# Rolling Ball and its angular momentum

Everything ist idealized. This means there exists no friction or other forces which would ruin the 'Beauty and simplicity' of this question which are aiming to demonstrate a specific law or rule.

Questions A:

A Ball with a moment of intertia $$I$$ and a linear speed (speed of the Center of mass) $$V$$ Rolls without any skipping or whatsoever (just roll movement so linear and rotation) and then this balls climbs an inclined plane. The question is: At the top of the plane, the ball finds it self in position of rest (thus no movement and no rotation ) how do you justify this in relation of the conservation of angular momentum?

So personally i do not understand this question, since there is obviously a a Torque being applied by the gravitational force while climbing the plane and thus it reduces its angular velocity to 0

(this is a question from an exam so i guess i misunderstood it)

Where as we know that Angular momentum $$L$$ is given to be $$L = \omega *I$$ How is it conserved and How do you answer this question.

• Angular momentum about which axis? May 2, 2020 at 6:23
• "there exists no friction" -> won't rotate at all May 3, 2020 at 4:56

You are absolutely right in saying that the angular momentum isn't conserved. Let's say we observe the ball rising up the inclined plane, from its instantaneous axis of rotation, then we would see that the ball experiences a torque due to gravity ($$|\boldsymbol{\tau}|=mgR$$, where $$m$$ is the mass of the ball and $$R$$ is its radius). This torque reduces the ball's angilar momentum and thus the ball comes to rest.

Just in case, if the friction is absent on the inclined plane, then the ball's $$\boldsymbol{\omega}$$ will not change when it reaches at the top on the inclined plane. Onky the velocity will change. Thus in the center of mass frame, the angular momentum ($$I\boldsymbol{\omega}$$) will be conserved. However, angular momentum may not be conserved in any other general reference frame.

• I see. It was a trick question i guess!
May 2, 2020 at 6:17

When the ball is rolling up the pure inclined plane there is no friction so there is no torque about center of mass hence the ball pursues translational motion Ie no angular momentum

• The gravitational force produces a torque on the instantaneous axis of rotation.