**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.
**

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

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name: ______________________________________________________________________

Signature: ____________________________________________________________________

__Problem 1 (25 points)__

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.

__Problem 2 (25 points)__

a) Which of the following forces are conservative?

In these equations *a*, *b*, and *c* are constants. Note: your answer must be wel motivated.

b)
For
the conservative forces identified in step a), find the potential energy *U*.

__Problem 3 (25 points)__

Two masses *m*_{1} and *m*_{2} slide freely
on a horizontal frictionless track and are connected by a spring whose force
constant is *k*. Assume the
rest length of the spring is *L*.

a) Write down the differential equation that describes the acceleration of mass 1 in terms of the position of mass 1 and 2.

b) Write down the differential equation that describes the acceleration of mass 2 in terms of the position of mass 1 and 2.

c) Use the two differential equations obtained in parts a) and b) to obtain a differential equation that desrcribes the acceleration of mass 1 in terms of just the position of mass 1.

d) Find the frequency of the oscillatory motion for this system.

__Problem 4 (25
points)__

a.
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?

b. Which of the curves in the following figure shows the time dependence of the amplitude of an over-damped oscillator?

c.
Consider a damped and driven pendulum. The Poincare sections for seven
different driving force amplitudes are shown on the right: *F* = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 N (from top to bottom).

For each Poincare section, indicate if the motion is

1. Simple Harmonic.

2. Periodic.

3. Chaotic.

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. What is the nationality of Didi Gregorius?

___ USA

___ The Netherlands

___ South Africa

___ China

___ Brazil

How many times did the Yankees reach the post season since 1921?

___ Between 0 and 9

___ Between 10 and 19

___ Between 20 and 29

___ Between 30 and 39

___ Between 40 and 49

___ Between 50 and 59

___ Between 60 and 69

___ Between 70 and 79

___ Between 80 and 89

___ Between 90 and 99

First midterm exam in Phy 235, covering the material of Chapters 1 - 5.