Download Waves Practice Problem Set - Southington Public Schools

ACTIVITY
BASED
PHYSICS
SHM, WAVES,
THERMODYNAMICS AND
E & M PRACTICE
PROBLEM SETS
W
SOUTHINGTON HIGH SCHOOL
CCP PHYSICS
SHM, WAVES, THERMODYNAMICS AND E & M
PRACTICE PROBLEM SET
TABLE OF CONTENTS
UNIT 7 7.1 7.2 7.3 7.4 UNIT 8 8.1 8.2 8.3 8.4 UNIT 9 9.1 9.2 9.3 9.4 UNIT 10 10.1 10.2 10.3 UNIT 11 11.1 SHM, WAVES AND SOUND ...................................................................................... 1 Simple Harmonic Motion .......................................................................................................... 1 Sound......................................................................................................................................... 4 Doppler Effect ........................................................................................................................... 5 SHM & Sound Review ............................................................................................................... 6 OPTICS ................................................................................................................. 13 Reflection and Mirrors ............................................................................................................ 13 Refraction and Lenses ............................................................................................................. 19 Optics General Review ........................................................................................................... 23 Practice Ray Diagrams ........................................................................................................... 25 THERMODYNAMICS ............................................................................................... 27 Temperature and Heat ............................................................................................................ 27 Humidity .................................................................................... Error! Bookmark not defined. Heat Engines ........................................................................................................................... 32 General Thermodynamics Review .......................................................................................... 33 CHARGE, FIELDS, CURRENT POTENTIAL AND DC CIRCUITS ............................. 35 Coulomb’s Law ..................................................................................................................... 35 Equivalent Resistance and Circuit Analysis ......................................................................... 38 Electricity Review ................................................................................................................. 40 MAGNETISM ..................................................................................................... 42 Magnetism ............................................................................................................................. 42 MECHANICS PRACTICE PROBLEM SETS
1
Unit 7 SHM, WAVES AND SOUND
7.1
SIMPLE HARMONIC MOTION
7.1.1
Mass-Spring Systems
1. An ultrasonic transducer used for medical diagnosis oscillates with a frequency of 6.7
x 106 Hz. How much time does each oscillation take and what is the angular
frequency?
Answer: T=1.5 x 10-7 s;
ω =4.21 x 107rad/s
2. A body of unknown mass is attached to an ideal spring that is mounted horizontally
with its left end held stationary. The spring constant of the spring is 120 N/m and it
vibrates with a frequency of 6.00 Hz. Assuming that there is no friction, find:
a. The Period
Answer: 0.17 sec
b. The Angular Frequency
Answer: 37.7 rad/s
c. The Mass of the body
Answer: 0.084 kg
3. A spring stretches 0.200 m when a 0.600 kg mass is hung from it. The spring is then
stretched 0.150 m from this equilibrium point and released. Find:
a. The spring constant (k)
Answer: 29.4 N/m
b. The amplitude
Answer: 0.150 m
c. The total energy of the system.
Answer: 0.33 J
d. Maximum velocity (vo)
Answer: 1.05 m/s
e. The velocity when the mass is 0.050 m from equilibrium. Answer: .99 m/s
f. The equation for the position of the mass x (t) Answer: x (t) = 0.150cos(7t)
4. A proud deep-sea fisherman hangs a 65.0 kg fish from an ideal spring with a
negligible mass. The fish stretches the spring 0.120m. What is the period of
oscillation of the fish if it is pulled down and released? Answer: 0.70 sec
5. A 0.150 kg toy is undergoing simple harmonic motion on the end of a horizontal
spring with a spring constant of 300 N/m. When the object is 0.12m from
equilibrium, it has a speed of 0.300 m/s. Find:
a. The Total energy of the system
Answer: 2.17 J
b. The Velocity of the object at equilibrium
Answer: 5.4 m/s
c. The Amplitude of the motion.
Answer: 0.120 m
7.1.2
Simple Pendulum
1. What is the period of a pendulum at sea level with a length of 1.5 m?
Answer: 2.45 sec
a. What would the period of this pendulum be if the length were shortened to a
¼ of the original length?
Answer: 1.23 sec
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2. How long would a pendulum have to be if it has a frequency of 2 Hz?
Answer: 0.062 m
3. A simple pendulum with a bob mass of 0.5 kg and a string length of 1.0 m and is
taken to the moon, which has an acceleration due to gravity that is 1/6 that of the
acceleration due to gravity on Earth. If you want to maintain the same period on
Moon that you have on Earth, what adjustment will you have to make to the…
a. Mass of the pendulum bob?
Answer: None
b. Length of the string?
Answer: Shorten to 0.166 m
7.1.3
SHM Review Problems
1. (G1) When a 65-kg person climbs into a 1000-kg car, the car’s springs compress
vertically by 2.8 cm. What will be the frequency of vibration when the car hits a
bump? Ignore damping.
Answer: 0.74 Hz
2. (G7) A balsa wood block of mass 50 g floats on a lake, bobbing up and down at a
frequency of 2.5 Hz.
a. What is the value of the effective spring constant of the water?
Answer: 12 N/m
b. A partially filled water bottle of mass 0.25 kg and almost the same size and
shape of the balsa block is tossed into the water. At what frequency would you
expect the bottle to bob up and down? Assume SHM. Answer: 1.1 Hz
3. (G9) A 0.50-kg mass at the end of a spring vibrates 3.0 times per second with an
amplitude of 0.15 m. Determine:
a. The velocity when it passes the equilibrium point. Answer: 2.8 m/s
b. The velocity when it is 0.10 m from equilibrium. Answer: 2.1 m/s
c. The total energy of the system.
Answer: 2.0 J
d. The equation describing the motion of the mass. Assume φ = 0.
Answer: (0.15 m) cos [2π (3.0 Hz)t]
4. (G13) It takes a force of 80.0 N to compress the spring of a toy popgun 0.200 m to
“load” a 0.150-kg ball. With what speed will the ball leave the gun?
Answer: 10.3 m/s
5. (G15) A mass sitting on a horizontal, frictionless surface is attached to one end of a
spring; the other end of the spring is fixed to a wall. 3.0 J of work is required to
compress the spring by 0.12 m. If the mass is released from rest with the spring
compressed, it experiences a maximum acceleration of 15 m/s2. Find the value of:
a. The spring constant.
Answer: 4.2 x 102 N/m
b. The mass.
Answer: 3.3 kg
6. (G17) A 0.50-kg mass vibrates according to the equation x=0.45cos(8.40t), where x is
in meters and t is in seconds. Determine:
a. The amplitude.
Answer: 0.45 m
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b. The frequency.
Answer: 1.34 Hz
c. The total energy.
Answer: 3.6 J
d. The kinetic energy and potential energy when x = 0.30 m.
Answer: KE=2.0 J; PE=1.6 J
7. (G21) A 25.0-g bullet strikes a 0.600-kg block attached to a fixed horizontal spring
whose spring constant is 6.70 x 103 N/m and sets it into vibration with an amplitude
of 21.5 cm. What was the speed of the bullet before impact if the two objects move
together after impact?
