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EXAM II, PHYSICS 4304, November 28, 2007
Dr. Charles W. Myles
INSTRUCTIONS: Please read ALL of these before doing anything else!!!
PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!
PLEASE don’t write on the exam sheets, there won’t be room! If you don’t have paper, I’ll give you
some.
PLEASE show ALL work, writing down at least the essential steps in the problem solution. Partial
credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to
follow order will receive more credit than disorganized work.
The setup (PHYSICS) of a problem counts more heavily than the detailed mathematics of working it
out.
PLEASE write neatly. Before handing in your solutions, PLEASE: a) number the pages and put them
in numerical order, b) put the problem solutions in numerical order, and c) clearly mark your final
answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves.
NOTE: I HAVE 10 EXAMS TO GRADE!!! PLEASE HELP ME GRADE THEM
EFFICIENTLY BY FOLLOWING THESE SIMPLE INSTRUCTIONS!!! FAILURE
TO FOLLOW THEM MAY RESULT IN A LOWER GRADE!! THANK YOU!!
Note: Work any four (4) of the 5 problems. Each is equally weighted & worth 25 points for 100 points
total!
HINT on Problems 1. & 2! Recall that, for situations of high symmetry, field calculations are often much easier if
Gauss’s Law is used!
1. The following questions are about a sphere of radius R with a non-uniform volume mass density, which
depends on the distance r from the sphere’s center as ρ = ρ0(r4R-4), where ρ0 is a constant. Please read the
HINT above before starting this! Calculate the following:
a. The gravitational field g outside the sphere & a distance r  R from the center. (6 points)
b. The gravitational field g inside the sphere and a distance r  R from the center. (6 points)
c. The gravitational potential at all points in space. Take the zero of potential to be at r = . (8 points)
A point mass m is placed a distance r  R away from the center of the sphere. Calculate:
d. The gravitational force between m & the sphere. (3 points)
e. The gravitational potential energy for the point mass in part d. (2 points)
The following about a sphere of radius r might be useful: Surface area: A = 4πr2. Volume: V = (4πr3). This
integral might be useful: ∫rn dr = (rn+1)/(n+1); n is any power (n  -1). Don’t forget the lower limit!!!
NOTE!!!! Work any four (4) of the five problems.
The following questions are about a solid, infinitely long right circular cylinder of radius R which has a
uniform volume mass density ρ. Please read the HINT before Problem 1, before starting this! Calculate the
following:
a. The gravitational field g outside the cylinder & a distance r  R from the cylinder axis. (7 points)
b. The gravitational field g inside the cylinder & a distance r  R from the cylinder axis. (8 points)
c. The gravitational potential at all points in space. Take the zero of potential at r = R. (10 points)
The following about a cylinder of radius r & length ℓ might be useful: Lateral surface area: A = 2πrℓ. Volume: V
= πr2ℓ. This integral might be useful: ∫rn dr = (rn+1)/(n+1); n is any power (n  -1). Don’t forget the lower limit!!!
2.
NOTE!!!! Work any four (4) of the five problems.
3.
NOTE: These are questions about the (fictitious!) Klingon Home Planet! (Called planet K in what
follows). Astronauts have visited K & have found that the gravitational force F there does not obey
Newton’s Universal Gravitation Law! They found that F on a point mass m varies as the INVERSE
FOURTH POWER of the distance r from the center of K as F = -mCr-4. Data needed to obtain the
required numerical results: Force constant: C = 3.0  1028 Nm4/kg. Assume that K is a uniform sphere of
radius RK = 5.0  106 m.
a.
b.
c.
d.
PROBLEM 3 IS CONTINUED ON THE NEXT PAGE!
PROBLEM 3 CONTINUED
A particle of mass m is shot from K’s surface at a speed v0 = 7.0  103 m/s. Calculate the
maximum height which it will reach before beginning to fall back to the surface. A numerical
result is wanted & needed! (4 points)
A particle of mass m is dropped from rest at a height h = 8.0  105 m above K’s surface. Calculate
the speed with which it will hit the surface. A numerical result is wanted & needed! (3 points)
Calculate the escape velocity for a particle of mass m on the surface of K. (That is, find the minimum
velocity needed to “escape” from K’s gravitational field). A numerical result is wanted & needed! (4
points)
A spherical satellite of uniform density & mass m = 2,500 kg is in a circular orbit of radius
r = 2.5  107 m about K. Calculate the gravitational force between the satellite & K, the speed of
the
satellite in orbit, & the period of the satellite’s orbit. Calculate the effective gravitational
acceleration (in m/s2) experienced by the satellite. Numerical results are wanted & needed! (4
points)
e. A particle of mass m is dropped from rest from a large height h above K’s surface. By using
conservation of total mechanical energy, derive an expression for the time t it takes to fall to the
Earth’s surface. This will be in the form of a messy integral. Leave it in integral form! (10 points)
NOTE!!!! Work any four (4) of the five problems.
NOTE: SOME PORTIONS OF PROBLEMS 4 & 5 CAN BE DONE WITHOUT THE USE OF LAGRANGE’S
METHOD. HOWEVER, SINCE THIS IS AN EXAM WHICH IS PARTIALLY OVER THIS
METHOD, NO CREDIT WILL BE GIVEN FOR SOLUTIONS WHICH DON’T USE THIS
METHOD!!!
4. See figure. A point mass m is attached to one end of a massless, inextensible rod of
length b. The other end of the rod is attached to a frictionless hinge on a horizontal
ceiling. The hinge allows movement in both the plane of the figure (characterized by
angle θ in the figure) & in a plane perpendicular to both the vertical direction & the
plane of the figure (characterized by angle φ in the figure). The only force acting is
gravity. This is called a spherical pendulum.
a. Write expressions for the potential energy, the kinetic energy, & the Lagrangian
for
this system. How many degrees of freedom are there? (6 points)
b. Are there constraints? If so, what are they? If so, write the equations of constraint. If so, what is the
physical significance of the constraint forces? (6 points)
c. Use Lagrange’s equations to derive the equations of motion. (9 points)
d. What are the constants of the motion? That is, what physical quantities are conserved? (4
points)
e. 5 POINT BONUS! Assume a small amplitude of oscillation for θ. Write the equations of motion
under that assumption. Go as far as you can solving these equations. Solving them in this case will
obtain the oscillation frequency of this pendulum.
NOTE!!!! Work any four (4) of the five problems.
5. See figure. A point mass m slides without friction down a wedge of mass M
which is on a frictionless horizontal surface. The surface of the wedge on which m
is sliding is shaped like a quadrant of a circle of radius R. At the same time, the
wedge slides without friction along the horizontal surface at constant velocity v.
R
a. Write expressions for the potential energy, the kinetic energy, & the Lagrangian for this system.
How many degrees of freedom are there? (6 points)
b. Are there constraints? If so, what are they? If so, write the equations of constraint. If so, what is the
physical significance of the constraint forces? (6 points)
c. Use Lagrange’s equations along with the Lagrange multiplier method to derive the equations of
motion. (9 points)
d. What are the constants of the motion? That is, what physical quantities are conserved? (4
points)