- #1

FranzDiCoccio

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- Homework Statement
- a ramp with rolling friction faces another ramp without friction. A ball rolls down the first ramp and climbs up the second, sliding without rolling.

The problem asks for the height reached by the ball on the second ramp, and then the height it reaches when it climbs again up the ramp it originally came from

- Relevant Equations
- conservation of mechanical energy, made up of gravitational potential energy and kinetic energy, both translational and rotational (at least for the first question).

This image represents the ramp.

The first part is pretty easy.

The red part has friction, and the ball rolls down it. The blue part has no friction, and the ball climbs it only owing to the translational kinetic energy that it gained at the bottom of the red ramp, which is only a fraction of its initial potential energy. Thus the height it reaches on the blue slope is necessarily lower than the initial one. Specifically, the final height on the blue slope is [itex]h'=5/7 h[/itex], where [itex]h[/itex] is the initial height on the red ramp (here I assume a solid ball).

The angular velocity (and hence rotational kinetic energy) does not vary while the ball is on the blue slope.

The second question seems a bit harder. The book the problem came from claims that the ball gets back to the initial height, due to energy conservation.

This seems at least really counterintuitive to me. I would dare to say that it is probably plainly wrong.

At the bottom of the red ramp the ball is rolling clockwise. If the blue ramp had friction, the rotational kinetic energy would "help" the ball climb up to (ideally) the same height it fell from.

On the other hand the ball is rolling "the wrong way" at the start of the red slope. In order to climb the red slope, it should roll counterclockwise.

It seems to me that its rotational kinetic energy should hinder its progress on the red slope, and not favour it.

I'm not really sure what would happen when the ball reaches the red part falling down from the blue ramp while "rolling backwards".

My guess is that it cannot possibly roll without sliding, and if it slides it would dissipate some energy due to friction.

Hence the height on the return on the red slope cannot be the same as the initial one. How lower than that? It probably depends on the friction coefficient, which however is not specified in the problem.

It seems to me that the claim of the book is similar to claiming that an object would reach an height [itex]h = v^2/2g[/itex] irrespective of the angle formed by its initial velocity and the horizontal direction.

Am I missing anything? Could the solution proposed by the book be possibly right?

Thanks a lot for any insight

Franz