Answer: 557 m/s
8. (G29) You want to build a grandfather clock with a pendulum (a weight on the end of
a light cable) that has one second between its “tick” (swinging to) and its “tock”
(swinging fro). How long do you make the cable?
Answer: 0.993 m
9. (G30) What is the period of a simple pendulum 50 cm long on Earth?
Answer: 0.61 m
10. (G31) The length of a simple pendulum is 0.66 m, the pendulum bob has a mass of
310 grams, and it is released at an angle of 12° to the vertical.
a. With what frequency does it vibrate? Assume SHM. Answer: 0.61 Hz
b. What is the pendulum bob’s speed when it passes through the lowest point of
the swing?
Answer: 0.53 m/s
c. What is the total energy stored in this oscillation, assuming no loses?
Answer: 0.044 J
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7.2
4
SOUND
Unless stated otherwise, assume that T = 20°C.
1. Find the frequency of a sound wave moving in air at a temperature of 22°C with a
wavelength of 0.667 m.
Answer: f = 516 Hz
2. You hear the sound of the firing of a distant cannon 6.0 seconds after seeing the flash.
How far are you from the cannon?
Answer: d = 2.1 km
3. A sound wave with a frequency of 9800 Hz travels along a copper pipe. If the
wavelength is 0.370 m and the density of copper is 8.9 x 103 kg/m3, what is the elastic
modulus of copper?
Answer: E= 1.2 x 1011 N/m2
4. A certain instant camera determines the distance to the subject by sending out a sound
wave and measuring the time needed for the echo to return to the camera. How long
will it take the sound wave to return to the camera if the subject were 3.00 m away?
Answer: t = 0.017 s
5. If you drop a stone into a mineshaft that is 122.5 m deep, how soon after you drop the
stone do you hear it hit the bottom of the shaft? The temperature in the mineshaft is
10°C.
Answer: t = 5.36 s
6. With what tension must a rope of length 2.50 m and mass of 0.120 kg be stretched for
transverse waves of frequency of 40.0 Hz to have a wavelength of 0.750 m?
Answer: T= 43.2 N
7. A ship uses a sonar system to detect underwater objects in the ocean. The system
emits underwater sound waves and measures the time interval for the reflected wave
to return to the detector.
a. Determine the speed of sound in seawater if the bulk modulus is 2.2 x 109
N/m2.
Answer: v= 1465 m/s
b. The ship is on the continental shelf when it emits a sound wave. It takes the
echo 0.203 seconds to be picked up by the detector. What is the depth of the
ocean on the continental shelf?
Answer: depth = 150 m.
8. A tuning fork of frequency 262 Hz is sounded at the same time as another tuning fork
with a frequency of 257 Hz. What is the beat frequency that is heard?
Answer: fb = 5 Hz
9. A tuning fork with a frequency of 432 Hz is sounded at the same time as a guitar. If 6
beats are heard in 3 seconds, what are the possible frequencies of the guitar string?
Answer: f = 430 Hz or 434 Hz
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7.3
5
DOPPLER EFFECT
1. On a cold wintry day, Bob is late for work. He drives at a speed of 15 m/s toward the
factory where he works. The factory whistle is blown with a frequency of 800 Hz to
indicate the start of the workday. If it is -4°C…
a. What is the frequency that Bob hears when the whistle is blown?
Answer: f’= 837 Hz
b. What is the frequency that Bob hears when he passes the building and moves
away toward the parking lot behind the factory? Answer: f’= 763 Hz
2. While standing near a railroad crossing, a person hears a distant train horn. According
to the train’s engineer, the frequency of the horn is 262 Hz. If the train is traveling at
20.0 m/s toward the crossing and the speed of sound is 346 m/s…
a. What would the train horn’s wavelength be at rest?
Answer: λ = 1.32 m
b. By how much would the horn’s wavelength change as a result of the train’s
motion?
Answer: Δλ = 0.075 m
3. An ambulance with a siren emitting a whine at 1300 Hz races by a car that was pulled
off to the side of the road. After being passed, the driver of the park car hears a
frequency of 1220 Hz. How fast was the ambulance moving?
Answer: vs = 22.5 m/s
4. In 1845, French Scientist B. Ballot first tested the Doppler shift. He had a trumpet
player sound an A, 440 Hz, while riding on a flatcar pulled by a locomotive. At the
same time, a stationary trumpeter played the same note. If the locomotive was
moving toward Ballot at a speed of 5.0 m/s, what beat frequency would Ballot have
heard?
Answer: fb = 6.5 Hz
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7.4
6
SHM & SOUND REVIEW
7.4.1
Sound
1. (G11.35) A sound wave in air has a frequency of 262 Hz and travels at a speed of 330
m/s. How far apart are the wave crests (compressions)? Answer: 1.26 m
2. (G11.37) Calculate the speed of longitudinal waves in:
a. Water
Answer: 1.4 x 103 m/s
b. Granite
Answer: 4.1 x 103 m/s
c. Steel
Answer: 5.1 x 103 m/s
3. (G11.39) A cord of mass 0.55 kg is stretched between two supports 30 m apart. If the
tension in the cord is 150 N, how long will it take a pulse to travel from one support
to the other?
Answer: 0.33 sec
4. (G11.41) A sailor strikes the side of his ship just below the surface of the sea. He
hears the echo of the wave reflected from the ocean floor directly below 3.0 s later.
How deep is the ocean at this point?
Answer: 2.3 km
5. (G12.1) A hiker determines the length of a lake by listening for the echo of her shout
reflected by the cliff at the far end of the lake. She hears the echo 1.5 s after shouting.
Estimate the length of the lake.
Answer: 2.6 x 102 m
6. (G12.5) A person sees a heavy stone strike the concrete pavement. A moment later
two sounds are heard from the impact: one travels in the air and the other in the
concrete, and they are 1.4 sec apart. How far away did the impact occur?
Answer: 5.4 x 102 m
7. (G12.6) A fishing boat is drifting just above a school of tuna on a foggy day. Without
warning, an engine backfire occurs on another boat 1.0 km away. How much time
elapses between the instant when the backfire is heard…
a. by the fish?
Answer: 0.69 sec
b. by the fishermen?
Answer: 2.9 sec
8. (G12.7) The sound from a very high burst of fireworks takes 4.5 s to arrive at your
eardrums. The burst occurred 1500 m above you and traveled vertically through two
stratified layers of air, the top one at 0°C and the bottom one at 20°C. How thick is
each layer of air?
Answer: 1200 m, 300 m
9. (G12.43) What will be the “beat frequency” if middle C (262 Hz) and C# (277 Hz)
are played together?
