A ball of mass M is attached to a string of length R and negligible mass. The ba

Question

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 above. when the ball is at point p, the string is horizontal. Point Q is at bottom of 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. On the figure, draw and label all the forces exerted on the ball when it is at points. P and Q. respectively. Derive an expression for v_min, the minimum speed the ball can have at point Z without leaving the circular path. The maximum tension the string can have without breaking is T_max. Derive an expression for V_max, the maximum speed the ball can have at point Q without breaking the string. 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.

Explanation / Answer

a) At point P, mg acts downward and tension t acts inward in +x- direction

At point Q, mg acts downward and tension t acts inward in +y- direction

b) Fc = t-mg   but the tension is zero ===> Fc = mg .............. (1)

F = ma =mac , Fc = mv^2/r ....................... (2)

Equating (1) and (2), v = sqrt(rg)

c) The Tension in the string is the upward force and the weight is the downward force, the sum of these two forces is ma so:

ma = Tmax-mg   

But the acceleration acting on the system is the centripetal acceleration.

mv^2/r = Tmax-mg

v = sqrt [ (R/M)*(Tmax -mg)]

d) The ball would go straight up, slowing down as it is accelerated downward by gravity, it will stop, then fall down, accelerating as it goes. Its motion is tangential to the path, which is straight up.

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