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# Ball rolling down an incline, then through a loop to loop; find speed to complete the loop.

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A solid steel ball of radius r rolls down an incline into a loop-the-loop of radius R. What is the minimum speed the ball must have at the top of the loop in order to stay on the track? At what vertical height on the incline, in terms of the radius of the loop, must the ball be released in order for it to have the required minimum speed at the top of the loop, neglecting any frictional losses?

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With good explanations and calculations, the problem is solved in this solution.

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I would recommend drawing a free-body diagram of the ball at the top of the loop. Assume the ball stays in contact with the track, and let's look at the forces involved:

1) Force of gravity (mg) downwards
2) Normal force of contact from the track (F_n) downwards

Assuming there's no friction, there aren't any forces left. If we now use Newton's laws and equate F_net to ma,

we get F_n + mg = ma

Now if we want to find the ...

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