1. A physical pendulum consists of a uniform rod of mass M and length L. The pendulum is pivoted at a point that is a distance x from the center of the rod, so the period for oscillation of the pendulum depends on x: T(x).

(a) What value of x gives the maximum value for T? (Use any variable or symbol stated above as necessary.)

(b) What value of x gives the minimum value for T? (Use any variable or symbol stated above as necessary.)

2. A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M and length L that is pivoted freely about one end, with a solid sphere of the same mass, M, and a radius of L/2 centered about the free end of the rod?

(a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of M and L. (Use any variable or symbol stated above along with the following as necessary: g.)

(b) Obtain an expression for the period of the pendulum for small oscillations. (Use any variable or symbol stated above along with the following as necessary: g.)

(c) Determine the length L that gives a period of T = 3.1 s.

3. A 3-kg mass attached to a spring with k = 157 N/m is oscillating in a vat of oil, which damps the oscillations.
(a) If the damping constant of the oil is b = 14 kg/s, how long will it take the amplitude of the oscillations to decrease to 1% of its original value?

(b) What should the damping constant be to reduce the amplitude of the oscillations by 97% in 3 s?

4. Consider a guitar string stretching 76.0 cm between its anchored ends. The string is tuned to play middle C, with a frequency of 256 Hz, when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced 2.08 mm at its midpoint and released to produce this note, what are the wave speed, v, and the maximum speed, Vmax, of the midpoint of the string?