Phys 325 Midterm #1 Fall 2015
Fall 2015 – Physics 325 Midterm Exam #1
Thursday Oct 1, 11:00 am – 12:30 pm
This is a closed book exam. No use of calculators or any other electronic devices is allowed. Work the
problems only in your answer booklets only. The exam questions will not be collected at the end, so
anything you write on these question pages will NOT be graded
You have 90 minutes to work the problems.
At the beginning of the exam:
1) Write your name and netid on your answer booklet(s).
2) Turn your cell phone off.
3) Put away all calculators, phones, computers, notes, and books.
During the exam:
1) Show your work and/or reasoning. Answers with no work or explanation get no points. But ...
2) Don’t write long essays explaining your reasoning. We only need to see enough work to confirm that you
understand what you’re doing and are not just guessing. Also a good annotated sketch is often the best
explanation of all!
3) All question parts on this exam are independent: you can get full points on any part even if your answers
to all the other parts are incorrect. Therefore you should attempt all the question parts! If you get stuck,
move on to the next one and come back later. The worst thing you can do is stall on one question and not
get to others whose solution may be very simple.
4) Partial credit will be given for incorrect answers if the work is understandable and some of it is correct.
IMPORTANT: If you think you’ve made a mistake but can’t find it, explain what you think is wrong
➔ you may well get partial credit for noticing your error!
5) It is fine to leave answers as radicals or irreducible fractions (e.g. 10 3 or 5/7).
6) Specify the units for all numerical answers.
Write everything in your answer booklet → that is all we will grade.
When you’re done with the exam:
Turn in your answer booklet ; keep the exam questions and formula sheets.
Academic Integrity:
The giving of assistance to or receiving of assistance from another person, or the use of unauthorized materials
during University Examinations can be grounds for disciplinary action, up to and including expulsion from the
University.
Please be aware that prior to or during an examination, the instructional staff may wish to rearrange the
student seating. Such action does not mean that anyone is suspected of inappropriate behavior.
Phys 325 Midterm #1 Fall 2015
Phys 325 Midterm #1 Fall 2015
You must derive any result that is not on the formula sheet.
You may of course use from memory any formulae from a previous class, e.g. 211, 225.
You may use without proof these Moments of Inertia:
2
• Thin rod of uniform mass m and length d rotating around its center: I = md /12
2
• Thin rod of uniform mass m and length d rotating around its end: I = md / 3
In both of the above cases, the axis of rotation ω̂ is perpendicular to the rod.
Problem 1 : A Bar and a Spring
When the system is in static equilibrium (i.e. when nothing is moving), it is found
that the spring is compressed to a length x = h / 2. Calculate the mass M of the
steel bar in terms of the given values d, k, h, and/or g.
PIVOT!
h
x
!
AR
LB !
d
EE
ST ngth
le
A heavy, uniform steel bar of mass M and length d is attached to a vertical wall
with a pivot. Under the influence of uniform gravity g, the bar tilts downward,
but it is preventing from falling too far by a horizontal spring. This spring is
attached to the wall a distance h below the pivot and its unstretched length is also h.
The magnitude of the spring force is thus k | h – x | where x is the length of the
spring and k is the spring constant. The contact between the spring and the steel
bar is frictionless.
SPRING!
Problem 2 : Motion in Cylindrical Coordinates
A particle of mass m follows this trajectory, in cylindrical coordinates: s(t) = b e− kt , φ (t) = ω t , z(t) = u t

where b, k, ω, and u are constants. Calculate the force field F(s,φ , z) that is causing this motion.
Note: the force field may not depend on all three coordinates, it may depend on only one or two of them.
Problem 3 : Non-Uniform Gravity
The gravitational force near the Earth's surface is almost constant, but not quite. Take y = 0 to be some location
a bit above the Earth's surface and let the +y axis point downwards. The force of gravity on a mass m is

y⎞
⎛
F grav ≈ ŷ mg ⎜ 1+ ⎟
⎝ b⎠
for
y
1.
b
The constant b is half of the earth's radius, so (y / b)  1 is an excellent approximation for "everyday" y values.
At time t = 0 a mass m is dropped from rest from position y = 0. Calculate time t as a function of the mass's
position y. Throughout this problem, use Taylor approximations wherever possible so that you only keep terms
that are linear in (y / b) , i.e. drop all terms of order (y / b)2 or higher.
Phys 325 Midterm #1 Fall 2015
Problem 4 : Rocket in a Nebula
A rocket ship of mass m0 is sits at rest in a nebula: an interstellar cloud of gas, where the gas is also at rest.
The ship is far from any stars or planets so ignore gravity. At time t = 0, the ship switches on both its rocket
engines and its gas-collector device, and begins the following operation:
● It collects gas from the nebula at a constant rate k (in kg/sec).
● Half of the collected gas is immediately stored in the ship for later use.
● The other half of the collected gas is immediately ejected out the back at speed u (relative to the ship)
Calculate the rocket's speed v as a function of time.
Problem 5 : Dumbbell in a Non-Uniform Force Field
A dumbbell consists of masses m– and m+ connected by a massless rigid rod of length 2b. The two masses are
identical : m– = m+ = m. (We are just labelling them differently for convenience.) At time t = 0, the dumbbell is
placed at rest along the x-axis with mass m– at the origin and
 mass m+ at x = +2b. No gravity is present …
instead, the masses experience a non-uniform force field F(r ) = m k x ŷ where k is a positive constant.
The dumbbell's motion is described by the coordinates X(t) and Y(t)
of its center-of-mass and the angle φ(t) that it makes with the x-axis.
These three coordinates are shown in the figure at right.
y
b
(a) Calculate the coordinates X(t) and Y(t) of the dumbbell's CM
for times t > 0.
(b) Obtain but DO NOT SOLVE a differential equation for the
angle φ(t) between the dumbbell and the x-axis, for times t > 0.
b
m−
φ
m+
(X, Y )
x