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Answer to Question #6228 in Mechanics | Relativity for Stephen

Question #6228
A ball rolls off a frictionless sphere with neglible inital velocity, at what point does the ball fall off?
Expert's answer
The forces on the ball include the centripetal force due to gravity, the centrifugal force due to motion along the curve and the resulting normal force
<img src="data:image/png;base64,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" alt="">
The gain in kinetic energy as the bug slides is provided by the loss in
potential energy. Since
<img src="data:image/png;base64,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" alt="">
we have
<img src="data:image/png;base64,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" alt="">
At the moment the ball leaves the ball, the normal force Fn becomes zero. We can now substitute the value of the centrifugal force
<img src="data:image/png;base64,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" alt="">

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