Answer: 15 Hz
a. What if each was played two octaves lower (each frequency reduced by a
factor of 4)
Answer: 3.8 Hz
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10. (G12.51) The predominant frequency of a certain police car’s siren is 1800 Hz when
at rest. What frequency do you detect if you move with a speed of 30.0 m/s…
a. Toward the car
Answer: 1950 Hz
b. Away from the car
Answer: 1640 Hz
11. (G12.53) In one of the original Doppler experiments, one tuba was played on a
moving platform car at a frequency of 75 Hz, and a second identical one was played
on the same tone while at rest in the railway station. What beat frequency was heard
if the train approached the station at a speed of 10.0 m/s?
Answer: 2 Hz
12. (G12.78) The frequency of a steam train whistle as it approaches you is 522 Hz. After
it passes you, its frequency is measured as 486 Hz. How fast was the train moving?
Assume constant velocity.
Answer: 12.3 m/s
13. (G12.71) A tight guitar string has a frequency of 600 Hz as its third harmonic. What
will be its fundamental frequency if it is fingered at a length of only 60% of its
original length?
Answer: 333 Hz
14. (G12.73) The string of a violin is 32 cm long between fixed points with a
fundamental frequency of 440 Hz and a linear density of 5.5 x 10-4 kg/m.
a. What are the speed and tension in the string?
Answer: 2.8 x 102 m/s, 44 N
b. What is the frequency of the first overtone?
Answer: 880 Hz
Density of Substances
Material
Iron, cast
Steel
Brass
Aluminum
Concrete
Brick
Marble
Granite
Wood (pine)
Parallel to Grain
Perpendicular to
Grain
Nylon
Bone (limb)
Material
Water
Mercury
Alcohol (ethyl)
Air
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Elastic
Modulus
100 x 109 N/m2
200 x 109 N/m2
100 x 109 N/m2
70 x 109 N/m2
20 x 109 N/m2
14 x 109 N/m2
50 x 109 N/m2
45 x 109 N/m2
10 x 109 N/m2
1 x 109 N/m2
5 x 109 N/m2
15 x 109 N/m2
Bulk Modulus
2.0 x 109 N/m2
2.5 x 109 N/m2
1.0 x 109 N/m2
1.01 x 105 N/m2
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7.4.2
8
Mass-Spring System
1. When a family of four with a total mass of 200 kg gets into their 1200 kg car, the
car’s springs compress 5 cm. What is the spring constant of the car’s springs
assuming they act as one single spring?
Answer: 39,200 N/m
2. Imagine that you videotape the motion of a mass attached to a spring and measure the
displacement x from the equilibrium position as a function of time t. When you plot
position, velocity and acceleration as a function of time you get the following graphs.
a. Find the amplitude.
b. Find the period.
c. Find the angular frequency.
Answer: ω = 4 rad/s
d. Find the magnitude of the maximum velocity.
e. Find the magnitude of the maximum
acceleration.
3. When a 0.50 kg-object is attached to a vertically supported spring, it stretches 0.10 m.
It is then pulled another 0.10 m from its new equilibrium position. Find the period,
angular frequency, total energy, and displacement equation for this mass-spring
system.
Answer: T = 0.63 sec,
ω = 9.9 rad/s,
ET= .25 J,
x(t) = 0.10 cost (9.9t)
4. An object with mass m = 0.60 kg attached to a spring with k = 10 N/m vibrates back
and forth along a horizontal frictionless surface. If the amplitude of the motion is
0.050 m, what is the velocity of the object when it is 0.010 m from the equilibrium
position?
Answer: 0.20 m/s
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7.4.3
9
Simple Pendulum
1. What length of string would be necessary to make a simple pendulum with a period
of 10.0 s?
Answer: 24.8 m
2. On a planet with an unknown value of g, the period of a 0.65 m long pendulum is 2.8
s. What is g for the planet?
Answer: 3.27 m/s2
3. A pendulum bob of mass m attached to a string of length L vibrates back and forth
along a circular arc. (a) Draw a free-body diagram for the bob showing all the forces
acting on it. (b) What is the frequency of its motion using the symbols provided and
universal constants? (c) What is the total energy of the pendulum at its highest point
using the symbols provided and universal constants?
Answer:
a. See notes for FBD
L
b.
c. E = mg(L-LcosΘ )
7.4.4
General Problems
1. Given x(t) = 0.01 m cos(0.02π t - π/2). Find (a) the amplitude, (b) the period, (c) the
frequency, and (d) the initial phase of the motion.
Answer: xm = 0.01m,
T = 100 s, f = 0.01 Hz,
φ = π /2 = equilibrium
2. A particle is executing simple harmonic motion. The displacement x as a function of
time t is shown in the figure below. Find (a) the period, (b) amplitude,(c) equation of
motion, (d) maximum velocity and (e) maximum acceleration.
Answer: T = 4.00 s, xm = 10 cm
x(t) = .10 m cos(1.57t)
vmax = 0.16 m/s, amax = 0.25 m/s2
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3. A mass-less spring with spring constant k = 10.0 N/m is attached to an object of mass
m = 0.300 kg. One third of the spring is cut off. What is the frequency of the
oscillations when the "new" spring-mass is set into motion?
Answer: f = 0.92 Hz
4. Figure 6 below is a plot of the potential energy of a mass-spring system. The total
mechanical energy ET of the system = 0.200 J. Find (a) the potential energy PE and
(b) the kinetic energy KE at x = 0.025 m. Find (c) the spring constant k and (d) the
speed of the particle when x = 0.025 given that the mass of the object m = 0.30 kg.
Find (e) the amplitude of the motion and (f) the maximum velocity of the object.
Answer: PE = 0.050 J, KE = 0.15 J, k = 160 N/m, v = 1 m/s, xm = 0.0500 m, vmax = 1.15 m/s
5. The motion of a particle is given by x(t) = 4.0 cm cos(πt - π/6). Find the particle’s
velocity when x = 2.0 cm.
Answer: v = -10.9 cm/s
6. A wave travels along a stretched rope. The wavelength is 2.0 m. The wave period is
0.1 s. What is the speed of this wave?
Answer: 20 m/s
7. A wave has a frequency of 58 Hz and a speed of 31 m/s. What is the wavelength of
this wave?
Answer: 0.53 m
8. A clothesline with a mass of 0.750 kg is 3.00 meters long. How much tension do you
have to apply to produce the observed a wave speed of 12.0 m/s? Answer: 36 N
9. Determine the wavelength of a 6000-Hz sound wave traveling in Alcohol.
Answer: 0.19 m
10. A sound wave produced by a clock chime is heard 501 meters away, 1.50 sec later.
a. What is the temperature of the air through which the sound travels?
Answer: T = 5°C
b. How long would it take this sound to travel 501 meters away in freshwater?
Answer: t = 0.35 s
11. If you clap your hands and hear the echo from a distant wall 0.20 seconds later, how
far away is the wall?
Answer: d = 34.3 m
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12. Billy Bob Joe whistles at 441 Hz and Mary Jane whistles at 462 Hz. What beat
frequency does Mary Jane hear?
