**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 projectile, launched vertically upward to a height *h* above the Earth’s surface at a Southern latitude *l*.

a)
Define the rotating coordinate system (*x*, *y*, *z*) in which you will view the motion
of this projectile.

b) Determine the acceleration of the projectile as function of time. Specify the components of the acceleration along the three coordinate axes of your rotating coordinate system; make sure to include the sign of the acceleration to fully specify the direction of the acceleration.

c) How far from its launch position does the projectile hit the ground? Specify the magnitude of the displacement and the direction (N, NE, E, SE, S, SW, W or NW)?

Neglect air resistance and consider only small vertical heights *h*. The rotational velocity of the Earth around its axis is *w*.

__Problem 2 (25 points)__

A smooth rope of length *L* is
placed above a hole in a table, as shown in the Figure below.

One end of the rope falls through the hole at *t* = 0 s, pulling steadily on the remainder of the rope.

Find the acceleration and velocity of the rope as a function of the
distance to the end of the rope *x*.

Ignore frction.

__Problem 3 (25 points)__

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 therms 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?

__Problem 4 (25
points)__

a. Consider the Hofman transfer to travel to Mars. What quantity is minimized if we use this method to travel Mars?

b. You observe the following deflection of air that approaches a low-pressure system. On which hemisphere is this low-pressure system located? The Northern or the Southern hemisphere? You need to motivate our choice.

c.
Consider the elastic scattering of two particles
with mass *m*_{1} and *m*_{2}. Mass *m*_{2} is initially at rest in the laboratory system. It is observed that after the collision,
the two masses travel at right angles with respect to each other. Two possibilities are shown in the
Figure below.

Based on the information provided, what can you say about the two masses?

d.
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?

e. What is the significance of November 16, 1776?

___ The date on which the Yankees won their first world series.

___ The date on which the AJAX soccer team was created.

___ Your teacher’s birthday.

___ The date on which the Netherlands acknowledged the independence of the American colonies.

___ Wait a moment; I expected a baseball question, not a history question.

Last updated on Wednesday, December 20, 2017 14:45

Second midterm exam in Phy 235, covering the material of Chapters 6 - 10.