A 17.00 kg lead sphere is hanging from a hook by a thin wire 3.20 m long, and is

Question

A 17.00 kg lead sphere is hanging from a hook by a thin wire 3.20 m long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a 2.50 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?

I know I am supposed to use conservation of momentum for the collision and I know that velocity is a minimum when the radius of the circle is small or the period is large. But I don't know how to solve for those.

Explanation / Answer

Mass of the lead sphere, m1 = 17 kg Mass of the steel dart, m2 = 2.5 kg Length of the wire, r = 3.2 m Let Initial velocity of the steel dart , v = ? Minimum velocity of the combination to complete circular loop:  v' = (5 r g )     = ( 5 * 3.2 * 9.8 )     = 12.52 m/s From the law of conservation of momentum, m2 *  v = ( m1 + m2 ) * v' 2.5 * v = ( 17 + 2.5 ) * 12.52 So, v = 97.7 m/s                                                                    

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