A ring is attached at the center of the underside of a trampoline. A sneaky teenager crawls under the trampoline and uses the ring to pull the trampoline slowly down while his 61-kg mother is sleeping on it. When he releases the trampoline, she is launched upward. As she passes through the position at which she was before her son stretched the trampoline, her speed is 2.0 m/s How much elastic potential energy did the son add to the trampoline by pulling it down? Assume the interaction is nondissipative.

122 J

Explanation:

The motion is a simple harmonic motion. At any point during the motion, the sum of the kinetic energy and the elastic potential energy is constant. At the equilibrium position (the point from which the mother was before her stretched the trampoline), the velocity is at a maximum; hence, kinetic energy is maximum. Potential energy is 0 because the displacement from the equilibrium point is 0. The energy at the equilibrium is, therefore, wholly, kinetic. Since this energy must be the same at any point, it follows that this is the initial elastic potential energy added to the trampoline by the son. Now this kinetic energy is given by

where is the mass and is the velocity.

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