Rotational Motion: Energy and Momentum Conservation

[Prannoy Mehta]

Homework Statement

A child with mass m is standing at the edge of a merry go round having moment of inertia I, radius R and initial angular velocity x as shown. (The figure shows a disc moving anticlockwise, with the velocity v (Mentioned at the end) pointing upwards to the right most edge of the disc. ) The child jumps off the edge of the merry go round with tangential velocity v, w.r.t. the ground. The new angular velocity of the merry go round is.

Homework Equations

Basic Rotational motion equations. Mainly rotational energy, and angular momentum.

The Attempt at a Solution

The conservation of angular momentum yielded the correct result for the problem stated. But when I do this using conservation of energy it does not (I have missed something out, I need to know what). Here is what I have done.

0.5 * m *v^2 + 0.5 * I * y^2 = 0.5 (I + mr^2)*x^2

Taking y as the final angular momentum. The result is something else. The answer according the reference book is given as:

((I + mr^2)*x^2 - mvr)/IThank you for the support and all your help.

 

[haruspex]

If a child jumps, does the child use energy?

 

[Prannoy Mehta]

Yes, he takes a part of the energy of the entire system.

 

[haruspex]

Yes, he takes a part of the energy of the entire system.

I would not call that jumping. Sounds more like falling off. What does a jump involve?

 

[Prannoy Mehta]

The velocity with he lands with ? Not even that. I am not sure then. He would a apply a backward force on the ride and then use the force to jump forward. Action reaction. As it is internal forces acting energy conservation and momentum conservation can be applicable. I am not sure what you mean ?

 

[haruspex]

The velocity with he lands with ? Not even that. I am not sure then. He would a apply a backward force on the ride and then use the force to jump forward. Action reaction. As it is internal forces acting energy conservation and momentum conservation can be applicable. I am not sure what you mean ?

Jump up in the air. Think about energy.

 

[Prannoy Mehta]

Are you referring to Gravitational Potential energy ? mgh? I can take take that as the velocity just as he leaves. Assuming the disc to be my frame of reference..
I did not get you..

 

[haruspex]

Are you referring to Gravitational Potential energy ? mgh? I can take take that as the velocity just as he leaves. Assuming the disc to be my frame of reference..
I did not get you..

If you jump up in the air, where does the energy come from to do that?

 

[Prannoy Mehta]

The energy comes from the ground to me ? (Conservation of Energy, I push the ground back, and then the ground pushes me back in front)

 

[haruspex]

The energy comes from the ground to me ? (Conservation of Energy, I push the ground back, and then the ground pushes me back in front)

No, that's not conservation of energy, that's action and reaction being equal and opposite.
You do not get free energy from the ground. Why do you have muscles? Why do you need to eat?

 

[Prannoy Mehta]

Muscles do provide energy. But then do we consider that while solving a numerical ?

 

[Prannoy Mehta]

So there is practically no way to do energy conservation here ? Taking this to be ideal scenario ?

 

[haruspex]

So there is practically no way to do energy conservation here ? Taking this to be ideal scenario ?

The boy jumped, he didn't fall. You have no idea (a priori) what energy he supplied in doing so.
Instead, you can solve the problem using conservation of angular momentum, then deduce the work done by the boy.

 

[Prannoy Mehta]

Yes, thank you :)

 

[Prannoy Mehta]

Just another question, if he did manage to fall out of the merry go round. Then we would conserve energy, not momentum.. ?

 

[haruspex]

Just another question, if he did manage to fall out of the merry go round. Then we would conserve energy, not momentum.. ?

Both would be conserved, but then the given information would not be feasible. The numbers would be different.

 

[Prannoy Mehta]

Makes more sense, thank you.