Fall 2000 Professor Ellen D. Williams |
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Assignment 2 | Problem Set Solutions - Link Deleted | Assignment 3 | Assignment 4 | Grade Change Policy |
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Physics 171/171H - Spring 2000
(171H will meet separately on Thursdays with Prof. Goldenbaum)
Instructor: Professor E. Williams
Study Sessions: 2:30-5:30 Thursday, Conference Room - 2120 PhysicsOffice: Room 2332 Physics
Phone: 301-405-6156
e-mail: edw@physics.umd.edu
These are informal group study sessions, where the professor or TA will be available to answer questions individually. You should feel free to stop by briefly at any time during the study session to ask a few questions, or to stay there to study or participate in discussions of other students' questions as well as your own.
Course Textbook: Physics, 4th Edition, Volume I, Resnick, Halliday and Krane
(not all material in each chapter will be covered)
Six Ideas that Shaped Physics: Unit N and Unit C, T. A. Moore
2. Attendance and quizzes
3. In Class Exams: Exam 1: Tues., Oct. 3
Exam 2: Mon., Oct. 30
Exam 3: Tuesday, Nov. 21
4. Final Exam: Friday, December 15, 1:30 PM
In-Class Exams 50% (40% for 171H)
Final Exam 25%
Miscellaneous Course Information
Homework assignments will be posted on the course web site. If you didn't get the assignment when announced in class, or if you lost it, you can find it on the site.
Homework, Quiz and Exam solutions will be posted on the course web site only.
Quizzes will be given usually once a week. The quiz problems will be examples worked in class during the previous lecture, or sample problems from the assigned reading in the text unless otherwise announced.
Assistance outside of the course is available from several sources:
Working together on problem sets is strongly encouraged. However,
each student is expected to work through the problem independently, and
hand in his or her own work for grading. For any questions about
academic honesty, see University policies at:
Quizzes and Exams
If you have a question about grading of a quiz or exam, please see
Dr. Williams immediately after the class when the paper was returned.
Homework
The primary purpose of grading the homework is to give you a
way of finding out how well you really understand the material. If
you miss points on a problem, that alerts you that you should rework the
problem to make sure you understand how to do it properly. If you
feel that you did the problem correctly (and presented your work clearly)
and would like to get your homework re-graded, here is the procedure:
1) Use some common sense. Any given homework problem is about 1/10 of 1/13 of 1/5 of your grade. Think about that before spending a lot of time disputing over partial credit.
2) Don't ask for consideration of changed credit without first looking at the posted solution sets and making sure you really DID do the problem properly.
3) IF after considering points 1 & 2, you still have a question,
then you can write a note explaining the question, attach it, and a
copy of the relevant page from the solution set, to your HW set and
give it to Dr. Williams, who will forward it to the TA for consideration.
You should circle on your solution set, in a different color ink, the answer
that you are worried about.
OR
3') You can ask the TA for help understanding the problem during
the study session. In this case, please make sure you have a copy
of the solution set with you, and have marked the answers you are concerned
about on both the solution set and your homework set.
Changes in working exam:
Repeat problems 1, 3 and 4 as on the original exam. The corrected figure for problem 3 is:
(As far as I can tell, everyone who did the problem assumed that this was what I meant.)
Problem 2: Instead of parts a, b and c, do the following:
Calculate the height mass m2 rises before its speed reaches
zero, using:
a) the work-kinetic energy theorem
b) Newton's laws (F=ma) directly
(You should get the same answer both ways.)
Remember:
Work-Kinetic Energy Theorem: The TOTAL work done
on a particle is equal to the change in the particle's kinetic energy.
If some of the forces doing work are conservative forces where we know the dependence of force on distance, we can calculate a potential energy change and relate its negative to the work done by that force. Then we have: This approach is needed for problem 2 and problem 3c.
Reading and Problem Assignments
listed in reverse chronological order
Suggested study plan: Begin by reading over the assigned
problems for each chapter. Then look through the figures, tables
and sample problems in the chapter. This will give you an idea of
the material is organized and help you focus as you read the chapter.
Then read through the chapter, keeping in mind the questions that have
been assigned for homework. Work through the example problems carefully.
At the end of each section, see if you can do the assigned problems that
correspond to that section. Mark down questions if you get stuck
and ask questions in class. Don't wait until the night before the
homework is due to begin working the problems!! (You'll also get
a lot more out of the study sessions if you've looked at the problems before
you walk in the door.)
Reading: Ch. 23, Sections 2,3,4, 6
Ch. 24, Sections 1-4
Problem Set: Due at the beginning of class, Dec. 12 (note due
date is a Tuesday!!)
