Physics 105 Problem Set 8
Due: Wednesday, November 26, 2007, by 10 am in 208 Jadwin.

This week and next we will be studying oscillations (reading from K&K, backed up by Knight) and, briefly, waves (reading from Knight).

Required reading for this week (some of these assigned last week):

-- K&K's treatment of simple harmonic motion is actually dispersed throughout the book. To follow the thread, look at Examples 2.17, 2.18, and 4.2 (for a different way to solve the SHO differential equation -- we don't recommend doing it this way!), Section 4.10, including Examples 4.15 and 4.16, and Example 9.3. For a note on viscous forces, which we will use in damped oscillations, see pp. 95-7.
-- K&K 6.6
-- K&K Chapter 10, including Note 10.1 (work through the examples on pp. 434-5 to work on your complex arithmetic) and Note 10.2 (very important).

Optional

-- Knight Chapter 14.
-- K&K Note 6.2
-- It is a very good idea to warm up by doing problems in 103 Learning Guide 9 after you've done the required reading. Be sure to take a look at the notes at the end of LG 9.

Problems to turn in.

Problem 8.1 a) Dig a narrow tunnel straight through the center of the Earth. What is the force F(x) as a function of distance x from the center of the Earth? Recall that in gravity, only the mass interior to your position contributes to the force.

b) Now dig a straight tunnel from arbitrary point A on the surface of the Earth to any other arbitrary point B on the surface, not exactly opposite A -- choose a point B which is 1000 km (by land) away from A. What is the component Fx(x) of the force along the tunnel as a function of position x from the center of the tunnel?

c) What is the period of the oscillation in each tunnel?

Problem 8.2 K&K 6.17. Note: this problem is a bit ambiguous. You SHOULD ASSUME that the equilbrium position is the one shown in the figure, namely, the stick pointing straight up. The problem is also quite coy about the relaxed lengths of the springs. It turns out that this does not matter, in the sense that any relaxed lengths CONSISTENT WITH THE EQUILIBRIUM POSITION BEING STRAIGHT UP will give the same answer. You MAY ASSUME the relaxed lengths of the springs are exactly that needed to reach the center, that is, both springs are relaxed in the figure. However, you should take a look at the case where you have to stretch (or compress) the springs to attach them to the stick.

And yes, there is gravity in this problem!

Problem 8.3 A great problem from the physics GRE, of all places. A particle of mass m moves in one dimension with a potential energy U(x)=-ax2+bx4, where a and b are positive constants. What is the angular frequency of small oscillations about the equilibrium points of this system?

The reason this problem is so cool is that, in the Standard Model of high energy physics, this potential is used to represent the energy in the so called "higgs field" that gives mass to all elementary particles. In that sense, the form of this equation gives rise to all existence. Whoa! (And the "small oscillations", when quantized, are the higgs particles that we are building huge accelerators to look for.)

Problem 8.4 A system whose natural frequency in the absence of damping is 4 rad/s is subject to a damping force such that b/m=10 s-1.

a) Show that the system is over-damped and find the general solution for the displacement.
b) If the mass is initially at x= +0.5 m, sketch the displacement as a function of time for initial velocities of -2 m/s and -6 m/s (both toward the origin.)

Problem 8.5 A seismograph can be modeled as a mass on a spring with a fluid damper. A (bad) seismograph is designed to be underdamped with a natural oscillation period of 2 s, and, when set swinging, its amplitude decays with an exponential time constant of 20s, that is, its amplitude of oscillation is proportional to e-t/20.

(a) If an earthquake provides a sinusoidal driving acceleration with a period of 0.1 s and an amplitude of 1 cm s-2, what is the amplitude of the steady-state response of the seismograph mass? Use complex numbers to get your answer.
(b) How far off (in percentage) would your result for the amplitude have been if you had used the approximation that the driving frequency is much much higher than the natural frequency?