Physics 321 -- Spring 2004
Homework #4, due at
beginning of class Wednesday Feb. 11.
1. [3pts] Marion & Thornton, problem 9-40.
Hint: Read pg. 364 carefully. The component of velocity parallel to the surface is unchanged by
the collision. The perpendicular
component is not only reversed, but is reduced according to the coefficient of
restitution.
2. [4pts] Consider a one-dimensional (head-on) elastic collision
between a particle of mass m1 traveling initially with velocity u1,
incident on a particle of mass m2 initially at rest.
(a) Analyze this problem using the center of mass frame of
reference. First, find the initial
velocities in the CM frame, u1’ and u2’. Second, find the final velocities in the CM
frame, v1’ and v2’.
Third, transform those final velocities back to the lab frame to get v1
and v2.
(b) Check that your answers obey conservation of momentum and kinetic energy in the
lab frame. (You could have done the
whole problem in the lab frame to begin with, but the algebra is easier using
the CM frame, and the physics is clearer.)
3. [4pts] Marion & Thornton, problem 9-34.
Hint: There are solutions both for
alpha > 0 and for alpha < 0.
4. [5pts] Marion & Thornton, problem 9-41.
Hint:
After you have written down the two components of conservation of momentum and the
relation between initial and final kinetic energy, you have some serious algebra
to do. A good approach is to begin by eliminating the unknown
angle by using cos2z + sin2z = 1.
This leaves only two equations; you can solve
them and then plug back into one of the original equations to find the desired angle.
(You may prefer to use Mathematica or one of its competitors in place of hand tools
for this.)
5. [4pts] Marion & Thornton, problem 9-42.
(Assume the diameter of the initial coil is negligibly small,
so the rope is vertical and angular momentum can be ignored.)
6. [4pts] Marion & Thornton, problem 9-19.
Hint: You can use Newton’s 2nd Law for the center of mass motion.
The motion itself is simple: assume that the links of the chain fall freely
when they are above the table, and that they stop without bouncing when they
reach it.
7. [6pts] In class, we solved for the motion of a chain that
was initially hanging over the edge of a frictionless horizontal surface.
Generalize that problem by assuming that the surface slopes downward toward
the edge at an angle theta, and has a coefficient of friction mu.
As in the original problem, assume that there is a frictionless barrier in
place to keep the part of the chain that is off the table vertical.
(a) Apply Newton's 2nd Law to the vertical
part of the chain, to relate the tension at the corner to the acceleration.
(b) Apply Newton's 2nd Law to the
part of the chain that is still on the surface, to relate the tension at the
corner to the acceleration.
(c) Equate your results from parts (a) and (b) to find
the equation of motion of the chain.
(d) Calculate the work done by friction during the time
that the length of chain hanging over increases from x0 to x.
(e) Calculate the gravitational potential energy as a
function of x.
(f) Use your results from parts (d) and (e) to write a
conservation-of-energy equation.
(g) Take the time derivative of your result from part (f)
to obtain the equation of motion. If it doesn't agree with your result
from part (c), go back and find the mistake!
(h) Solve the equation of motion, assuming the chain starts
from rest with a length x0 hanging down.
(i) Check that your answer to (h) is correct in the limit
that the angled surface becomes vertical.