Supplemental Problems S21, S22, S23
Ch. 23, # 20, 26, 34, 50
Ch. 24 # 6, 14, 22, 24
S.21. A sample of hydrogen gas (hydrogen = H2, molecular weight
= 2.0 grams/mole, distance between H atoms in the molecule is 0.074 nm)
is at 500K. At this temperature, rotations of the molecule are allowed
but vibrations are not.
a) What are the average total translational and rotational kinetic
energies of a hydrogen molecule? What is the angular frequency of
rotation around one axis?
b) What is the average force (.e. the average over time of many collisions)
with which a molecule hits a wall of the container if the walls are 1m
apart?
.
S.22 A cubical box with length-of-side L contains N particles
of which
N1 = 1.0 x 1020 have speed 2.0x 102 m/s
N2 = 5.0x1020 have speed 4.5x102 m/s
N3 = 3.x1020 have speed 7.5x102 m/s
The particles are moving randomly within the box and rebound elastically
in collisions with each other or the walls.
a) What are the average and rms speeds of the particles
b) What is the temperature of the gas of particles if all the particles
have mass 75x10-27 kg?
c) If the molecular diameter of the particles is 2.5x10-10m,
find the mean free path in the gas. Use the average speed for your
calculation.
S23. A diatomic ideal gas is held in a box of volume 9 m3,
at a temperature
of 500K and a pressure of 0.1 atm. The
atoms in the molecules can rotate,
but not vibrate at this temperature.
a) How much gas is in the box?
b) The gas is heated to 600K without
changing volume. How much heat
was transferred to the gas?
c) Then the gas is slowly allowed to
expand at constant temperature
(600K) until it reaches its original
pressure. What is the final volume of
the gas? How much work is done by the
gas during this expansion
d) Draw a pressure-volume diagram for
the two step expansion of the gas
described in parts b and c. On the same
diagram, draw the
pressure-volume line for a one step
heat transfer from 500K to 600K at
constant pressure. Explain the difference
in the work available from the
two processes.
S24. A monoatomic ideal gas is confined in a two-dimensional container
(only x and y motion allowed) and equilibrated
at a temperature of 350K.
a) What is its internal energy? (Hint:
how many translational degrees of
freedom are there ?)
b) If the atomic mass of the atoms in
the gas is 28.1 g/mole, what is the
root mean square velocity of an atom?
c) The distribution of speeds of the
atoms is found to obey the equation:
N(v) = (2pAv)exp{-mv2/2kT)
Find an expression for A in terms of
the macroscopic parameters of the
ideal gas.
Reading: Ch. 17 Sections 1-3
Ch. 22, Sections 3, 5
Ch. 23, Sections 1, 5
Ch. 25, Sections 1-4
Problem Set: Due at the beginning of class, MONDAY Dec. 4
Ch. 17, #4, 12
Ch. 22, # 2, 10
Ch. 23, # 2, 4, 36, 40
Ch. 25 # 2, 12, 26
S16
Extra Credit Problems (up to 5 points each on homework score)
S17, S18
S.16 Two skaters, each of mass 50.0 kg, approach each other along parallel pathes separated by 3.0 m. One skater has velocity 10 m/s and the other has velocity -10 m/s both in the x direction. The first skater carries a massless pole of length L = 3.0 m. The second skater grabs the end of it as he passes by. Assume frictionless ice and treat the skaters as point particles.
a) What is the velocity of the system center of mass (system = 2 skaters + pole)?
b) What are the translational and angular velocities of each of the skaters after they are connected by the pole?
c) By pulling on the pole the skaters reduce their separation to 1.0 m. What are their translational and angular velocities then?
d) Compare the kinetic energies of the skaters before the collision, after the collision, and after they have reduced their separation. Where do the changes come from?
e) What quantities are conserved in parts b-d?
S.17 Do problem 21 in Chapter 13 of the book.
a) What is the difference in the mass for d = 0 and
for d = L/2?
b) What is the ratio of rotational to translational
kinetic energy after the collision for d = L/4 and for d = L/2?
S.18 Consider the same configuration as for problem S.17 (a point
particle hitting a thin rod normal to its length), but now the collision
is inelastic. Compare the change in kinetic energy during the collision
for d = 0 and for d = L/2. Use the values L = 2.2 m, m1 (the
point particle ) = 2.0 kg, m2 (the rod) = 2.5 kg, and vo (the initial speed
of the point particle) = 1 m/s.
If you are having trouble with concepts or problem solving in chapter 10, you might find it helpful to see a different approach in the supplemental material:
Circular motion, rotational motion, angular momentum: Reif Chs.
8, 17, 18
Rotational Energy and Angular Momentum: Moore, Volume C, Chs.
