Object on a Rope: Momentum & Kinetic Energy

[fizics]

Homework Statement

A point object of mass m is connected to a cylinder of radius R via a massless rope. At time t = 0 the object is moving with an initial velocity v perpendicular to the rope, the rope has a length L, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds Tmax.
I wonder if the momentum and kinetic energy of the object are conserved in this case.

Homework Equations

The Attempt at a Solution

 

[Doc Al]

What do you think? Is there more to the problem than your question? (What are they asking you to find?)

 

[fizics]

The problem actually has two questions,but both of them involve whether the kinetic energy or momentum is conserved.So after knowing whether they are conserved,I would solve the problem.The answer suggests that kinetic energy is conserved,while the angular momentum is not.But I can't get why.

 

[Doc Al]

Is work being done?

Is a torque being exerted?

 

[fizics]

I think there is work and no torque.

 

[Doc Al]

I think there's work done on it

What's doing the work?

 

[fizics]

The tension exerted by the cord is pulling the object closer to the cylinder,so it's doing work;but on the other hand,since both the mechanical energy and potential energy doesn't change,the kinetic energy should be conserved.I am confused.

 

[Doc Al]

The tension exerted by the cord is pulling the object closer to the cylinder,so it's doing work;

The cord is definitely exerting a force on the object, but to see if work is being done you must compare the direction of the force to the direction of the instantaneous velocity of the object.

Consider the other end of the rope. Is any work being done there?

 

[fizics]

The force is always perpendicular to the direction of velocity.Can I divide the process to many many uniform circular motions,where the radius is decreasing?Thus the kinetic energy is conserved.
And what about the angular momentum?I think there's no torque acting on it,so it's conserved as well.

 

[Doc Al]

Yes, kinetic energy is conserved since no work is done.

And what about the angular momentum?I think there's no torque acting on it,so it's conserved as well.

Consider the angle that the rope makes with respect to the radial direction.

(If both kinetic energy and angular momentum are conserved, you'll have some explaining to do since the radius decreases.)

 

[fizics]

Thank you.I know why I was confused.I was always thinking in a reference system where the tangential point made by the rope and cylinder is stationary,but actually it should be in one where the cylinder is stationary.

 

[fizics]

I looked at the question concerning the momentum again,it says:
"What is the angular momentum of the object with respect to the axis of the cylinder at the instant that the rope breaks?"
I paid too little attention too these words.

 

[compwiz3000]

Can't we treat the cylinder as part of the system? If so, then wouldn't the torque be internal and thus conserve angular momentum?

 

[Doc Al]

Can't we treat the cylinder as part of the system? If so, then wouldn't the torque be internal and thus conserve angular momentum?

Since the cylinder is kept stationary, there are external forces acting. But if you included the cylinder+attached Earth as part of the system, then you'd be correct that total angular momentum would be conserved.