A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is

Question

A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance x from one end of the rod, along the axis of the rod.

Part B
Use Fx = -dU/dx to find the magnitude of the gravitational force exerted on the sphere by the rod.
Express your answer in terms of the variables m, M, L, x and appropriate constants.

Part C
Find the direction of the gravitational force exerted on the sphere by the rod.

1. the force is directed to the rod

2. the force is directed out of the rod

Explanation / Answer

Call the height of the pivot L, so that the rod can just swing down to height = 0. Horizontal: kinetic energy = 0 potential energy = MgL Vertical: kinetic energy = 1/2 I?^2 potential energy = MgL/2 I = ?r^2 dm dm = M/L dr I = M/L ?r^2 dr from r = 0 to L I = 1/3 ML^2 gain in kinetic energy = loss of potential energy 1/2 * 1/3 ML^2 * ?^2 = MgL/2 ? = v(3g/L) note that the speed of a pendulum does not depend on mass, just length. v_t = ?r = v(3g/L)*L = v(3gL)

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