Answer: fb = 21 Hz
13. A railroad train is traveling at 25.0 m/s in still air. The frequency of the note emitted
by the locomotive whistle is 400 Hz.
a. What is the frequency heard by a stationary observer standing in front of the
locomotive?
Answer: f’ = 431 Hz
b. What is the frequency heard by a stationary observer standing behind the
locomotive?
Answer: f’ = 373 Hz
c. The train comes to a full stop at the nearest station so that new passengers can
get onto the train. When it has come to rest, it blows its whistle at another
train coming toward the station at 10 m/s. What frequency does a passenger
sitting on the moving?
Answer: f’ = 412 Hz
14. A train moving at a constant speed is passing a stationary observer on a platform. On
one of the train cars, a flute player is continually playing a note known as concert A,
which has a frequency of 440 Hz. After the flute has passed, the observer hears the
sound as a G, which has a frequency of 392 Hz. What is the speed of the train?
Answer: 42 m/s
15. A guitar string has a fundamental frequency of 256 Hz. What is the frequency of the
3rd harmonic?
Answer: 768 Hz
a. What would the length of the guitar string have to be to produce a transverse
wave in the string with a speed of 405 m/s at the fundamental frequency?
Answer: 0.79 m
16. A string that is 2.00 meters long and has a mass of .0025 kg.
a. If the fundamental frequency is 120 Hz, what are the frequencies of the first
four harmonics? Answer: 120 Hz, 240 Hz, 360 Hz, 480 Hz
b. What must the tension be for the string to vibrate at the 4th harmonic?
Answer: 288 N
17. The speed of sound in a certain metal block is 3.00 x 103 m/s. The graph shows the
amplitude in meters of a wave traveling through the block versus time in
milliseconds. What is the wavelength of this wave?
Answer: 6 m
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18. A piano tuner stretches a steel piano wire with a tension of 800 N. The steel wire is
0.400 m long and has a mass of 3.00 grams.
a. What is the frequency of its fundamental mode of vibration?
Answer: 408 Hz
b. What is the number of the highest harmonic that could be heard by a person
who is capable of hearing frequencies up to 10,000 Hz?
Answer: 24th harmonic
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Unit 8 OPTICS
8.1
REFLECTION AND MIRRORS
8.1.1
Plane Mirrors
1. Draw the reflected light ray(s) and position of the observer’s eye where it can see
the reflected ray.
2. A bulb is placed in front of a plane mirror.
a. Use a ruler and a protractor to construct four rays that travel from the bulb
to the mirror and reflect. Include eyes at positions that could see the
reflected rays.
b. Extend the reflected rays with dotted lines behind the mirror to locate the
virtual image.
c. Measure and compare the image distance to the object distance.
3. The ray diagram below shows where Observer 1 sees the virtual image of the
bulb. Show where, if at all, Observer 2 sees the virtual image.
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4. The ray diagram below shows where Observer 1 sees the virtual image of the
bulb. Show where, if at all, Observer 2 sees the virtual image.
5. The ray diagram below shows where Observer 1 sees the virtual image of the
bulb. Show where, if at all, Observer 2 sees the virtual image.
6. A top view of a mirror and an arrow is shown below.
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a. Draw a ray diagram that shows how light from both ends of the arrow
reach the observer.
b. Locate and sketch the image of the arrow.
7. How does the size of the smallest mirror you would need to see your entire body
at one time compare to your height? Make a ray diagram to prove it. Mr. Eye-foot
represents a simplified body.
8. Would the length of the mirror needed to see your entire body change if you
moved farther away from the mirror? Draw a ray diagram to support your answer.
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9. One of Cinderella’s stepsisters, who is 1.60 m tall, is looking at herself in a plane
mirror when she spots one of the mice helping Cinderella at the very bottom of
the mirror. If the mouse is 5.0 cm away from the wall where the mirror is hung,
and the mirror’s edge is 60 cm from the floor, how far away is she from the
mouse? Make sure to draw a ray diagram to help solve this problem!
Answer: 3.3 cm from the mouse
8.1.2
Spherical Mirrors
1. An action figure that is 8.0 cm tall is placed 23.0 cm in front of a convergent
mirror with a focal length of 10.0 cm.
a. Draw a ray diagram for the image of the action figure.
b. What are the three characteristics of the image?
Answer: Inverted, M<1, Real
c. What is the image distance?
Answer: 17.7 cm
d. What is the image height?
Answer: -6.15 cm
2. If the 8.0 cm tall action figure from Question 1 is now placed 6.0 cm in front of a
divergent mirror with a radius of 24.0 cm.
a. Draw a ray diagram for the image of the action figure.
b. What are the three characteristics of the image?
Answer: Upright, M<1, Virtual
c. What is the image distance?
Answer: -4.0 cm
d. What is the image height?
Answer: 5.3 cm
3. The focal point of a divergent mirror is 20.0 cm behind the mirror. An object is
placed 12 cm from the mirror.
a. Draw a ray diagram for the image of the object.
b. What is the image distance?
Answer: -7.5 cm
c. What is the magnification of the image?
Answer: M = 0.625 x
4. A magnified, inverted image is located a distance of 32.0 cm from a spherical
mirror with a focal length of 12.0 cm.
a. Is it a convergent or divergent mirror? Explain.
b. Is the image real or virtual?
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c. How far was the object placed in front of the mirror? Answer: 19.2 cm
5. An inverted image has a magnification of 2 when the object is placed 22 cm from
a convergent mirror. What are the image distance and the focal length of the
mirror?
Answer: di = 44 cm
f = 14.7 cm
8.1.3
Reflection and Mirror Review
1. (G1) Suppose that you want to take a photograph of yourself as you look at your
image in a flat mirror 1.5 m away. For what distance should the camera lens be
focused?
Answer: 3.0 m
2. (G4) A person whose eyes are 1.62 m above the floor stands 2.10 m in front of a
vertical plane mirror whose bottom edge is 43 cm above the floor, as shown. What is
the horizontal distance x to the base of the wall supporting the mirror of the nearest
point on the floor that can be seen reflected in the mirror?
Answer: 76 cm
3. (G5) Two mirrors meet at a 135° angle. If light rays strike one mirror at 40° as
shown, at what angle do they leave the second mirror? Answer: 5°
4. (G9) A solar cooker, really a convergent mirror pointed at the sun, focuses the Sun’s
rays 17.0 cm in front of the mirror. What is the radius of the spherical surface from
which the mirror was made?
Answer: 34.0 cm
5. (G12) How far from a convergent mirror (radius 27.0 cm) must an object be placed if
its image is to be at infinity?
Answer: 13.5 cm
6. (G12) If you look at yourself in a shiny Christmas tree ball with a diameter of 9.0 cm
when your face is 30.0 cm away from it, where is your image located?
Answer: -2.09 cm
a. What is the image’s magnification?
Answer: +0.070
b. Is it real or virtual? Is it upright or inverted?
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7. (G13) A mirror at an amusement park shows an upright image of any person who
stands 1.3 m in front of it. If the image is 3x the person’s height, what is the radius of
curvature?
