**Do not turn
the pages of the exam until you are instructed to do so.**

** **

**Exam
rules:** You
may use *only* a writing instrument while taking this test. You may *not*
consult any calculators, computers, books, nor each other.

Problems 1 and 2 must be answered in exam booklet 1. Problems 3 and 4 must be answered in exam booklet 2. The answers need to be well motivated and expressed in terms of the variables used in the problem. You will receive partial credit where appropriate, but only when we can read your solution. Answers that are not motivated will not receive any credit, even if correct.

At the end of the exam, you
need to hand in your exam, the blue exam booklets, and the equation
sheet. All items must be clearly labeled with your name, your student ID
number, and the day/time of your recitation. **If any of these items are
missing, we will not grade your exam, and you will receive a score of 0 points.**

**You are required to complete the
following Honor Pledge for Exams. Copy and sign the pledge before
starting your exam.**

ÒI affirm that I will not give or receive any unauthorized help on this exam, and that all work will be my own.Ó

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name: ______________________________________________________________________

Signature: ____________________________________________________________________

**PROBLEM
1 (25 POINTS) ANSWER
IN BOOK 1**

Consider
a thin uniform disk of mass *M* and radius *a*. The center of
the disk is located at the origin of our coordinate system, and the disk is
located in the *xy* plane.

a.
Find the potential *F* on the *z* axis, a distance *z* from the center
of the disk.

b.
Find the graviational
field on the *z* axis, a distance *z* from the center of the
disk. Indicate the direction and magnitude of the gravitational field.

c.
Find the
gravitational force on a mass *m* located at this position. Indicate
the direction and magnitude of the gravitational force.

**Your
answers must be well motivated and expressed in terms of the variables
provided.**

**PROBLEM
2 (25 POINTS) ANSWER
IN BOOK 1**

Consider a
sherical pendulum of mass *m* and lengh *b*, as shown in the Figure
below. For a spherical pendulum, both *q* and *f*
can depend on time.

a.
Express the kinetic
energy and the potential energy of the pendulum in terms of the generalized
coordinates *q* and *f*.

b. Calculate are the generalized momenta.

c. Determine the Hamiltonian for this system.

d. Determine the equations of motion.

e. Which of the generalized coordinates or momenta are constant?

**Your
answers must be well motivated and expressed in terms of the variables
provided.**

**PROBLEM
3 (25 POINTS) ANSWER
IN BOOK 2**

Consider
a particle travelling in a constant force field (e.g. the gravitational field),
directed along the x axis. The particle starts from rest at (0,0) and
moves to a position (*x*_{2}, *y*_{2}).

a.
How does the velocity
of the particle depends on *x*?

b.
Express the time
required for the particle to move from the initial position to the final
position, following a path *y*(*x*), in terms of an integral over *x*
between *x* = 0 and *x* = *x*_{2}.

c. Find the path that allows the particle to accomplish this movement in the least amount of time.

**Your
answers must be well motivated and expressed in terms of the variables
provided.**

**PROBLEM
4 (25 POINTS) ANSWER
IN BOOK 2**

a.
Consider a disk
rolling without slipping on an inclined plane, as shown in the Figure
below. Consider the coordinates *y* and *q*.

What is the equation of constraint you would use for this system?

b. Consider the orbital speed of the observed mass as function of the distance from the center of the Andromeda galaxy, shown in the following Figure.

The
dashed line represents the 1/Ã*R* dependence expected from the Keplerian
result of NewtonÕs laws. What can account for the difference between the
expected orbital speed and the observed orbital speed?

c.
Consider light
passing from one medium with index of refraction *n*_{1} into
another medium with index of refraction *n*_{2}, as shown in the
following Figure.

What parameter do we minimize when we derive the law of refraction?

d.
What two conditions
must be met for the Hamitonian *H* to be equal to the total energy *E*?

e. What is the favorite color of your instructor?

A. Black

B. Red

C. Green

D. Yellow

E. Blue

F. Orange

G. Pink

H. White

I. Yankees

J. The Netherlands