9 and 13
S13 A particle of mass m is attached to a string which
passes through a hole in a table. The particle rotates without friction
on the table around the hole with speed v and radius R.
a) What is the tension in the string?
b) If one pulls on the string until the mass moves inward to 1/4 the
original radius, what is the new speed of rotation? What is the new
tension in the string?
c) Show two different ways to determine the amount of work
done in moving the mass inward, and confirm that the two approaches give
the same answer.
S14 A small mass, m, is moving in the vicinity of a large
mass M. The large mass is so much larger that it undergoes no observable
acceleration due to its gravitational interaction with the small mass.
(Bold symbols, for instance r2, indicate vectors.
Non-bold symbols, for instance r2, indicate the magnitude of
the vector.) The zero of the coordinate system is the center of the
large mass.
a) The small mass is observed at time t1 to be at a location
r1
and traveling with speed v1 . At a later time, t2
, it is at a location r2 . The magnitude of r2
is r2 = 0.1r1 . What is the speed of
the particle at the later time? Express your answer in terms of G,
M, r1 and v1 .
b) The velocity at time t2 is observed to be perpendicular
to the location vector r2 . What is the angle
between the location vector and the velocity vector at time t1
? Draw a diagram indicating clearly the angle you have determined.
S15 A large solid sphere of mass m2 = 5.0 kg and radius R = 0.50
m is suspended from a pivot by a rigid massless rod. The distance
of the center of mass of the sphere from the pivot is L = 1.2
m. A point particle of mass m1 =2.5 kg and a horizontal initial velocity
of v0 = 6.0 m/s hits the large sphere head-on and sticks to it.
a) What is the angular velocity of the combined masses just after the
collision?
b) Assume the collision lasts 0.0020s, what is the average force on
m1 during the collision? What is the torque exerted on m1?
c) Show that the final angular velocity of m1 is consistent with the
torque exerted on it.
If you are having trouble with concepts or problem solving in chapter 10, you might find it helpful to see a different approach in the supplemental material:
Collisions: Reif Ch. 16, Section C
Collisions and Center of Mass: Moore, Volume C, Sections 4.5
5.5, 12.7
Problem Set: Due at beginning of Class, Nov. 10
Chapter 10, # 8, 10, 26, 28, 32, 46
Chapter 11, #26, 32
Suppl. Problem S10, S11
Extra Credit (up to 20% extra on this hw score) S 12
S10. A stationary nucleus of mass 107mp, where mp is the rest mass of
a proton, breaks apart (decays) into two particles. One of the particles
has rest mass m1 = 20mp and speed v1 = 0.89c.
a) Find the rest mass m2 and speed v2 of the other particle.
Express your answers in terms of mp and c.
b) Kinetic energy has increased in this decay. Explain why.
S11. If a relativistic collision occurs elastically, that is with no change in the total kinetic energy of the particles involved, what is the change in the total mass of the particles involved?
EXTRA CREDIT:
S12. In an isolated system (no external forces) of two particles,
mass m1 is moving at velocity
and
mass m2 is moving at velocity .
The vector between the two particles is
a) Find an expression for the center of mass position in terms of
b) Find expressions for the total momentum of this system in:
i) the given reference frame
ii) the reference frame in which m1 is initially stationary
iii) the reference frame in which the COM is stationary.
If you are having trouble with concepts or problem solving in chapter 9, you might find it helpful to see a different approach in the supplemental material:
Momentum and Center of Mass: Reif Ch. 15
Momentum and Center of Mass: Moore, Volume C, Ch.4 & 5
If you are having trouble with concepts or problem solving in chapter 8, you might find it helpful to see a different approach in the supplemental material:
Potential Energy: Reif Ch. 14
Kinetic and Potential Energy: Moore, Volume C, Ch.6 & 7
If you are having trouble with concepts or problem solving in chapter 7, you might find it helpful to see a different approach in the supplemental material:
Work and kinetic Energy: Reif Ch. 13
An object of mass 0.6 kg is sliding around the inside of a cylinder of radius 1.8 m. One half of the cylinder’s inner surface is frictionless, and the other half is rough with kinetic coefficient of friction mk = 0.2. The mass initially is moving on the smooth surface with constant speed of 2 m/s.
a) Draw a force diagram for the object after it has entered the rough part of the cylinder. What is the magnitude of the frictional force just after the object has entered the rough part?b) What is the velocity of the particle one second after it has entered the rough part of the cylinder? (Use Dv = aDt).