Answer: 3.9 m
8. (G15) Some rearview mirrors produce images of cars to your rear that are a bit
smaller than they would be if the mirror were flat. Is this mirror convergent or
divergent?
a. What type of image is produced?
b. What would the height of the image be for a car that was 1.3 m high and 15.0
m behind you, assuming the mirror’s radius of curvature is 3.2 m?
Answer: 0.13 m
9.
(G17) Where should an object be placed in front of a convergent mirror so that it
produces an image at the same location?
Answer: At C
a. Is the image real or virtual?
b. Is the image inverted or upright?
c. What is the magnification of the image?
Answer: -1
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REFRACTION AND LENSES
8.2.1
Refraction
1. Calculate the index of refraction for a clear plastic material if the velocity of light in
that material is 2.5 x 108 m/s.
Answer: 1.20
a. A ray of light in air is incident at an angle of 40.8° on the surface of this clear
plastic material. What is the angle of refraction in the plastic? Answer: 33°
2. A ray of light passes from kerosene to crown glass (n = 1.52) at an angle of incidence
of 45.2°. If the angle of refraction in the glass is measured to be 41°, what is the index
of refraction for kerosene?
Answer: 1.39
3.
A ray of light passes from air into a glass prism at an angle of incidence of 35°. If the
angle of refraction in the glass is 23.7°, what is the speed of light in the glass?
Answer: 2.1 x 108 m/s
4. What is the critical angle for a light ray passing into air from polystyrene that has an
index of refraction of 1.60?
Answer: 38.7°
5. A light source is located 2.0 m below the surface of a swimming pool and 1.5 m from
the edge. The pool is filled to the top with water.
a. At what angle does the light reaching the edge of the pool leave the water?
Answer: 53.0° from the normal
b. Does this cause the light viewed from this angle to appear deeper or shallower
than it actually is. Explain your answer using a diagram!
6. A fiber optic cable (n=1.50) is submerged in water (n=1.33). What is the critical angle
for the light to stay inside the cable?
Answer: 62.4°
7. A certain kind of glass has an index of refraction of 1.65 for blue light and an index
of refraction of 1.61 for red light. If white light (containing all colors) is incident on
the glass at an angle of 30°, what is the angle between the red and blue light inside the
glass?
Answer: 0.45°
8. A laser is incident hits a glass prism that is the shape of an equilateral triangle as
shown below. Draw a ray diagram to show what happens to the light ray when it
interacts with the prism.
a. Calculate the angle at which the laser beam leaves the prism.
Answer: 75°
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Thin Lenses
1. An object 8.0 cm high is 80.0 cm in front of a converging lens that has a focal length
of 25 cm.
a. Draw a ray diagram for the described situation.
b. What are the three characteristics of the image?
c. What is the image distance?
d. What is the image height?
Answer: Inverted, M<1, Real
Answer: 36 cm
Answer: -3.6 cm
2. A light 10.0 cm high is placed 60.0 cm in front of a diverging lens that has a focal
length of 20.0 cm.
a. Draw a ray diagram for the described situation.
b. What are the three characteristics of the image?
Answer: Upright, M<1, Virtual
c. What is the image distance?
Answer: -15.0 cm
d. What is the image height?
Answer: 2.5 cm
3. An object is placed in front of a converging lens with a focal length of 10.0 cm such
that a clear image appears on a screen 5.1 m away.
a. How far away from the lens was the slide placed?
Answer: 10.2 cm
b. If the object’s height is 12.5 mm, what is the magnification of the image on
the screen?
Answer: M = -50 x
4. A lens forms an image of an object that is 16.0 cm from the lens. The image is 12.0
cm from the lens on the same side as the object.
a. What is the focal length of the lens?
Answer: -48 cm
b. Is it a convergent or divergent lens? Explain.
c. If the object is 8.50 mm, how tall is the image? Answer: 6.38 mm
d. Draw a ray diagram for the described situation.
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5. A photographic slide is to the left of a lens. The lens projects an image of the slide
onto a wall 6.00 m to the right of the slide. The image is -80.0 times the size of the
slide.
a. How far is the slide from the lens?
Answer: 7.4 cm
b. What is the focal length of the lens?
Answer: 7.3 cm
c. Is it a convergent or divergent lens? Explain.
6. An object 8.0 cm high is placed 12.0 cm to the left of a converging lens of focal
length 8.0 cm. Find the position and height of the image produced by this lens.
Answer: di = 24 cm; hi = -16 cm
a. A second converging lens of focal length 6.0 cm is placed 36.0 cm to the right
of the first lens. Both lenses have the same optic axis. If the image from the
first lens acts as the object for the second lens, find the position and height of
the image produced by the second lens.
Answer: di = 12 cm from second lens; hi = 16 cm
b. What are the three characteristics of the final image compared to the original
object?
Answer: upright, M>1, Real
8.2.3
Refraction and Lenses Review
1. (G27) The speed of light in ices is 2.29 x 108 m/s. What is the index of refraction of
ice?
Answer: 1.31
2. (G33) Rays of the Sun are seen to make a 21.0° angle to the vertical beneath the
water. At what angle above the horizon is the Sun?
Answer: 61.5°
3. (G35) In searching the bottom of a pool at night, a
watchman shines a narrow beam of light from his
flashlight, 1.3 m above the water level, onto the surface of
the water at a point 2.7 m from his foot at the edge of the
pool. Where does the spot of light hit the bottom of the
pool, relative to the edge, if the pool is 2.1 m deep?
Answer: 4.6 m
4. (G37) An aquarium filled with water has flat glass sides
whose index of refraction is 1.52. A beam of light from
outside the aquarium strikes the glass at a 43.5° angle to
the perpendicular.
a. What is the angle of this light ray when it enters
the glass? Answer: 26.9°
b. Then what is the angle of this light ray when it
enters the water? Answer: 31.2°
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c. What would be the refracted angle if the ray entered the water directly?
Answer: 31.2°
5. (G41) The critical angle for a certain liquid-air surface is 44.7°. What is the index of
refraction of the liquid?
Answer: 1.42
6. (G49) Sunlight is observed to focus at a point 18.5 cm behind a lens.
a. What kind of lens is it?
b. What is its power in diopters?
Answer: 5.41 D
7. (G50) A certain lens focuses an object 2.25 m away as an image 48.3 cm on the other
side of the lens. What type of lens is it and what is its focal length? Answer: 0.13 m
a. What is the magnification of the image?
Answer: 0.13 m
8. (G53) The Sun’s diameter is 1.4 x 106 km and it is 1.5 x 108 km away. How large is
the image of the Sun on the film used in a camera with…
a. a 28-mm focal length lens?
Answer: -0.26 mm
b. a 50-mm focal length lens?
Answer: -0.47 mm
c. a 200-mm focal length lens?
Answer: -1.9 mm
9. (G55) A -6.0 diopter is held 14.0 cm from an ant 1.0 mm high. What is the position,
type, and height of the image?
