doing wrong?

 

[Doc Al]

Due to gravity on m2, both ropes are being pulled equal distances, so the magnitude of acceleration of m1 is twice that of m2.

It sounds good in my head, but the answer says that acceleration of m1 is THREE times that of m2.

What am I doing wrong?

Your error is in thinking that if one end of the rope is pulled down by a distance x, the other end must be pulled up by that same distance. This is only true with respect to the pulley that the rope attaches to. For details, see post #11 in the thread I referred to above.

 

[arildno]

As Doc Al explained, your error lies in confusing relative and absolute quantities (in this case, accelerations).
Now, if M2 moves downwards with acceleration a2, movable pulley P must move upwards with acceleration a2 in order of keeping rope 2 of constant length.
Hence, P and M2 moves away from each other with acceleration 2*a2.
Hence, rope 1's segment length between P and M2 increases with accelaration 2*a2.
In order to keep rope 1's total length constant, the segment of rope 1 between P and M1 must decrease with acceleration 2*a2.
But this means, that the relative acceleration of M1 to P is 2*a2; or M1's absolute accelaration is a1 is equal to 3a2