4. If only ball
A
is set at
φ
=
φ
(0)
initially, obtain the speed of each ball at the
bottom after the
n
-th collision. What is the speed of each ball as
n
→ ∞
?
Problem 45.
1992-Fall-CM-U-2.
ID:CM-U-498
A toy consists of two equal masses (
m
) which hang from straight massless arms (length
l
) from an essentially massless pin. The pin (length
L
) and the arms are in plane.
Consider only motion in this plane.
1. Find the potential and kinetic energies of the masses as a function of
θ
, the
angle between the vertical and the pin, and the time derivatives of
θ
. (Assume
the toy is rocking back and forth about the pivot point.)
2. Find the condition in terms of
L
,
l
, and
α
such that this device is stable.
3. Find the period of oscillation if
θ
is restricted to very small values.
Classical Mechanics
QEID#15664097
February, 2018

Qualification Exam
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Problem 46.
1992-Fall-CM-U-3.
ID:CM-U-510
A homogeneous disk of radius
R
and mass
M
rolls without slipping on a horizontal
surface and is attracted to a point
Q
which lies a distance
d
below the plane (see
figure). If the force of attraction is proportional to the distance from the center of
mass of the disk to the force center
Q
, find the frequency of oscillations about the
position of equilibrium using the Lagrangian formulation.
Problem 47.
1992-Spring-CM-U-1
ID:CM-U-517
For a particle of mass
m
subject to a central force
F
= ˆ
r
F
r
(
r
), where
F
r
(
r
) is an
arbitrary function of the coordinate distance
r
from a fixed center:
1. Show that the energy
E
is s constant. What property of the force is used?
2. Show that the angular momentum
L
is s constant. What property of the force
is used?
3. Show that, as s consequence of the previous part, the motion of the particle is
in a plane.
4. Show that, as a consequence, the trajectory of motion in polar coordinate can
be solved by quadrature. (
i.e.
, the time-dependence of the coordinates can be
expressed as integrals, which you should express, but which you cannot evaluate
until the function
F
r
(
r
) is specified.) For this part it will be useful to introduce
an
effective potential
incorporating the angular momentum conservation.
5. Suppose
F
r
(
r
) is attractive and proportional to
r
n
, where
n
is an integer. For
what values of
n
are stable circular orbits possible?
[
Hint:
Use the effective potential defined before, make a rough drawing of the
different possible situations, and argue qualitatively using this drawing.]
Classical Mechanics
QEID#15664097
February, 2018

Qualification Exam
QEID#15664097
21