Physics 2030
October 27, 2011
Midterm Exam
30 points total
8:30 am – 10:20 am
1. (10 points total)
A point mass m is suspended from the ceiling by a rigid rod. Assume the mass of the rod is
negligible, has a fixed length R, and is attached to the ceiling using a frictionless bearing. The mass
can swing in any direction so long as the length of the rod remains R. Work in spherical coordinates
(x = R sin(θ) cos(φ), y = R sin(θ) sin(φ), and z = R cos(θ)). Do not worry about the rotation of the
Earth, only its gravitational field.
(a) (3 points) What is the Lagrangian for this system?
(b) (2 points) Show that a conservation law is present whenever a Lagrangian does not depend on
one of the coordinate variables. Use this result to find a conserved quantity (other than the energy)
for the motion of the mass.
(c) (3 points) Find the equations of motion for the mass.
(d) (2 points) What is the Hamiltonian for this system?
2. (10 points total)
A disk spins about its axis at a constant angular velocity ω. A frictionless ball is shot out from the
center of the disk with speed v at time t = 0, initially along the x̂-axis of the disk. Its distance from
the center of the disk is therefore r = vt and the angle it makes with the disk’s x̂-axis is θ = −ωt.
Thus the ball has Cartesian coordinates on the disk given by:
x(t) = +vt cos(ωt)
y(t) = −vt sin(ωt)
(a) (6 points) Working in the (non-inertial) frame of the rotating disk, write down the equations of
motion for the ball in Cartesian coordinates. Make no approximations. The following formula may
be useful:
d
dt
=
inertial
d
dt
+ ω
~×
body
Hint: You must consider both the Coriolis force, and the centrifugal force.
(b) (4 points) Show that x(t) and y(t) given in the premise of the problem solve the equations of
motion that you found in part (a), demonstrating that you can work either in the inertial or the
rotating disk frames and arrive at the same result.
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3. (10 points total)
Prove the virial theorem for a system of N particles labeled by index i and subjected to forces F~i by
following these steps:
(a) (3 points) Introduce the quantity
G≡
X
p~i · ~ri
i
and show that
X
d
G = 2T +
F~i · ~ri
dt
i
where T is the total kinetic energy.
(b) (2 points) Now argue that the time average of dG
dt = 0 in the limit of infinitely long averaging
time, making the assumption that G itself is bounded (doesn’t get arbitrarily large in size).
(c) (2 points) For forces that are due to a potential V (r) = arn show that T =
n
2
V.
(d) (3 points) If the force on particle i is not due to a single fixed potential, but rather is the sum of
the forces from all the other particles in the system:
F~i =
X
F~ji
j6=i
do you still expect the virial theorem to hold? Why or why not?
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