1.

In the figure, a solid cylinder attached to a horizontal spring (k = 7.00 N/m) rolls without slipping along a horizontal surface. If the system is released from rest when the spring is stretched by 0.390 m, find (a) the translational kinetic energy and (b) the rotational kinetic energy of the cylinder as it passes through the equilibrium position. (c) If the cylinder has mass M = 7.30 kg, find the period of the simple harmonic motion that the cylinder's center of mass executes. (Hint: Find the time derivative of the total mechanical energy.) M kA damped harmonic oscillator consists of a block (m = 2.07 kg), a spring (k = 10.8 N/m), and a damping force (F = -bv). Initially, it oscillates with an amplitude of 27.6 cm; because of the damping, the amplitude falls to 0.639 of the initial value at the completion of 3 oscillations. (a) What is the value of b? (Hint: Assume that b2