Instructions:
á Do not turn this page until you are instructed to do so.
á Each answer must be well motivated. You will need to show your work in order to get full credit.
á If we can not read what you wrote, we can only assume it is wrong.
á Useful equations are attached to the exam and a copy of Appendices D, E, and F is available as a separate handout.
á The total number of points you can get on this exam is 105 (you get 5 bonus point if you answer question 7 correctly).
NOTE: Answer Questions 1  3 in Exam book 1 and
Questions 4  7 in Exam book 2.
Problem 1 (15 points)
Consider a string of length L, kept fixed at both ends. The string has a density r and a tension t. The condition of the string at time t = 0 s is specified by the following boundary conditions:
and
a) Find the characteristic frequencies.
b) Calculate the amplitude of the n^{th} mode.
Problem 2 (20 points)
A pendulum is constructed by attaching a mass m to a string of length l. The upper end of the string is connected to the uppermost point on a vertical disk of radius R (R < l/p) as shown in the Figure below. As the pendulum swings back and forth, its contact point with the disk changes continuously.
a)
Define the angle q
as the angle between the vector from the center of the disk to the contact
point of the string and the positive vertical direction. Define a Cartesian coordinate system
such that its origin coincides with the center of the disk. Express the x and y
coordinates and the x and y components of the velocity of the mass m in terms of the angle q.
b)
Express the Lagrangian for this system in terms of the angle q.
c)
What is the equation of motion for the angle q?
d)
Assume that the pendulum is making small oscillation around
the equilibrium angle q_{0}. Define the angle e as q
 q_{0}. Rewrite the equation of motion for q in terms of e. Use the
small angle approximation and assume q_{0}
= p/2.
e)
What is the angular frequency of the oscillatory motion?
Problem 3 (15
points)
Consider a pendulum, shown in the Figure below,
composed of a rigid rod of length b with
a mass m_{1} at its
end. Another mass m_{2} is placed halfway down the rod. Consider the motion of the pendulum
when the oscillations are small, and assume that the pendulum swings in the
plane of the Figure.
a)
What is the inertia tensor?
b)
Express the angular momentum of the pendulum in terms of its
angular velocity.
c)
Use the external torque acting on the pendulum to determine
the equation of motion for the angle q.
d)
What is the angular frequency of the system?
e)
What is the Lagrangian of the system?
f)
Use the Lagrange equation of motion to determine the angular
frequency of the system.
Problem 4 (15
points)
Two particles of mass m_{1} and m_{2}
are moving under the influence of their mutual gravitational force in perfect
circular orbits with a period t. The masses are suddenly stopped in
their orbits and allowed to gravitate toward each other.
a)
What is the distance between the particles when they are
orbiting in their circular orbits?
b)
What is the time interval between the moment the particles are
stopped in their orbits and the moment they collide?
Problem 5 (20
points)
A smooth rope, of length L and density r,
is placed above a hole in a table (see Figure). One end of the rope falls through the hole at time t = 0, pulling steadily on the remainder of the
rope. Ignore all friction.
a)
At time t a length x of the rope has passed through the hole. Assume the velocity of the rope at this
time is v. Use the relation F = dp/dt
to obtain a differential equation relating x and v
(and/or their derivatives).
b)
Solve the differential equation found in a) by using v = ax^{n}
as a trial function.
c)
Determine the velocity of the rope as function of x.
d)
Determine the acceleration of the rope as function of x.
Problem 6
(15 points)
Consider the coupled pendulum shown in the Figure
below.
Each pendulum has a mass m attached to a massless string of length b. The
spring connecting the two masses has a spring constant k and is unstretched in the
equilibrium position. Assuming
that the oscillations are small, we can make the following approximations:
a)
Express the kinetic energy of the system in terms of the
generalized coordinates q_{1}
and q_{2}.
b)
Express the potential energy of the system in terms of the
generalized coordinates q_{1}
and q_{2}.
c)
What are the eigen frequencies of the system?
d)
What are the normal modes of the system?
Problem
7 (5 points)
a) Match the pictures of the ball parks to the team names listed below.
A 
B 
C 
D 


E 

___ 1. Yankees
___ 2. White Sox
___ 3. Mets
___ 4. Cubs
___ 5. Red Sox
b) What is the best baseball team in the world?
___ 1. Yankees
___ 2. Yankees
___ 3. Yankees
___ 4. Yankees
___ 5. Yankees
Useful Relations
Euler's equations without external constraints:
Euler's equations with external constraints g{y; x} = 0:
where l(x) is the Lagrange undetermined multiplier.
Hamilton's principle:
The Lagrange equations of motion:
Lagrange equations of motion with undetermined multipliers:
Definition of the Hamiltonian:
Hamilton's equations of motion:
Orbital
motion:
Effective
potential energy:
Center
of Mass:
Linear
momentum of a system of particles:
Angular
momentum of a system of particles:
Scattering
cross sections:
Rutherford
scattering:
Rocket equations:
R u = M a
Fixed and rotating coordinate systems:
Effective
forces in rotating coordinate systems:
Kinetic
energy of a rigid object:
where the quantity I_{ij} is called the inertia tensor, and is a
3 x 3 matrix:
Transformation of the inertia tensors for rotations:
The angular momentum of a rotating
rigid body is equal to
The Euler equations in a forcefree
field:
The
Euler equations in a force field:
Kinetic
energy of a system of oscillators:
Potential
energy of a system of oscillators:
where
Equations
of motion of coupled oscillators:
Solution
of the loaded string:
where
Solution
of the continuous string:
where
and
Kinetic
and potential energy of the string:
The
ideal wave equation:
The
real wave equation:
The
phase velocity of a wave:
The wavelength of a wave: