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.

 

[Nickie]

You are on the right track by considering the law of conservation of angular momentum. In this system, the total angular momentum remains constant before and after the mass m4 disconnects. This means that the initial angular momentum of the system, Ʃmiviri, must equal the final angular momentum, ωI, where I is the moment of inertia of the system.

To calculate the moment of inertia, you can use the parallel axis theorem, which states that the moment of inertia of a system about a point is equal to the moment of inertia about the center of mass plus the mass of the system times the square of the distance between the two points. In this case, the moment of inertia about the center of mass would be the moment of inertia of the cross, which can be calculated using the formula for a rectangular plate.

Once you have calculated the moment of inertia, you can set the initial and final angular momentums equal to each other and solve for ω, the final angular velocity of the system. This will give you the answer to the question of what the radial velocity of the system is after m4 disconnects.