TEST BANK FOR Elementary Mechanics and Thermodynamics By Prof. John W. Norbury (Solutions Manual)

SOLUTIONS MANUAL
for elementary mechanics &
thermodynamics
Professor John W. Norbury
Physics Department
University of Wisconsin-Milwaukee
P.O. Box 413
Milwaukee, WI 53201
November 20, 2000
Contents
1 MOTION ALONG A STRAIGHT LINE 5
2 VECTORS 15
3 MOTION IN 2 & 3 DIMENSIONS 19
4 FORCE & MOTION - I 35
5 FORCE & MOTION - II 37
6 KINETIC ENERGY & WORK 51
7 POTENTIAL ENERGY & CONSERVATION OF ENERGY 53
8 SYSTEMS OF PARTICLES 57
9 COLLISIONS 61
10 ROTATION 65
11 ROLLING, TORQUE & ANGULAR MOMENTUM 75
12 OSCILLATIONS 77
13 WAVES - I 85
14 WAVES - II 87
15 TEMPERATURE, HEAT & 1ST LAW OF THERMODYNAMICS
93
16 KINETIC THEORY OF GASES 99
3
Chapter 1
MOTION ALONG A
STRAIGHT LINE
5
6 CHAPTER 1. MOTION ALONG A STRAIGHT LINE
1. The following functions give the position as a function of time:
i) x = A
ii) x = Bt
iii) x = Ct2
iv) x = Dcos !t
v) x = E sin !t
where A;B;C;D;E; ! are constants.
A) What are the units for A;B;C;D;E; !?
B) Write down the velocity and acceleration equations as a function of
time. Indicate for what functions the acceleration is constant.
C) Sketch graphs of x; v; a as a function of time.
SOLUTION
A) X is always in m.
Thus we must have A in m; B in msec¡1, C in msec¡2.
!t is always an angle, µ is radius and cos µ and sin µ have no units.
Thus ! must be sec¡1 or radians sec¡1.
D and E must be m.
B) v = dx
dt and a = dv
dt. Thus
i) v = 0 ii) v = B iii) v = Ct
iv) v = ¡!D sin !t v) v = !E cos !t
and notice that the units we worked out in part A) are all consistent
with v having units of m¢ sec¡1. Similarly
i) a = 0 ii) a = 0 iii) a = C
iv) a = ¡!2Dcos !t v) a = ¡!2E sin !t
7
i) ii) iii)
x
t
v
a
x x
v v
a a
t t t
t t t
t t
C)
8 CHAPTER 1. MOTION ALONG A STRAIGHT LINE
iv) v)
0 1 2 3 4 5 6
t
-1
-0.5
0
0.5
1
x
0 1 2 3 4 5 6
t
-1
-0.5
0
0.5
1
x
0 1 2 3 4 5 6
t
-1
-0.5
0
0.5
1
v
0 1 2 3 4 5 6
t
-1
-0.5
0
0.5
1
v
0 1 2 3 4 5 6
t
-1
-0.5
0
0.5
1
a
0 1 2 3 4 5 6
t
-1
-0.5
0
0.5
1
a
9
2. The ¯gures below show position-time graphs. Sketch the corresponding
velocity-time and acceleration-time graphs.
t
x
t
x
t
x
SOLUTION
The velocity-time and acceleration-time graphs are:
t
v
t t
v
t
a
t
a
t
a
v
10 CHAPTER 1. MOTION ALONG A STRAIGHT LINE
3. If you drop an object from a height H above the ground, work out a
formula for the speed with which the object hits the ground.
SOLUTION
v2 = v2
0 + 2a(y ¡ y0)
In the vertical direction we have:
v0 = 0, a = ¡g, y0 = H, y = 0.
Thus
v2 = 0¡ 2g(0 ¡ H)
= 2gH
) v =
p
2gH
11
4. A car is travelling at constant speed v1 and passes a second car moving
at speed v2. The instant it passes, the driver of the second car decides
to try to catch up to the ¯rst car, by stepping on the gas pedal and
moving at acceleration a. Derive a formula for how long it takes to
catch up. (The ¯rst car travels at constant speed v1 and does not
accelerate.)
SOLUTION
Suppose the second car catches up in a time interval t. During that
interval, the ¯rst car (which is not accelerating) has travelled a distance
d = v1t. The second car also travels this distance d in time t, but the
second car is accelerating at a and so it's distance is given by