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EXAM II, PHYSICS 4304
November 30, 2005
Dr. Charles W. Myles
INSTRUCTIONS: Please read ALL of these before doing anything else!!!
1. PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!
2. PLEASE don’t write on the exam sheets, there won’t be room! If you don’t have paper, I’ll give you
some.
3. 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.
4. The setup (PHYSICS) of a problem counts more heavily than the detailed mathematics of working it
out.
5. PLEASE write neatly. Before handing in your solutions:
PLEASE: a) Number the pages & put them in numerical order. b) Put the problem solutions in
numerical order. 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! & d) Staple the pages together.
NOTE: I HAVE 20 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 four (4) of the six problems. Each is equally weighted & worth 25 points for a total
of 100 points.
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 solid, infinitely long right circular cylinder of radius R which has
a non-uniform volume mass density, which depends on the distance r from the cylinder’s axis as
ρ = ρ0(r/R), where ρ0 is a constant density. Please read the HINT, just before this problem, before
starting it! Calculate the following:
a. The gravitational field g outside the cylinder & a distance r  R from the axis. (7 points)
b. The gravitational field g inside the cylinder & a distance r  R from the 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  might be useful: Lateral surface area: A =
2πr.Volume: V = πr2. This integral might be useful: ∫rn dr = (rn+1)/(n+1); n is any power (n  -1).
Don’t forget the lower limit!!!
2. The following questions are about a sphere of radius R which has uniform volume mass density.
Please
read the HINT, just before Problem 1, before starting this problem! In parts a, b, & c, calculate:
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 six problems.
3. NOTE: Parts a., b., c. & d. are (obviously!) independent parts! Work each with Newton’s
Universal Gravitation Law (or results obtained from it in Ch. 5) & not by assuming a constant
gravitational acceleration! All parts concern a fictional planet X. Assume that X is a sphere
with uniform mass density. Data to obtain the required numerical results: Gravitation constant
G = 6.67  10–11 Nm2/kg2, Planet Mass MX = 4.0  1024 kg, Planet Radius RX = 4.5  106 m.
a. A particle mass m is shot from X’s surface at a speed of v0 = 8.0  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)
b. A particle of mass m is dropped from rest at a height h = 7.0  105 m above X’s surface. Calculate
the speed with which it will hit the surface. A numerical result is wanted & needed! (3 points)
c. Calculate the escape velocity for a particle of mass m on the surface of X. (That is, find the minimum
velocity needed to “escape” from X’s gravitational field). A numerical result is wanted & needed! (4
points)
d. A spherical satellite of uniform density & mass m = 2,000 kg is in a circular orbit of radius
r = 2.0  107 m about X. Compute the gravitational force between the satellite & X, the speed of
the
satellite in orbit, & the period of the satellite’s orbit. Compute 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 X’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: SOME PORTIONS OF PROBLEMS 4,5 & 6 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
TO THESE PROBLEMS WHICH DON’T USE THIS METHOD!!!
4. See figure. Two masses, m1 & m2 (m1 > m2) are connected by a
massless, inextensible string of length , which is put over a
massless, frictionless pulley at the top of a wedge, as shown.
The
masses are allowed to slide under the influence of gravity on the
two frictionless inclined planes. Use the generalized coordinates
suggested in the figure to solve this problem. Hint: Obviously
the string length,  = 1 + 2, where 1 & 2 are shown in the figure.
a. Write expressions for the kinetic energy, the potential energy, and the Lagrangian for this
system. How many degrees of freedom are there? (5 points)
b. What is the constraint? Write the equation of constraint. What is the physical
significance of the constraint force? (3 points)
c. Use Lagrange’s equations with the method of Lagrange multipliers to derive the
equations of motion. (5 points)
d. What are the constants of the motion? That is, which quantities are conserved? (4 points)
e. Using the results of part c, along with the equation of constraint, go as far as you can
towards finding the force of constraint and solving the equations of motion. (4 points)
f. Under what conditions will this system be in equilibrium? Find the equilibrium position. (4 points)
NOTE!!!! Work any four (4) of the six problems.
NOTE: SOME PORTIONS OF PROBLEMS 4,5 & 6 CAN BE DONE WITHOUT THE USE
OF LAGRANGE'S METHOD. HOWEVER, SINCE THIS IS AN EXAM WHICH IS
PARTIALLY OVER THESE METHODS, NO CREDIT WILL BE GIVEN FOR SOLUTIONS
TO THESE PROBLEMS WHICH DON’T USE THIS METHOD!!!
5. See figure. A particle of mass m is attached to a rigid ceiling by an ideal
spring of spring constant κ. At equilibrium, this combination hangs vertically
downward. To this mass-spring combination is attached a 2nd identical massspring oscillator. The spring of the 2nd oscillator is connected to the mass of
the 1st oscillator. Consider vertical motiononly. Use the generalized
coordinates suggested in the figure to solve this problem.
a. Write expressions for the kinetic energy, the potential energy, & the
Lagrangian for this system. Hint: These oscillators are obviously coupled,
so the potential energy must have a coupling term (that is, cross terms in x1 &
x2!) in it! How many degrees of freedom are there? (7 points)
b. Use Lagrange’s equations to find the equations of motion for this system. (7 points)
c. What are the constants of the motion? That is, what physical quantities are conserved? (5
points)
d. Using the results of part b, go as far as you can towards solving the equations of motion.
Hint: The following might help in their solution: Define a variable u = x2 – x1 & get its
equation of motion. This should be a familiar equation for which you should know the solution!
Solve it with the initial conditions that at time t = 0, u = u0 (a constant) & (du/dt) = 0. (6 points)
6. See figure. A large mass, M, is constrained to move on a smooth,
horizontal frictionless track. A small mass, m, is connected to M by a
massless, inextensible string of length . Use the generalized coordinates
suggested in the figure to solve this problem.
a. Write expressions for the kinetic energy, the potential energy, & the
Lagrangian for this system. How many degrees of freedom are there?
(7 points)
b. Use Lagrange’s equations to find the equations of motion for this
system. (7 points)
c. What are the constants of the motion? That is, what physical quantities are conserved? (5 points)
d. Starting with the results of part b, assume that the angle θ is small enough that, in the
equations of motion you can make the replacements (θ in radians): sinθ ~ θ & cosθ ~ 1.
Using this approximation, solve the equations of motion. Hint: In this approximation, the
θ equation should be a familiar one for which you should know the solution! Solve it
with the initial conditions that at time t = 0, θ = θ0 (a constant) & (dθ/dt) = 0. (6 points)