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.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

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

2. Relevant 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[itex]^{c}_{zz}[/itex]w)[itex]_{f}[/itex] -(I[itex]^{c}_{zz}[/itex]w)[itex]_{i}[/itex] [for the pulley of mass M]

3. The attempt at a solution

This is what I tried to do

(all initial velocities and angular velocities are zero)

For monkey A:

∫[itex]^{2}_{0}[/itex](T[itex]_{1}[/itex] - Mg) dt = M*y'[itex]_{A}[/itex][itex]_{f}[/itex]

-> T[itex]_{1}[/itex]*t - Mgt = M*y'[itex]_{A}[/itex][itex]_{f}[/itex]

-> 2T[itex]_{1}[/itex] -2Mg = M*y'[itex]_{A}[/itex][itex]_{f}[/itex]

For monkey B:

∫[itex]^{2}_{0}[/itex](T[itex]_{2}[/itex] - 2Mg) dt = 2M*y'[itex]_{B}[/itex][itex]_{f}[/itex]

-> T[itex]_{2}[/itex]*t - 2Mgt = 2M*y'[itex]_{B}[/itex][itex]_{f}[/itex]

-> 2T[itex]_{2}[/itex] - 4Mg = 2M*y'[itex]_{B}[/itex][itex]_{f}[/itex]

For the pulley:

∫[itex]^{2}_{0}[/itex](MgR - 2MgR) dt = (I[itex]^{c}_{zz}[/itex]w)[itex]_{f}[/itex]

-> -2MgR = (1/2)MR[itex]^{2}[/itex]w[itex]_{f}[/itex]

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'[itex]_{A}[/itex] = R*w[itex]_{f}[/itex], 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

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# Homework Help: Dynamics: impulse momentum

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