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).

**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 *F _{x}(x)* of the force

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)=-ax ^{2}+bx^{4},* where

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?