Physics 311
Fall 2008
Final Exam
1. Analyze the motion of a particle which is like a damped harmonic
oscillator except that the “damping” force is along, rather than opposite to,
the velocity. That is, instead of a resistance force –bv there is a force +bv.
Set up and solve the equation of motion and consider the possible range of
solutions. Show that there are two qualitatively different types of motion,
one oscillatory and one not. Sketch graphs of x(t) for each case. (There are
situations such as this in acoustics and electromagnetic theory where
external interactions may be other than resistive).
2. Suppose a particle moving in one dimension is subject to the force
F(x)=−kx +kx
3
α
2
where k and α are constants and k is positive (this is called an anharmonic
oscillator and is important in some molecular problems). (a) Determine the
potential energy for this particle, sketch its graph, and discuss all possible
motions for all possible values of the energy. If there are any stable
equilibria, determine where they are and the frequency of small oscillations
around them. (b) Determine the position x(t) for the case when E = ¼ kα2.
You may choose the initial conditions in any convenient way. Show that your
result agrees with the qualitative discussion in part (a).
3. In Physics 151, you worked a number of collision problems in one and
two dimensions. You never did a three dimensional problem. It is almost
always simplest to treat 3-D collisions in center of mass coordinates (that is,
you set the center of mass to be the origin of your coordinate system).
Consider two particles of mass m1 and m2 and velocities
v
1
and
v
2
measured in laboratory coordinates. (a) Determine the velocity of the
center of mass of the system in lab coordinates as well as the velocities of
m1 and m2 in the center of mass coordinate system. Show that the total
momentum of the system in center of mass coordinates is always zero. Does
this make sense physically?
Explain. (b) Suppose the
collision is elastic. Show that
the magnitudes of the
velocities in the center of
mass coordinates do not
change as a result of the
collision. This means that
elastic collisions can result
