Solving a Pulley System: Find Acceleration a

[bodensee9]

Hello:

I've attached a drawing of the problem (pretend that the circular things are pulleys, and the strings are straight). A person of mass M stands on a platform of mass m and pulls himself up by 2 ropes which hang over pulleys, as in the attachment. he pulls each rope with force F and accelerates upward with uniform acceleration a. Find a.

So I think we can say that the length of the strings over these pulleys can be

let D = distance between the pulleys
let R = radius of pulleys

X + D + 2piR + Z + X + D + 2piR + Y = L, where L, R, D are constant.
I think Z = (X - h), where h = the length of ceiling to pulley. So if we take the second derivative we have: 2*d2X/dt2 + d2Z/dt2 + d2Y/dt2 = 0. I am not sure if that's correct ... and not sure what to do after that? I think that the force is the same for both sides. How do I find the relationship between the acceleration of Z and Y? Would they be the same since the pulleys are massless and if they weren't the same the platform wouldn't be level? Thanks.

 

[LowlyPion]

Seems like a long way around the barn, when all you have to do is note that the system (M + m), if it is accelerating upward, will be (M + m)*(a + g) won't it?

If the Force is being applied by 2 ropes, then each rope carries F/2.

If the person's feet are not tied to the platform, it will however be a little difficult to generate this, since how can he pull harder than M*g on both ropes? If the platform m is heavier than he is, M, won't he just fly up as he holds the ropes?