**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.

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.
**

__Answer questions 1 - 2 in exam book
1, questions 3 – 4 in exam book 2, and questions 5 - 7 in exam book 3.
__

**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 (15
points) BOOK
1**

Consider a rail road car moving with a constant
acceleration *a* in the *x* direction. A simple pendulum of mass *m* and length *L* is placed inside the railroad car, as shown in the Figure below.

a) What is the Lagrangian of the pendulum?

b)
What is the equation of motion of the angle *q*?

c)
What is the equilibrium angle of the pendulum (*q*_{e})?

d) What is the frequency of small oscillations around the equilibrium angle?

**Problem 2 (15
points) BOOK
1**

Consider a string of length *L* with the following initial conditions:

and

a) Determine the characteristic frequencies of the string.

b)
Calculate the amplitude of the *n*^{th} mode.

**Useful relation:
**

**Problem 3 (15
points) BOOK
2**

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 4 (20
points) BOOK
2**

a)
A system with a spherically symmetric mass
distribution generates the gravitational potential and the gravitational field
shown in the figure below. Sketch
the mass density of this system as function of *R*.

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

c)
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?

d)
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.

e)
Consider the following phase diagram that describe
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. Sketch the potential associated with the phase diagram. In particular, pay attention to the symmetry of the potential around the equilibrium position.

**Problem 5 (20
points) BOOK
3**

Two masses of mass *M* are connected to each other by a spring with spring constant *k*_{12},
and by two springs with spring constant *k* to fixed positions, as shown in the
Figure below.

When the masses are in their equilibrium position, the springs are at their rest lengths.

a) Determine the coupled equations of motion of the two masses.

b) What are the characteristic frequencies of this system.

c) What is the general solution of the position of mass 1 and 2 as function of time?

d) What are the normal coordinates of this system?

e) Determine the equations of motion of this system in terms of these normal coordinates.

**Problem 6 (15
points) BOOK
3**

A flexible cord of length *L* slides from a frictionless table top as shown in the Figure
below.

The rope is released when a length *y*_{0} is hanging over the edge of the table.

a)
What
is the Lagrangian of this system when a length *y* is hanging over the edge of the table.

b) Use the Lagrangian to obtain the equation of motion.

c) At what time will the left end of the cord reach the edge of the table.

**Problem 7 (5
points) BOOK
3**

a) Which Yankee players are considered “young talent”?

___ 1. Derek Jeter

___ 2. Aaron Judge

___ 3. Didi Gregorius

___ 4. CC Sabathia

___ 5. Mariano Rivera

b) What is the best baseball team in the world?

___ 1. Yankees

___ 2. Yankees

___ 3. Yankees

___ 4. Yankees

___ 5. Red Sox

Last updated on Wednesday, December 20, 2017 14:46

Final exam for Phy 235, Fall 2017, covering the material of Chapters 1 - 13.