Answer: -7.6 cm, 0.54 mm
10. (G57) How far from a 50.0 mm focal length lens must an object be placed if its
image is to be magnified 2.00x and be real?
Answer: 75.0 mm
a. What if its image is to be virtual and magnified 2.00x? Answer: 25.0 mm
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OPTICS GENERAL REVIEW
1. A laser beam strikes a horizontal plane mirror at an angle of 36° to the horizontal. It
then can be seen as a spot on a vertical screen that is placed 15 m away from the point
of incidence. How high up along the screen is the spot located? Answer: 10.9 m
2. An object 5.0 cm tall is placed in front of a convergent mirror of focal length 4.0 cm.
If the object is 12.0 cm from the mirror, find the image position and image height. Is
the image upright or inverted? Is it real or virtual? Check your results by completing a
ray diagram. Answer: di = 6.0 cm; hi = -2.5 cm
3. A divergent spherical mirror forms a virtual image that is 0.8 times the size of the
object. If the image is 20 cm behind the mirror, determine:
a. The position of the object.
Answer: do = 25 cm
b. The radius of curvature of the mirror.
Answer: r = -200 cm
4. The refractive index of a certain type of glass is 1.55. What is the speed of light in
this type of glass?
Answer: v = 1.94 x 108 m/s
5. A narrow beam of light is incident on a diamond at an angle of 52°. What is the angle
of refraction?
Answer: θ 2 = 19°
6. What is the critical angle for a light ray hitting an interface with an index of refraction
of 1.49 on the incident side and an index of refraction of 1.37 on the refracted side?
Answer: θ c = 67°
7. A convergent lens with a focal length of 6.0 cm is held 4.0 cm from an insect that is
0.50 cm tall.
a. Where is the image located?
Answer: di = -12 cm
b. How tall will the insect appear to be?
Answer: hi = 1.5 cm
c. Is the image real or virtual?
d. Is the image upright or inverted?
8. A diverging lens has a focal length of 10 cm. Where should a 3.0 cm tall object be
placed to produce an image 5.0 cm from the lens?
Answer: do = 10 cm
a. What is the height of the image?
Answer: hi = 1.5 cm
b. Is it upright or inverted?
9. Two plane mirrors are inclined at an angle with each other. A ray of light is incident
on the first mirror at an angle of 25° and reflects toward the second mirror. The ray of
light reflects off the second mirror at an angle of 55° with respect to the mirror itself.
What is the angle between the two mirrors?
Answer: 60°
10. Baldwin Young stands 68 cm from his dresser mirror, inspecting his scalp. How far is
the image of his scalp located from his actual scalp?
Answer: d = 136 cm
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11. A person measures the width of a swimming pool to be 4.50 meters. This same
person notices that the bottom edge of the pool is just visible at an angle of 12.0°
above the horizontal. Knowing that the nwater = 1.33 and nair = 1.33, calculate the depth
of the pool.
Answer: depth = 4.15 m
4.50 m
12. The critical angle of a certain piece of plastic in air is 30°. What is the critical angle if
the plastic is immersed in water?
Answer: θ 2 = 42°
13. In the Fall 2006, the Sky Mirror sculpture was opened in Rockefeller Center in New
York City. Standing three stories tall and weighing 23 tons, its convergent side faced
the Rockefeller Center and its divergent side faced Fifth Avenue.
a. A taxi on Fifth Avenue is located 38 m from the divergent side of the
sculpture and its image is one-fifth the size of the taxi. Determine the focal
length of the mirror.
Answer: f = -9.5 m
b. What is the image size and image distance of the 260-m tall Rockefeller
Center if it is located an estimated distance of 95 meters from the convergent
mirror surface. Assume the focal length of the two sides have the same
magnitude.
Answer: di = 11 m; hi = -29 m
14. The widest cinema screen in the world was reportedly constructed in New Zealand in
2007. The screen is 30.6 meters (100 feet) wide. Images from a 35-mm wide film are
projected onto this screen. Suppose that the screen in the theater is located a distance
of 46 m from the projector. Determine the magnification of the image and the focal
length of the lens system.
Answer: M = -870; f = 53 mm
15. The divergent lens has a focal point that is located 17.8 cm from the lens. A virtual
image is produced on the same side of the lens as the object at a distance of 7.23 cm
from the lens. Determine the object distance.
Answer: do = 12.2 cm
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25
PRACTICE RAY DIAGRAMS
Provide are some examples of ray diagrams that you can copy and practice your
geometric optics.
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Unit 9 THERMODYNAMICS
9.1
TEMPERATURE AND HEAT
1. Find the Celsius and Kelvin temperatures corresponding to:
a. A winter night in Seattle (41°F)
Answer: 5°C, 278 K
b. A hot summer day in Palm Springs (107.0°F) Answer: 41.67°C, 314.8 K
c. A cold winter day in northern Manitoba (-18.0°F) Answer: -27.8°C, 245 K
2. (G13.3) “Room temperature” is often taken to be 68°F; what is this on the Celsius
scale?
Answer: 20°C
a. The temperature of the filament in a light bulb is about 1800°C; what is this
on the Fahrenheit scale?
Answer: 3272°F
3. (G13.5) The highest and lowest recorded temperature were 136°F (in the Libyan
desert) and -129°F (in Antarctica). What are these temperatures on the Celsius scale?
Answer: 58°C, -89°C
4. (G13.8) At what temperature will the Fahrenheit and Centigrade scales yield the same
numerical value?
Answer: -40°
5. (G13.30) Typical temperatures in the interior of the Earth and Sun are about 4000°C
and 15 x 106°C, respectively. What are these temperatures in Kelvin?
Answer: 4273 K, 15 x 106 K
6. How much heat is generated when the brakes are used to bring a 1000-kg car from a
speed of 25 m/s to 15 m/s?
Answer: 200 kJ
7. A 340-kg marble boulder rolls off the top of a cliff and falls a vertical height of 140 m
before striking the ground. Estimate the temperature rise of the rock if 50% of the
heat generated remains in the marble boulder.
Answer: 0.80°C
8. A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature
of 20°C. What is its length on a hot summer day when the temperature is 35°C?
Answer: 50.009 m
9. During a bout with the flu an 80-kg man ran a fever of 2.0°C above normal, that is a
body temperature of 39.0°C instead of the normal 37.0°C. If the specific heat of the
human body is an average of 3470 J/kg•°C, how much heat is required to raise his
temperature by that amount?
Answer: 5.5 x 105 J
10. Suppose a copper rod that is 45.0 cm long and has a cross-sectional area of 1.25 x 104
m2 connects two reservoirs. The hot reservoir has a temperature of 100°C and the
cold reservoir is 0°C. What is the rate of heat flow by conduction through the copper
rod? The thermal conductivity of copper is 385.0 W/m•K.
