December 18, 2004: 4 pm - 7 pm

Instructions:

·      Do not turn this page until you are instructed to do so.

·      Each answer must be well motivated.  You will need to show your work in order to get full credit.

·      If we can not read what you wrote, we can only assume it is wrong.

·      Useful equations are attached to the exam and a copy of Appendices D, E, and F is available as a separate handout.

·      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 nth 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 q0.  Define the angle e as q - q0.  Rewrite the equation of motion for q in terms of e.  Use the small angle approximation and assume q0 = 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 m1 at its end.  Another mass m2 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 m1 and m2 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 = axn 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 q1 and q2.

b)    Express the potential energy of the system in terms of the generalized coordinates q1 and q2.

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 Iij 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 force-free 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:

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:

Practice Final Exam.

This exam was used in Phy 235 in Fall 2004.

You are encouraged to use this practice exam to determine how well prepared you are for the final exam.

Sit down for 3 hours and complete the practice exam. Then compare it with official solutions to determine how well you did.