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The component of ball's weight parallel to the track is


The ball is moved because of this component of ball's weight. But, as the ball moves, there are friction forces working on opposite direction.Hence, total force working on the ball that is parallel to the track is



And now consider the forces that is not parallel to the track. Because the ball moves parallel to the track, the resultant forces that is not parallel to the track is zero. See picture below.




Because the ball slips on one side of the track and almost slips on the another side, we can say that


and




Now use equation (1) to get acceleration of CM.

Substitute the equation above to equation (1), and also substitute f1 and f2 from equation (3) and (4),




Now, we need to find the angular acceleration. I will use picture to make it easier to imagine.
The angular acceleration of the ball is caused by two friction forces (f1 causes α1 and f2 causes α2), where

and


Substitute for f1 and f2, and we can get



The ball does not slip with the right plane, so acceleration of the contact point at the right plane must be zero. From picture, we can get the acceleration of the contact point A.



Hence,


Solving equation (8) using equation(5), (6), and (7), and we get the result:

where




That's my solution.
If you have any question, correction, or comment, please leave a comment below.
Thank you.

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1 comments

Anonymous said... @ November 27, 2010 at 2:20 PM

Don't know if you still update this, but I find the direction of the blue vector, Alpha_1, in the last picture to point in the opposite direction. If we assume friction pointing inward, should alpha_1 be such that the ball is rolling toward us, outward from the paper?

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