qualifier2019.pdf - QUALIFYING EXAMINATION Part 1 10:00 am...

QUALIFYING EXAMINATION, Part 1
10:00 am – 12:30 pm, Thursday August 29, 2019
Attempt all parts of both problems.
Please begin your answer to each problem on a separate sheet, write your 3 digit code
and the problem number on each sheet, and then number and staple together the sheets
for each problem. Each problem is worth 100 points; partial credit will be given.
Calculators and cell phones may NOT be used.
1

Problem 1: Classical Mechanics I
A compound Atwood machine is composed of three masses m
1
, m
2
, and m
3
attached to
two massless ropes through two massless pulleys A and B of radii r
A
and r
B
, respectively
(see figure). Pulley A is attached to a stationary ceiling. The lengths l
a
and l
b
of the
ropes around pulleys A and B are fixed, and the ropes do not slip as the pulleys rotate.
The system is in a constant gravitational field with acceleration g .
Take as two generalized coordinates x
1
and x
2
the distance from the center of pulley
A to mass m
1
and the distance from the center of pulley B to mass m
2
, respectively (see
figure).
(a) (25 points) Write down the vertical positions x
m
1
, x
m
2
, and x
m
3
of the three masses
(as measured from the center of pulley A) in terms of x
1
and x
2
, and find the total kinetic
energy T .
(b) (20 points) Find the gravitational potential energy V . Set the potential energy to
zero at the center of pulley A .
(c) (5 points) Write down the Lagrangian L of the system.
(d) (30 points) Derive the equations of motion for x
1
and x
2
. You do not need to solve
them explicitly.
(e) (20 points) Next assume that pulley A is a uniform disk with mass M (and radius
r
A
). Write down the Lagrangian L for this system.
Hint: the rope does not slip on the pulley.
2

Classical Mechanics II
࠵?
࠵?
⃗࠵?
k
k
l
z
1
z
2
z
eq
A bar of length d and mass M hangs under the
influence of gravity from two springs with equal un-
stretched lengths, l , and equal spring constants, k ,
as shown in the figure.
In the following parts, consider only motion of
the bar in the x - z plane and ignore any “swinging”
motions—i.e., consider only modes of small oscilla-
tions where the center of the bar does not move in
the x or y directions. In these modes, the bar center
and ends are free to move primarily in the z direc-
tion, including rotations in the x - z plane.
(a) (10 points) Find the equilibrium position of the
bar, z
eq
, relative to the position of the ends of the
unstretched springs.
For small oscillations around this equilibrium position, two independent modes are
possible:
1. A mode in which the center-of-mass oscillates in the z direction with the bar hori-
zontal
2. A mode in which the center-of-mass is fixed and the bar rotates in the x - z plane
(b) (30 points) Write down the equations of motion for the bar in each independent mode
by considering the corresponding forces and torques on the bar. What are the frequencies
of the oscillations in each of the modes?