Finding Displacement Using Impulse Momentum: A Problem with Monkeys and a Pulley

[sgtserious]

I am having a really hard time with dynamics of rigid bodies, but i think doing this problem will clear up some confusion i have.

Homework Statement

A massless rope hanging over a frictionless pulley of mass M supports two monkeys (one of mass M, the other of mass 2M). The system is released at rest at t = 0. During the following 2 sec, monkey B travels down 15 ft of rope to obtain a massless peanut at P. Monkey A holds tightly to the rope during these two sec. Find the displacement of A during the time interval. Treat the pulley as a uniform cylinder of radius R.
Relevant picture: diagram

Homework Equations

As mentioned in the topic title, the problem has to do with impulse momentum:
∫ƩFy = m(y'f - y'i) [for monkey A]
∫ƩFy = m(y'f - y'i) [for monkey B]
∫ƩM = (I$^{c}_{zz}$w)$_{f}$ -(I$^{c}_{zz}$w)$_{i}$ [for the pulley of mass M]

The Attempt at a Solution

This is what I tried to do
(all initial velocities and angular velocities are zero)

For monkey A:
∫$^{2}_{0}$(T$_{1}$ - Mg) dt = M*y'$_{A}$$_{f}$
-> T$_{1}$*t - Mgt = M*y'$_{A}$$_{f}$
-> 2T$_{1}$ -2Mg = M*y'$_{A}$$_{f}$

For monkey B:
∫$^{2}_{0}$(T$_{2}$ - 2Mg) dt = 2M*y'$_{B}$$_{f}$
-> T$_{2}$*t - 2Mgt = 2M*y'$_{B}$$_{f}$
-> 2T$_{2}$ - 4Mg = 2M*y'$_{B}$$_{f}$

For the pulley:
∫$^{2}_{0}$(MgR - 2MgR) dt = (I$^{c}_{zz}$w)$_{f}$
-> -2MgR = (1/2)MR$^{2}$w$_{f}$

From the conclusion of all three sets of equations, I have 5 unknowns, with only three equations
I figure from here I need to some kinematics:
By circular motion, the y'$_{A}$ = R*w$_{f}$, which helps eliminate one unknown
In addition, there is the length of rope connecting the three bodies, but considering the pulley has mass, I am not sure how to relate the velocities of point A and point B
I believe I get stuck a lot on the kinematic constraints, and here I am not sure how to solve the system of equations I developed

Any help is greatly appreciated

 

[rude man]

Seems to me this problem is solvable by elementary energy conservation principles.
Were you explicitly asked to analyze in terms of impulse momentum considerations?

 

[sgtserious]

Yes, the problem is an exercise in using impulse momentum. I know there are other ways to solve it, but I was hoping to clear up some confusion I have with using impulse momentum.