If you are having trouble with concepts or problem solving in chapter 6, you might find it helpful to see a different approach in the supplemental material:
Springs, Friction, Orbits: Reif Ch. 8, 12
Constrained Motion: Moore, N7-9
If you are having trouble with concepts or problem solving in chapter 5, you might find it helpful to see a different approach in the supplemental material:
Newtonian Mechanics: Reif Ch. 9-11
Force Diagrams: Moore, N3.7, 4.3, 4.4, 4.8
S5. Among the particles of high-energy physics are charged pions. They can be produced by bombarding a target in an accelerator with high-energy protons. The pions leave the target with speed close to that of light. The pions are radioactive, and when they are brought to rest, their half-life is measured to be 1.8x10-8 s. That is half of the number present at any time have decayed 1.8 x 10-8s later. In a collimated pion beam, of speed 0.99c, 50.0% of the pions have decayed by the time the beam has traveled to 38m from the target.
a) Are these results consistent if we ignore the effects of relativity? (Show reasoning.)
b) Show how the time dilation accounts for the measurements.
c) Show how the length contraction accounts for the measurements.
S6. Two spaceships, each of proper length 100m, pass near one another heading in opposite directions. An astronaut at the front of one ship (S) measures a time interval of 2.5x10-6 s for the second ship (S’) to pass.
a) What is the relative speed v of the two ships?
b) What time interval is measured on ship S for the front of ship S’ to pass from the front to the back of S?
S7. Observer S reports that an event occurred on his x axis at x = 3.0x108 m at a time t = 2.50s.
a) Observer S’ is moving in the direction of increasing x at a speed of 0.40c. What coordinates would he report for the event?
b) What coordinates would Observer S’ report if he were moving in the direction of decreasing x at this same speed?
S8. A clock moves along the x axis at a speed of 0.60c and reads zero as it passes the origin.
a) Use a Lorentz transform to determine the time it reads as it passes the 180m mark on this axis.
b) Explain why you can or cannot do this problem using the time dilation formula.
If you are having trouble with concepts or problem solving , you might find it helpful to see a different approach in the supplemental material:
Circular motion and relative motion: Reif, Ch. 8; Moore Unit N,
Ch. 9 and 10
S1. Vector description of circular motion.
Suppose we describe the position of a particle with a vector
R
= RcosQ i + RsinQ
j, where r is the distance of the particle from the origin of the reference
frame, Q is the angle measured counterclockwise
from the positive x axis, and i and j are the
unit vectors in the x and y directions respectively. If the particle
is moving in a circle with constant speed, the value of R remains constant,
and the angle Q changes linearly with time according
to Q = at.
Given a = 360o per second,
and R = 2m, make three plots:
Plot the x position of the particle vs. timeS2. A firecracker explodes 30 km away from an observer, who is sitting next to a clock A. The light from the firecracker explosion reaches the observer at exactly t = 0, according to clock A. Imagine that the flash of the explosion illuminates the face of another clock B which is sitting next to the firecracker. What time would clock B register at the moment of illumination, if it is correctly synchronized with clock A? Express your answer in milliseconds.
Plot the y position of the particle vs. time
Plot the trajectory of the particle (its path in the xy plane), labeling the positions corresponding to Q = 45o, 180o and 315o with the time at which each of those positions is reached. Draw arrows indicating the directions of the velocity vector and the acceleration vector at each of those three points as well.
S3. Imagine that you are in an inertial frame in empty space with a clock, a telescope and a powerful strobe light. A friend is sitting in the same frame a very large (unknown) distance from your clock. At precisely 12:00:00 noon, according to your clock, you set off the strobe lamp. Precisely 30.0 seconds later, you see in your telescope the flash of your friend’s clock illuminated by your strobe flash. How far away is your friend? What should you see on the face of your friend’s clock if that clock is synchronized with yours?
S4. Physical understanding of trajectories:
If you are having trouble with concepts or problem solving in chapters 3 and 4, you might find it helpful to see a different approach in the supplemental material:
Trig, vectors: Moore (Six Ideas, Unit C), p. 38, 54, 179-80
Calculus: Moore (Six Ideas, Unit N), p. 10, 54, 69-70
Projectile Motion: Reif, Ch. 7; Moore, Unit N, Ch. 11
If you are having trouble with concepts or problem solving in chapters 1 and 2, you might find it helpful to see a different approach in the supplemental material:
Graphical analysis of 1-dimensional Motion: Moore, Unit N, p.
51-52
Problem solving approaches for Motion in a Straight Line: Reif,
Chapters 4-5
Copyright (2000) University of Maryland, College Park.
All rights reserved. Permission to redistribute the contents without alteration
is granted to educational institutions for non-profit administrative or
educational purposes if proper credit is given to the University of Maryland,
College Park as the source.