A 20.00-kg lead sphere is hanging from a hook by a thin wire 2.80 m long and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00-kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

Explanation:

It is known that motion of objects on a vertical is an example of non-uniform motion.

When an object is at the highest point of the circle then for crossing highest point the centripetal force balances the weight of the object.

Therefore,    

At the highest point of the circle, the minimum speed is as follows.

                 v =

                    =

                    = 5.23 m/s

As the sphere falls from highest to lowest point of circle as follows. According to the law of conservation of energy,

Expression for potential energy is as follows.

where,    h = diameter of the circle (2r)

So,              

where,    u = velocity at the lowest time

Hence, the above equation will be as follows.

               u =

                  =

                  = 11.71 m/s

The dart is embedded into bullet and the collision is inelastic. From the law of conservation of momentum,

                v =

                   =

                   =

                   = 58.55 m/s

Thus, we can conclude that the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision is 58.55 m/s.

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