Conservation of angular momentum

[phyyy]

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:

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:
Ʃm_iv_i=0

Conservation of angular momentum:
Ʃm_iv_ir_i=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?

 

[azizlwl]

Relevant equations

Moment of Inertia =?
Angular Momentum=?

 

[phyyy]

Relevant equations

Moment of Inertia =?
Angular Momentum=?

I know that the moment of inertia is I=Ʃm_ir_i² and the angular momentum L can be expressed as ωI, so I tried:

L=Ʃm_iv_ir_i=ωI = ω(Ʃm_ir_i²) 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 r_i in this sum: Ʃm_ir_i² ?

 

[azizlwl]

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:

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:
Ʃm_iv_i=0

Conservation of angular momentum:
Ʃm_iv_ir_i=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.