Fall 2013 – Physics 325 Midterm Exam #1Thursday Oct 3, 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. Total of 50 points available.

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. 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.

Remember: There is a Table of Integrals provided ➔ do use it!

When you’re done with the exam: Turn in your answer booklet. (You can keep the exam questions).

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.

Midterm #1 Fall 2013

Midterm #1 Fall 2013

Problem 1 : Into the Swamp [7 points]

An abandoned canoe drifts into a swamp with initial velocity v0. Once in the swamp, the canoe experiences a linear drag force

f = −bv v̂ , where b is a constant with units of kg / s. How far does the canoe make it into the

swamp?

Problem 2 : A Ball Goes Berzerk [6 points]

A tiny ball of mass m is anchored to a pivot at the origin by a rigid rod of length R. Mysterious forces in the θ̂ and φ̂ directions cause the ball to move along this trajectory:

ball’s spherical coordinates: r(t) = R , θ(t) = π2+θ0 sin(ωt) , φ(t) = Ω t

where R, θ0, ω, and Ω are known constants. Calculate the ball’s kinetic energy at time t = 0.

Problem 3 : This Should Be Quick [6 points]

An electron of mass m travels along the y axis in the +y direction with constant speed v. At time t = 0 it crosses the origin. Calculate the angular momentum (magnitude only) of the electron around the point (x,y,z) = (3,9,4) as a function of time.

Problem 4 : Hovering Rocket [7 points]

Your company is about to run a test on its new rocket: when the rocket is on the launch pad, the engines will be turned on at t = 0 and the rocket will hover – be motionless – immediately above the launch pad for a few seconds. An engineer gives you this information: the mass of the empty rocket is mr, the mass of the fuel it will initially have in its tanks is also mr, and the rocket must hover until time t = T. Calculate the exhaust speed of the fuel (relative to the rocket) that is required for the rocket to hover until t = T without running out of fuel.

Midterm #1 Fall 2013

Problem 5 : Repulsion [8 points]

At time t = 0 a mass m is at rest at position x0 on the positive x axis. It then moves under the influence of a force field

F(x) = kx x̂ that pushes the mass away from the origin. Calculate x(t) for times t > 0.

Problem 6 : A Swinging Rod [(a) 7 points (b) 9 points]

A thin rigid rod has these properties: its length is 2b, its total mass is M, and its center-of-mass position is at the center of the rod. Although the CM position is nicely placed, the mass of the rod is non-uniformly distributed, in a way that is not specified. Instead, a black dot is painted on one end of the rod, and we are given the moment of inertia, I, for rotation of the rod around that black dot for any axis of rotation oriented perpendicular to the rod (i.e. for the rotation described in the next paragraph).

The end of the rod with the black dot is fixed to a frictionless pivot that allows the rod to swing freely in the xy-plane. We use the coordinate system shown in the figures below, with the y axis pointing downward = in the same direction as the uniform gravitational acceleration g. At some time t < 0 the rod is set to swinging in such a way that, at time t = 0, the rod is perfectly vertical and its outer endpoint (the end without the black dot) has velocity −v0 x̂ .

x

y

θ2b

PIVOT

t < 0

gravityg

x

y

2b

PIVOT

t = 0v0

(a) Let θ be the angle that the rod makes with the +y axis, as shown. Write down the differential equation of motion that you would have to solve in order to calculate the general solution θ(t) for the swinging rod in terms of the given parameters b, M, I, and/or g. DO NOT SOLVE THE EQUATION, just write it down. Note: the special situation at t = 0 is irrelevant here so ignore the right-hand figure for this part.

(b) At the instant t = 0, the pivot is released! The rod is now free and moves under the force of gravity alone. Calculate the coordinates x(t) and y(t) of the black dot for all t > 0 in terms of the given values b, M, I, v0 , and/or g. ** This last problem takes significantly longer than the others so leave it to the end. And don’t give up: you will get points for making progress.

Midterm #1 Fall 2013