Answer: 10.7 W
11. A hiker is wearing clothing that is 0.469-cm to keep warm. Her skin temperature is
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36.1°C and her body cross-sectional area is about 1.15 m2. If the thermal conductivity
of wool is 0.04 W/(m•K) and the rate of heat loss by conduction through her clothing
is 314 W, what is the temperature outside?
Answer: 4°C
12. The operating temperature of a tungsten filament in an incandescent light bulb is
2177°C. and its emissivity is 0.35. Find the surface area of the filament of a 150-W
light bulb if all the electrical energy is radiated by the filament as electromagnetic
waves into a room at 21°C.
Answer: 3.4 x 10-4 m2
13. A very thin square steel plate, 10 cm on a side, is heated in a blacksmith’s forge to a
temperature of 800°C. If the emissivity is 0.60, what is the total ideal rate of radiation
of energy?
Answer: 450 W
a. If thin square steel plate is put into a room that is 30°C, what is the net flow of
heat by radiation?
Answer: 448 W
11. (G13.9) A concrete highway is built of slabs 14 m long at 20°C. How wide should
the expansion cracks between the slabs be at 20°C to prevent buckling if the range of
temperatures is -30°C to +50°C? The coefficient of linear expansion for concrete is 12
x 10-6 °C-1.
Answer: 0.50 cm
12. (G14.E11) A major source of heat loss from a house is through the windows.
Calculate the rate of heat flow through a glass window of 2.0 m x 1.5 m that is 3.2
mm thick, if the temperatures of the inner and outer surfaces are 15.0°C and 14.0°C,
respectively.
Answer: 790 W
13. (G14.33) How much power is radiated by a tungsten sphere (e = 0.35) of radius 22
cm at a temperature of 25°C?
Answer: 95 W
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d. If the sphere is enclosed in a room whose walls are kept at -5°C, what is
the net flow rate of energy out of the sphere? Answer: 33 W
14. (G14.56) Estimate the rate at which heat can be conducted from the interior of the
body to the surface. Assume that the thickness of tissue is about 4.0 cm, that the skin
is at 34°C and the interior at 37°C, and the surface area is 1.5 m2. Answer: 22.5 W
Table 14-3 Latent Heats at 1 atm
Substance
Heat of Fusion (J/kg)
Oxygen
0.14 x 105
Nitrogen
0.26 x 105
Ethyl Alcohol
1.04 x 105
Ammonia
0.33 x 105
Water
3.33 x 105
Lead
0.25 x 105
Silver
0.88 x 105
Iron
2.89 x 105
Tungsten
1.84 x 105
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Heat of Vaporization (J/kg)
2.1 x 105
2.00 x 105
8.5 x 105
1.37 x 105
22.6 x 105
8.7 x 105
23 x 105
63.4 x 105
48 x 105
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PHASE CHA NGE AND SPECIFIC HEAT
Below is the saturation curve for temperatures ranging from -30°C to 40°C. Use this
information to solve the following problems. Do not write on the graph, you can use
the saturation curve in your activity guide.
Saturation
Diagram
Water Vapor in Air (gH20 vapor/kgAir)
50
Equilibrium Curve
40
30
Super-saturated
Region
20
Unsaturated
Region
10
0
0
10
20
30
40
Temperature (C)
1. Which contains more water vapor, air at 30°C with a relative humidity of 50% or air
at 5°C that is saturated? Answer: 30°C by about 7 g/m3
2. The air is currently at a temperature of 50°F and contains 4 g/m3 of water vapor. Will
the relative humidity be higher or lower if the temperature rises to 68°F and by what
percent? Answer: Lower by about 38%
3. The air temperature outside is 60°F and it has a relative humidity of 70%. What is the
dew point?
Answer: 9°C
4. A parcel of air is 20°C and contains 7 g/m3 of water vapor. What is the relative
humidity of this parcel of air?
Answer: 40%
a. The parcel of air rises into the atmosphere and cools at 7°C/km of altitude. At
what altitude would condensation begin to occur? Answer: 2.2 km
5. A geologist working in the field drinks her morning coffee out of an aluminum cup.
The cup has a mass of 0.120 kg and is initially at 20.0°C when she pours in 0.300 kg
of coffee initially at 70.0°C. What is the final temperature after the coffee and the cup
attain thermal equilibrium? Assume that coffee has the same specific heat capacity as
water and that the only heat exchange is between the coffee and the aluminum cup.
Answer: 66.0°C
6. (G14.1) How much heat is required to raise the temperature of 20.0 kg of water from
15C to 95°C?
Answer: 6.7 x 106 J
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7. (G14.3) To what temperature will 7700 J of work raise 3.0 kg of water initially at
10.0°C?
Answer: 10.6°C
8. (G14.8) How many kilocalories of heat are generated when the brakes are used to
bring a 1000-kg car to rest from a speed of 100 km/hr? Answer: 92 kilocalories
9. (G14.9) What is the specific heat of a metal substance if 135 kJ of heat is needed to
raise 5.1 kg of the metal from 20°C to 30°C?
Answer: 2.6 x 103 J/kg°C
10. (G14.15) A 1.20 kg head of a hammer has a speed of 8.0 m/s just before it strikes a
nail and is brought to rest. Estimate the temperature rise of a 14-g nail generated by
ten such hammer blows done in quick succession. Assume the nail absorbs all the
energy.
Answer: 61°C
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HEAT ENGINES
1. (G15.7) A heat engine rejects 8200 J of heat while performing 3200 J of useful work.
What is the efficiency of this engine?
Answer: 28%
2. (G15.18) A heat engine does 7200 J of work in each cycle while absorbing 12.0 kcal
of heat from a high-temperature reservoir. What is the efficiency of this engine?
Answer: 14 %
3. (G15.19) What is the maximum efficiency of a heat engine whose operating
temperatures are 580°C and 320°C?
Answer: 30.5%
4. (G15.20) The exhaust temperature of a heat engine is 230°C. What must be the high
temperature if the Carnot efficiency is to be 28 percent?
Answer: 426°C
5. (G15.21) A nuclear power plant operates at 75% of its maximum theoretical (Carnot)
efficiency between temperatures of 600°C and 350°C. If the plant produces electric
energy at the rate of 1.3 x 109 W, how much exhaust heat is discharged per hour?
Answer: 1.7 x 1013 J/h
6. (G15.25) A heat engine exhausts its heat at 350°C and has a Carnot efficiency of
39%. What exhaust temperature would enable it to achieve a Carnot efficiency of
50%?
Answer: 238°C
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GENERAL THERMODYNAMICS REVIEW
1. Convert -62.8°C to Fahrenheit and Kelvin:
Answer: -81.0°F, 210.4 K
2. Convert 41.0°F to Celsius and Kelvin:
Answer: 5°C, 278.2 K
3. One of the faces of a copper cube with a side of 7.7 cm is maintained at 100°C and
the opposite face is 30°C. If the thermal conductivity of copper is 385 W/(m•K),
calculate the rate of heat flow through the cube.
