3-4 In Fig. 15-60, a solid cylinder attached to a horizontal spring (k = 3.00 N/m) rolls without slipping along a horizontal surface. If the system is released from rest when the spring is stretched by 0.250 m, find (a) the translational kinetic energy and (b) the rotational kinetic energy of the cylinder as it passes through the equilibrium position. (c) Show that under these conditions the cylinder’s center of mass executes simple harmonic motion with period T = 2π(3M/2k)1/2
where M is the cylinder mass. (Hint: Find the time derivative of the total mechanical energy.) (HR 15-106)
Sol: (a) The potential energy at the turning point is equal (in the absence of friction) to the total kinetic energy (translational plus rotational) as it passes through the equilibrium position:
which leads to  = 0.125 J. The translational kinetic energy is therefore  .
(b) And the rotational kinetic energy is  .
(c) In this part, we use vcm to denote the speed at any instant (and not just the maximum speed as we had done in the previous parts). Since the energy is constant, then
which leads to
Comparing with Eq. 15-8, we see that  for this system. Since = 2/T, we obtain the desired result:  .
4-1 A uniform rope of mass m and length L hangs from a ceiling. (a) Show that the speed of a transverse wave on the rope is a function of y, the distance from the lower end, and is given by v = (gy)1/2. (b) Show that the time a transverse wave takes to travel the length of the rope is given by t = 2(L /g)1/2. (HR 16-25)
Sol: (a) The wave speed at any point on the rope is given by v =  , where is the tension at that point and is the linear mass density. Because the rope is hanging the tension varies from point to point. Consider a point on the rope a distance y from the bottom end. The forces acting on it are the weight of the rope below it, pulling down, and the tension, pulling up. Since the rope is in equilibrium, these forces balance. The weight of the rope below is given by gy, so the tension is = gy. The wave speed is  
(b) The time dt for the wave to move past a length dy, a distance y from the bottom end, is  and the total time for the wave to move the entire length of the rope is
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