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sgtserious
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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.
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
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]
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
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[itex]^{c}_{zz}[/itex]w)[itex]_{f}[/itex] -(I[itex]^{c}_{zz}[/itex]w)[itex]_{i}[/itex] [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:
∫[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