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in Physics by (70.7k points)

A ball of mass M is attached to a string of length R and negligible mass. The ball moves clockwise in a vertical circle, as shown. When the ball is at point P, the string is horizontal. Point Q is at the bottom of the circle and point Z is at the top of the circle. Air resistance is negligible. Express all algebraic answers in terms of the given quantities and fundamental constants.

a. On the figures below, draw and label all the forces exerted on the ball when it is at points P and Q, respectively.

b. Derive an expression for vmin the minimum speed the ball can have at point Z without leaving the circular path.

c. The maximum tension the string can have without breaking is Tmax Derive an expression for vmax, the maximum speed the ball can have at point Q without breaking the string.

d. Suppose that the string breaks at the instant the ball is at point P. Describe the motion of the ball immediately after the string breaks.

1 Answer

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b. The minimum speed occurs when gravity alone supplies the necessary centripetal force at the top of the circle (i.e. tension is zero and is not required). Therefore we have Mg = MVmim2/R which gives vmin = Rg.

c. At the bottom of the swing ΣF = ma becomes T – Mg = Mv2/R which gives Tmax - Mg = MVmax2/R and solving for vmax gives vmax = √(R/M(Tmax - Mg))

d. At point P the ball is moving straight up. If the string breaks at that point, the ball would continue to move straight up, slowing down until it reaches a maximum height and fall straight back to the ground.

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