Physics 111 HW27
assigned9 May 2011
PP03. To make a pendulum, you hang a uniform solid sphere of radius R on the end of a light string. The distance from the pivot to the center of the sphere is L.
a)Prove that the period of your pendulum is T = Tsp (1 + 2R2/5L2)1/2, where Tsp is the period of a simple pendulum of length L.
b)At what value of L is T only 0.10% larger than Tsp? (Your answer will be in terms of R.)
c)For a sphere of diameter 2.540 cm, what is the value of L in part (b)?
SHM01. A mass m is mounted between two springs of constants k1 and k2 as shown in the figure. When the mass is in the equilibrium position for one spring, it is also in the equilibrium position of the other. Determine the frequency (in Hz) of oscillations of this mass in terms of k1, k2, and m.
SHM02. (toughie!) A mass m is connected to two (“massless”) springs in this fashion:
Determine the frequency of oscillation in this case in terms of k1, k2, and m. (HINT: Displace the mass m a distance x to the right. Spring k1 will stretch a distance x1, and spring k2 will stretch a distance x2, where x = x1 + x2. If the springs are massless, the spring force on m must be equal to the tension in any point of the spring combination. Thus, the net spring force on m is the same as k1x1 and k2x2.)
SHM03. Two solid cylinders connected along their common axis by a short, light rod (hidden in diagram below) each have radius R and mass M. They are at rest on a horizontal tabletop. A spring with force constant k has one end attached to a clamp ant the other end attached to a frictionless ring at the center of mass of the cylinders/rod arrangement. The cylinders are pulled to the left a distance x, which stretches the spring, and released. Assume that there is sufficient friction between the cylinders and the tabletop so that there is no slipping. Determine the frequency of oscillation in terms of R, M, and k.
SHM04. Let’s pretend we’ve drilled a hole through the Earth, from one surface through the core to the surface on the opposite side. Let’s also pretend that the Earth does not rotate. The gravitational force on a mass m somewhere in this hole is roughly equal to mgr/RE, where g is our old friend (9.8 m/s2), r is the distance the mass is from the center of the Earth, and RE is the radius of the Earth. Assuming that air resistance is negligible, find how long the mass would take to oscillate one cycle about the center of the Earth. You should get a numerical answer.
SHM05. An 2cm by 2cm by 2cm ice cube has a mass of 7.36 g. It is placed in a glass of water. When it just floats there, the weight of it is exactly balanced by the buoyancy force. If you push it down a distance x, though, it bobs up and down with simple harmonic motion. The net force on it in this situation is - A x, where is the density of water (1000 kg/m3), A is the cross-sectional area of the cube, and x is how far the cube is above or below the equilibrium position. Determine the frequency of oscillation of the cube.