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?

 

[nlightner]

Hello,

Thank you for providing the details of the pulley system problem. I would approach this problem by first identifying all the known variables, such as the mass of the person (M), mass of the platform (m), force applied on each rope (F), and the distance between the pulleys (D). Then, I would use the Newton's second law of motion, which states that the net force on an object is equal to its mass times its acceleration (F=ma), to find the acceleration (a) of the person and the platform.

Since the person is pulling themselves up with the ropes, the net force on the person and the platform would be equal to the force applied on the ropes (F) minus the weight of the person and the platform (M+m) times the acceleration due to gravity (g). This can be represented as F-(M+m)g = (M+m)a.

Next, I would use the geometry of the pulley system to find the relationship between the acceleration (a) of the person and the platform and the acceleration of the strings (Z and Y). As you mentioned, the strings would have the same acceleration since the pulleys are massless and the platform would not be level if they were different. Therefore, we can say that a = d2Z/dt2 = d2Y/dt2.

Finally, I would solve for the acceleration (a) by substituting the relationship between the accelerations of the person and platform and the strings into the equation F-(M+m)g = (M+m)a. This would give us the final answer for the acceleration (a) of the person and the platform.

I hope this helps in solving the pulley system problem. Please let me know if you have any further questions.

Best,