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EXAM II, PHYSICS 4304
April 20, 2004
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 problem setup (PHYSICS) will count more heavily than the detailed mathematics of working it
out.
5. 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 20 EXAMS TO GRADE!!! PLEASE HELP ME GRADE
THEM EFFICIENTLY BY FOLLOWING THE ABOVE SIMPLE
INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY RESULT IN A
LOWER GRADE!!
THANK YOU!!
NOTE!!!! Work any four (4) of the six problems. Each problem is equally weighted and
worth 25 points for a total of 100 points on this exam.
1. A mass m, confined to the x-axis, experiences a conservative force derivable from a
potential energy function given by U(x) = (½)kx2 – (⅓)λx3 (k & λ are positive constants).
a. Find the zeros and the equilibrium points of U(x). Are the equilibrium points positions of
stable or unstable equilibrium? Use these results to sketch U(x). (5 points)
b. Make a sketch of the phase diagram for this nonlinear oscillator for the following total
mechanical energies: E = (⅛)(k3/λ2); E = (⅓)(k3/λ2). Are these two phase diagrams
qualitatively different? If so, how? (5 points)
c. Calculate the restoring force F(x) and (by combining this with Newton’s 2nd Law) derive the
differential equation of motion for this mass. (Don’t try to solve it yet!) (5 points)
d. Suppose the term – (⅓)λx3 U(x) is much smaller than (½)kx2, so that this mass is
almost a harmonic oscillator. Assume initial conditions that at t = 0, m is at rest at x = A.
Use the method of successive approximations (as follows) to find an approximate
solution, x(t), to the nonlinear equation obtained in c. As an initial approximation, take
the harmonic oscillator solution (solution when λ = 0) for x(t) which satisfies these
initial conditions. Use this approximation in the equation of motion to obtain the next
approximate solution. Make sure that this new solution satisfies the initial conditions!
Assume that λ is small enough that this process can be stopped after one iteration. How
does this non-linear solution differ from that for the simple harmonic oscillator? If you
can, discuss this physically, as well as mathematically. (10 points)
The following integrals might be useful: ∫cos(ut)dt = u–1 sin(ut). ∫sin(ut)dt = - u–1 cos(ut).
Don’t forget the integration constant!!!
The following identity might be useful: cos2(u) = (½) + (½)cos(2u)
NOTE!!!! Work any four (4) of the six problems.
2. Parts a. and b. are (obviously!) independent of each other!
a. By a successive approximation procedure, use your calculator to solve the equation
x - 1 = 2 sin(x). Obtain a result accurate to 4 significant figures. Use either direct iteration
or Newton’s Method. I recommend the latter! Hint: Your initial “guess” should be some
x > 0. Before choosing this initial guess, it might be useful to make a sketch of x - 1 and
2 sin(x) on the same graph to give you an idea where the two functions cross. (13 points)
b. Using complete, grammatically correct English sentences, define the following: chaos,
limit cycle, Poincar section, mapping, bifurcation, Lyapunov exponents. (12 points)
3. A sphere of radius R has a non-uniform volume mass density, which depends on the distance r
from the sphere’s center as ρ = ρ0 r3R-3, where ρ0 is a constant. Hint: Recall that, for problems of
high symmetry, gravitational calculations are often much easier if GAUSS’S LAW is used!
a. Calculate the gravitational field g outside the sphere, a distance r  R from the center. (6 points)
b. Calculate the gravitational field g inside the sphere, a distance r  R from the center. (6 points)
c. Calculate the gravitational potential at all points in space. Take the zero of potential to be at r = .
(9 points)
d. A point mass m is placed a distance r  R away from the sphere. Compute the gravitational force
between m and the sphere. (2 points)
e. Compute the gravitational potential energy for the point mass in part d. (2 points)
The following integral might be useful: ∫rn dr = (rn+1)/(n+1), where n is any power (n  -1). Don’t
forget the lower limit!!
4. These are questions about the (fictitious!) Klingon Home Planet! Astronauts visited it and found
that the gravitational force there does not follow Newton’s Law of Gravitation! Instead, they found
that the gravitational force on a point mass m varies as the INVERSE CUBE of the distance from
the center of the planet as F = -mkr-3. Data needed to obtain the required numerical results:
Force constant: k = 2.33  1021 Nm3/kg; Planet Radius: RP = 5.0  106 m. Assume that the
planet is a sphere of radius RP.
a. Calculate the gravitational potential energy function U(r) for this planet. (4 points)
b. A particle of mass m is dropped from rest from a height h = 6  105 m above the Planet’s surface.
Compute the speed with which m will hit the ground. A numerical result is wanted & needed! (4
points)
c. Calculate the escape velocity for a particle of mass m on the surface of the planet. That is, find the
minimum velocity needed to “escape” from the Planet’s gravitational field (described by the above
force!). A numerical result is wanted & needed! (4 points)
d. A satellite of mass m = 3000 kg is in a circular orbit of radius r = 3.5  107 m about the planet.
Treat the satellite as a point mass. Compute the gravitational force between the satellite and the
planet, the speed of the satellite in orbit, and the period of the satellite’s orbit. Numerical results
are wanted & needed! (4 points)
e. A particle of mass m is dropped from rest from a large height h above the Planet’s surface. Use
conservation of total mechanical energy to derive an expression for the time t it takes to fall to the
Planet’s surface. This should be in the form of a messy integral. Leave it in integral form! (9
points)
NOTE!!!! Work any four (4) of the six problems.
5. See figure. Two masses, m1 and m2 are connected by a
massless, inextensible cord of fixed length  that passes
through a hole in a horizontal table. m1 moves without
friction on the table in a curved path as shown, while m2
oscillates like the bob of a pendulum. (The oscillation
takes place in a plane.) Gravity acts downward. Use the
generalized coordinates suggested in the figure to solve
this problem.
a. Write expressions for the kinetic energy, the potential energy, and the Lagrangian for this
system. How many degrees of freedom are there? (7 points)
b. What is the constraint? Write the equation of constraint. What is the physical
significance of the constraint force? (7 points)
c. Use Lagrange’s equations with the method of Lagrange multipliers to derive the
equations of motion. (7 points)
d. What are the constants of the motion? (That is, which quantities are conserved?) (4 points)
6. See figure. A bead of mass m, subject to no external force, is constrained to move on a
straight wire rotating at a constant angular speed ω about an axis through O and
perpendicular to the wire. Note: “no external force” means that there is NO GRAVITY!
The bead rotates with the wire, but is also allowed to slip on the
wire.
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? (5 points)
c. Use Lagrange’s equations along with the method of Lagrange
multipliers to derive the equations of motion. (5 points)
d. Are there any constants of the motion? (That is, are there any physical quantities are
conserved?) If so, what are they? (5 points)
e. Solve the equations obtained in part c to obtain both the constraint force and the position
of the bead as a function of time. Assume initial conditions that, at t = 0, the bead is on
the x-axis at position x = A, with zero radial velocity component. (5 points)