**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)__

a.
Consider
the following phase diagram that describes the one-dimensional motion of an
object of mass *m*.

For this
phase diagram answer the following questions:

1.
What
is the location of the equilibrium position? You need to motivate your answer!

2.
Is
the equilibrium a stable or an unstable equilibrium? You need to motivate your answer!

3.
Rank
the three energies, *E*_{1}, *E*_{2}, and *E*_{3}, from largest to smallest.

Sketch the
potential associated with the phase diagram. In particular, pay attention to the
symmetry of the potential around the equilibrium position.

b.
Consider
the following phase diagram that describes the one dimentional motion of an
object of mass *m*.

For this
phase diagram answer the following questions:

1.
What
is the location of the equilibrium position? You need to motivate your answer!

2.
Is
the equilibrium a stable or an unstable equilibrium? You need to motivate your answer!

3.
Rank
the three energies, *E*_{0}, *E*_{1}, and *E*_{2}, from largest to smallest.

Sketch the
potential associated with the phase diagram. In particular, pay attention to the
symmetry of the potential around the equilibrium position.

c.
Consider
the following phase diagram that describes the motion of a plane pendulum of
mass *m* and length *l*.

Derive the
expression for the phase path of the plane pendulum (angular velocity as
function of angle) if the total energy *E*
exceeds *E*_{0} = 2*mgl.*

__Problem 2 (25 points)__

Consider the system of pulleys, masses,
and string shown in the Figure below.

A massless string of length *b* is attached to point *A*, passes over a pulley at point B
located a distance 2*d* away, and
finally attaches to mass *m*_{1}. Another pulley with mass *m*_{2} attached,
passes over the string, pulling it down between *A* and *B*. Assume that the radius of the pulley
holding mass *m*_{2} is small
so that we can neglect it. Assume
also that the pulleys are massless.
The distance *c* is constant.

a)
What
is the potential energy of the system in this configuration (when mass *m*_{1} is located a distance *x*_{1} below the level defined by
*A* and *B*)?

b)
Determine
the position of mass *m*_{1}
for which the system will be in equilibrium.

c)
Determine
if the equilibrium position obtained in part b) is a stable or an unstable
equilibrium position.

__Problem 3 (25 points)__

A particle of mass *m* is at rest at the end of a spring with spring constant *k*, hanging from a fixed support. At time *t* = 0, a constant downward force *F* is applied to the mass and acts until time *t* = *t*_{0}.

a)
Write
down the equation of motion for times between *t* = 0 and *t* = *t*_{0}.

b)
Write
down the equation of motion for times *t*
³ *t*_{0}.

c)
What
is the general solution for times between *t*
= 0 and *t* = *t*_{0}. Note:
you do not yet need to determine the two integration constants in this
solution.

d)
What
is the general solution for times *t* ³
*t*_{0}. Note: you do not yet need to determine
the two integration constants in this solution.

e)
What
are the initial conditions at time *t*
= 0? Use these intial conditions to
determine the integration constants in the general solution found in 3c for times
between *t* = 0 and *t* = *t*_{0}.

f)
Find
the displacement of the mass as function of time after the force *F* has been removed (*t* ³ *t*_{0}).

__Problem
4 (25 points)__

a.
Consider
two systems, each consisting of a particle of mass *m*. The forces acting on
these particles are shown in the following figure. The forces deviate from a linear
dependence on *x* at large |*x*|.
Which of the two forces will produce a soft system?

b.
For
each of the two forces shown in the figure above, sketch the potential energy
as function of *x*. In the same figures, also sketch
expected dependence of the potential energy on *x* for a linear system.

c.
Consider
the Lyapunov exponent as a function of *a* for the logistic equation map, shown in the Figure
below. How many solutions are
possible when *a* = 3.5?
What is the smallest value of *a* for which chaotic motion will occur?

d.
Consider
a system that can be described by a second order differential equation. Consider the following three second
order differential equations:

The constants in these differential
equations are positive constants. The
mass *m* is at rest at time *t* = 0 s and located at *x* = 1 m.

Consider the following phase
diagrams. Match each phase diagram
to one of these three differential equations.

**Phase diagram # 1.**

**Phase diagram # 2.**

**Phase diagram # 3.**

e.
At
what airport was this Boeing 747 of the Koninklijke Luchtvaart Maatschappij landing?

___ Amsterdam

___ Rochester

___ Orlando

___ JFK

___ Toronto

___ Chicago

___ Dalfsen

___ Atlanta

___ Boston

___ Montreal