Angular Momentum Conservation in Impulsive Blow: Rolling and Climbing Cylinder

[mooncrater]

Homework Statement

The question says:
A uniform solid cylinder rolling with angular velocity $\omega$ along a plane surface strikes a vertical rigid wall. With what angular velocity the cylinder begins to roll up the wall because of impulsive blow? It is observed it rolls without sliding after striking the wall.

1. $\omega$/2
2. $\omega$/3
3. $\omega$/5
4. $\omega$/4

Homework Equations

$I_1\omega _1=I_2\omega _2$

The Attempt at a Solution

Can we conserve angular momentum along the point at which the cylinder strikes the wall? I don't think so because that point gives an impulse to the cylinder.

 

[haruspex]

Can we conserve angular momentum along the point at which the cylinder strikes the wall? I don't think so because that point gives an impulse to the cylinder.

what moment would that impulse have about that point?

 

[mooncrater]

what moment would that impulse have about that point?

Since it passes through it, zero?

 

[haruspex]

Since it passes through it, zero?

Yes.

 

[ehild]

Friction plays the main role. During the collision, some impulsive force of friction is exerted on the cylinder. That force causes upward acceleration and its torque changes the initial angular velocity. Both the momentum and the angular momentum change.

 

[haruspex]

Friction plays the main role. During the collision, some impulsive force of friction is exerted on the cylinder. That force causes upward acceleration and its torque changes the initial angular velocity.

Indeed, but that also has no moment about the point of contact, so the OP's proposed method works fine.

 

[ehild]

The cylinder slides during the contact, it does not roll.

 

[haruspex]

The cylinder slides during the contact, it does not roll.

Not according to the OP.

 

[mooncrater]

Okay then, I am a little confused about the conditions for applying conservation of angular momentum. Is it that the torque around it should be zero?

 

[ehild]

Not according to the OP.

When the rolling cylinder hits the wall, it rubs the wall. There is sliding friction. The collision takes some time when the impulsive force and impulsive torque take effect. As a result, the cylinder looses its horizontal velocity and gains an upward velocity, and also its angular velocity changes.

 

[ehild]

Okay then, I am a little confused about the conditions for applying conservation of angular momentum. Is it that the torque around it should be zero?

You can apply the the torque equation with respect of a fixed axis or with respect to the CM. Some Δt time is needed to stop the horizontal motion of the cylinder. During that time, the surface of the cylinder slides on the wall. There is no instantaneously fixed axis of rotation.

 

[haruspex]

When the rolling cylinder hits the wall, it rubs the wall. There is sliding friction.

I see no reason to assume that. If the normal impulse is J, the frictional impulse due to static friction is up to $\mu_sJ$. If that is enough to provide the vertical velocity consistent with the angular momentum conservation then no slipping.
If we do allow slipping on contact, is there enough information to solve the problem? I doubt it.

 

[ehild]

I see no reason to assume that. If the normal impulse is J, the frictional impulse due to static friction is up to $\mu_sJ$. If that is enough to provide the vertical velocity consistent with the angular momentum conservation then no slipping.
If we do allow slipping on contact, is there enough information to solve the problem? I doubt it.

Have you solved the problem? Have you got one of the given values?

 

[haruspex]

Have you solved the problem? Have you got one of the given values?

Yes.

 

[ehild]

Well, you are right, angular momentum is conserved with respect to the contact point at the wall. The result is the same obtained with my method.

 

[mooncrater]

Ok now it's clear to me. Thanks haruspex and ehild. :).