Conservation of angular momentum

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Homework Statement


There's a system of 4 masses, all connected to a cross which has a negligible mass, and which is positioned on a smooth surface. The distance of each mass from the center of the cross is L and the cross spins around its center in a constant radial velocity of ω0 rad/sec:
dPcyf.gif

Now mass m4 disconnects from the cross.

What is the the radial velocity of the system after m4 disconnected, considering m1=m3 and m2=m4=M?

Homework Equations


Conservation of momentum:
Ʃmivi=0

Conservation of angular momentum:
Ʃmiviri=ωI

The Attempt at a Solution


I calculated using conservation of momentum that the linear velocity of the system after m4 disconnected was v2=Mω0L/(M+2m)

Now I think I should use the law of conservation of angular momentum but I'm not sure how. I think that the center of mass is L/2 to the right from the center of the cross so the distance of m1 and m3 from the center of mass is √((0.5L)2+L2). What should I do next?
 

Answers and Replies

  • #2
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Relevant equations

Moment of Inertia =?
Angular Momentum=?
 
  • #3
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Relevant equations

Moment of Inertia =?
Angular Momentum=?
I know that the moment of inertia is I=Ʃmiri2 and the angular momentum L can be expressed as ωI, so I tried:

L=Ʃmiviri=ωI = ω(Ʃmiri2) and I can get the value of ω this way, but I'm not sure what the the distance from each mass to the center of mass is. I mean, what are the values of ri in this sum: Ʃmiri2 ?
 
  • #4
1,065
10

Homework Statement


There's a system of 4 masses, all connected to a cross which has a negligible mass, and which is positioned on a smooth surface. The distance of each mass from the center of the cross is L and the cross spins around its center in a constant radial velocity of ω0 rad/sec:
dPcyf.gif

Now mass m4 disconnects from the cross.

What is the the radial velocity of the system after m4 disconnected, considering m1=m3 and m2=m4=M?

Homework Equations


Conservation of momentum:
Ʃmivi=0

Conservation of angular momentum:
Ʃmiviri=ωI

The Attempt at a Solution


I calculated using conservation of momentum that the linear velocity of the system after m4 disconnected was v2=Mω0L/(M+2m)

Now I think I should use the law of conservation of angular momentum but I'm not sure how. I think that the center of mass is L/2 to the right from the center of the cross so the distance of m1 and m3 from the center of mass is √((0.5L)2+L2). What should I do next?
You have to start with conservation of energy.
All masses have equal tangential velocity.
As mass m4 detached from the cross(it follows a tangential path), the total energy of the system remains.

Using consevation of momentum requires the momentum of detached mass m4, which follows a straight line.
 

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