Math Is Fun Forum

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

The beauty of physics (reasoning without data)!!

A bug has just lost its footing near the top of a bowling ball. It slides down the ball without significant friction. Deduce (showing the details) the angle it makes with the vertical when it loses contact with the surface?

Note: this problem ought to be solved by physicists or mathematicians with sufficient ground in physics (if you are not, don't try it...?). I have more then one solution.

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: The beauty of physics (reasoning without data)!!

Let the mass of the bug be m, the normal reaction by the surface on it be R, the radius of the ball be r, the velocity of the bug be v, and the angle with the vertical be θ.

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The centripetal force on the bug is given by

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When contact with the surface is lost, R = 0; then

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At that point the potential energy lost by dropping the vertical height is equal to the kinetic energy gained:

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[1] & [2] ⇒

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Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: The beauty of physics (reasoning without data)!!

Good!

Alternative solution:

If we let θ to be the angle which the bug made with the vetical when it lost contact with the surface. Then the Work done by the weight of the bug will be:

This Work will be converted into kinetic energy hence:

But at the moment when the bug lost contact with the surface the vetical component of the weight was equal to the centripetal force preduced by the circular motiom on the surface. Therefore:

From the previous two arguments we have:

Wence:

Q.E.D

Last edited by Grantingriver (2016-02-19 06:17:24)