A pendulum, comprising a light string of length L and a small sphere, swings in

Question

A pendulum, comprising a light string of length L and a small sphere, swings in a vertical plane. The string hits a peg located a distance d below the point of suspension.

(a) Show that if the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. (Do this on paper. Your instructor may ask you to turn in this work.)

(b) Show that if the pendulum is released from the horizontal position (? = 90

Explanation / Answer

A) The maximum heigh reached by the pendulum occurs when its velocity is zero, i.e., its energy is only potential. The initial state has only potential energy too. Then both height must be equal. The only condition is that there must exist a final state with null velocity, and this is true because the position where the potential energy is equal to the initial one is below the peg. B) The initial energy is (potential) m g L (if we put the zero potencial located at the rest point of the pendulum). The contition for the pendulum to swing around the peg is that the tension of the string is always positive, then the centripetal force must be greater of equal to the weight. Then m vmin²/(L-d) = m g On the other hand, the minimum kinetic energy is m g L - 2 m g (L-d) = m vmin²/2 then m g L - 2 m g (L-d) = m g (L-d)/2 => L - 2 L + 2 d = L/2 - d/2 => (5/2) d = (3/2) L => d = (3/5) L

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