A 29.00-kg ball (treat it as a point mass) is hanging on end of a massless rope

Question

A 29.00-kg ball (treat it as a point mass) is hanging on end of a massless rope that is 2.8 m long and is free to swing in a complete vertical circle. Suddenly it is struck horizontally by a 5 kg putty (also a point mass) that sticks to the ball. What is the minimum speed with which the putty must strike the ball so that the combination (of ball, string and putty) is just barely able to make a complete circular loop? (Note that the rope is very strong (so it won't snap/break) but it is after all a rope and can become slack). Express your answer with the appropriate units.

Explanation / Answer

for the complete circle

velocity needed to complete circle

m * v^2/r = m * g

v^2 = r * g

Now, for the velocity needed at the bottom is u

Using conservation of energy

0.50 * m * u^2 = 0.50 * m * v^2 + m * g * (2r)

0.50 * u^2 = 0.50 * r * g +g * (2r)

u^2 = 5 * r * g

u = sqrt(5 * 2.8 * 9.8)

u = 11.71 m/s

Now, for the minimum speed is ui

Using conservation of momentum

11.71 * (29 + 5) = 5 * ui

ui = 79.7 m/s

the minimum speed with which putty strike is 79.7 m/s

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