A ball rolls off a frictionless sphere with neglible inital velocity, at what point does the ball fall off?
1
Expert's answer
2012-02-02T11:03:52-0500
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="">
Numbers and figures are an essential part of our world, necessary for almost everything we do every day. As important…
APPROVED BY CLIENTS
"assignmentexpert.com" is professional group of people in Math subjects! They did assignments in very high level of mathematical modelling in the best quality. Thanks a lot
Comments
Leave a comment