Answer: 2075 W
4. A fluorescent light bulb contains about 0.10 grams of mercury, which needs to be
vaporized to allow the light to work. If the mercury in the light bulb starts at 20°C,
and its boiling point is 357°C, how much energy is required to vaporize all of the
mercury. The specific heat of mercury is 140 J/(kg•°C) and the latent heat of
vaporization is 2.95 x 105 J/kg.
Answer: 34 J
5. What is the relative humidity of an air parcel that has 2.4 g/kg of water vapor and has
a temperature of 50°F? Use the saturation curve in your lab manual!
Answer: About 30%
6. An air parcel at 40°C has a relative humidity of 10%. What is the dew point of this air
parcel? Use the saturation curve in your lab manual! Answer: About 8°C
7. A 4 cm diameter and 6 cm long cylindrical rod at 1000 K emits a 385 kJ of radiation
in 20 minutes. What is its emissivity?
Answer: 0.56
a. If the rod is in surroundings that are 293 K, what would the difference in
radiation emitted be in the same 20 minutes?
Answer: 4.8 kJ less
8. A typical doughnut contains 2.0 g of protein, 17.0 g of carbohydrates, and 7.0 g of fat.
The average food-energy values of these substances are 4.0 kcal/g for protein, 4.0
kcal/g for carbohydrates, and 9.0 kcal/g for fat.
a. During heavy exercise, an average person uses energy at a rate of 510 kcal/h.
How long would you have to exercise to work the doughnut off?
Answer: 16.4 min
b. If the energy in the doughnut could somehow be converted into kinetic energy
of your body as a whole, how fast could you move after eating the doughnut?
Take your mass to be 60 kg.
Answer: 139 m/s
9. A diesel engine performs 2200 J of mechanical work and discards 4300 J of heat each
cycle.
a. How much heat must be supplied to the engine in each cycle? Answer: 6500 J
b. What is the thermal efficiency of the engine? Answer: 34%
10. A refrigerator takes heat from Qcold, has a work input of |W|, and discards heat Qhot at
a warmer place. Refrigerators are described by their coefficient of performance K,
which is defined as:
K = Qcold/W
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MECHANICS PRACTICE PROBLEM SETS
34
A refrigerator has a coefficient of performance of 2.10. In each cycle it absorbs 3.40x104
J of heat from the cold reservoir.
a. How much mechanical energy is required each cycle to operate this
refrigerator?
Answer: 16200 J
b. During each cycle, how much heat is discarded to the high-temperature
reservoir?
Answer: 50,200 J
11. A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat
at this temperature in each cycle and gives up 335 J to the low-temperature reservoir.
a. How much mechanical work does the engine perform during each cycle?
Answer: 215J
b. What is the thermal efficiency of the cycle?
Answer: 39%
c. What is the temperature of the low temperature reservoir? Answer: 378 K
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MECHANICS PRACTICE PROBLEM SETS
Unit 10
35
CHARGE, FIELDS, CURRENT POTENTIAL AND DC CIRCUITS
10.1
COULOMB’S LAW
1. What is the magnitude of the force of a 10.0-µC charge on a 3.0-µC charge 2.0 m
away?
Answer: 0.068 N
2. How far apart must two electrons be if the force between them is 2.0 x 10-12 N?
Answer: 1.1 x 10-8m
3. A +2.2 X 10-9 C charge is on the x-axis at x= -1.5 m and a +5.4 x 10-9 C charge is
on the x-axis at x= 2.0 m. Find the net force exerted on a +3.5 x 10-9 C located at
the origin?
Answer: 1.2 x 10-8N to the left
4. Three charges are shown
figure
below.
Find the
magnitude
andFind
direction
of
13. in the
Three
charges
are shown
in the
figure below.
the magnitude
an dire
-5
the electrostatic force at thethe
5-nC
charge.
Answer:
1.4
x
10
N
electrostatic force of the 5 nC charge.
at 257.5°
5. A fly accumulates 3.0 x 10-10 C of positive charge as it flies through the air. It
settles on a leave. What is the magnitude and direction of the electric field at a
location 2.0 cm away from the fly?
Answer: 6750 N/C
6. When a TV set is turned on to watch cartoons, an electron beam in the TV tube is
steered across the screen by an electric field between two charged plates. If an
electron experiences a force of 3.0 x 10-6 N, how large is the field between the
deflection plates?
Answer: 1.9 x 1013N/C
7. A tiny ball with a mass of 0.012 kg carries a charge of -18-µC. What is the
magnitude and direction of the electric field needed to cause the ball to float
above the ground? Answer: 6533 N/C downward
14.
Three charges are shown below. Find the magnitude and direction of the
electrostatic force on the 6 nC charge.
8. A charged particle of +12 nC is located 0.10 m to the left of another charged
particle of -12 nC. What is the net electric field at a point that is located between
the two charges at a distance of 6.0 cm to the right of the positive charge?
Answer: 9.8 x 104 N/C to the right
a. What is the net electric field at a point that is located 4.0 cm to the left of
the positive charge? Answer: 6.2 x 104 N/C to the left
9. What is the electric potential 0.50 m away from a 4.5 x 10-4 C point charge?
Answer: 8.1 x 106 V
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MECHANICS PRACTICE PROBLEM SETS
36
10. How much energy is acquired by an electron in moving through a potential
difference of 2.5 x 104 V?
Answer: 4.0 x 10-15 J
11. How far apart are two parallel plates if a potential difference of 600 V produces
an electric field intensity of 1.2 x 104 N/C between them?
Answer: 0.05 m
12. What is the speed of an electron that has been accelerated from rest through a
potential difference of 80.0 kV? The mass of an electron is 9.11 x 10-31 kg.
Answer: 1.7 x 108
m/s
13.
(G16.3) Two charged balls are 20.0 cm apart. They are moved, and the force on
each of them is found to have been tripled. How far apart are they now?
Answer: 11.5 cm
14.
(G16.5) What is the magnitude of the attractive electric force between an iron
nucleus (q = +26e) and its innermost electron if the distance between them is 1.5
x 10-12 m?
Answer: 2.7 x 10-3 N
15. (G16.7) What is the magnitude of the force a +15-µC charge exerts on a +3.0-mC
charge 40 cm away? (1-µC = 1 x 10-6 C, 1-mC = 1 x 10-3 C)
Answer: 2.5 x 103 N
16.
(G16.11) Three particles, Q1 = +70 µC, Q2 = +48 µC, and Q3 = -80 µC, are
placed in a line as shown. The center one is 0.35 m from each of the others.
Calculate the net force on each charge due to the other two. Answer: -1.4 x 102 N
+ 5.3 x 102N
-3.9 x 102 N
17. (G16.13) A charge of 6.00 mC is placed at each corner of a square 1.00 m on a
side. Determine the magnitude and direction of the force on each charge.
Answer: 6.20 x 105N away from the square’s center
18. (G16.23) A proton is released in a uniform electric field and it experiences an
electric force of 3.2 x 10-14 N toward the south. What are the magnitude and
direction of the electric field?
Answer: + 2.0 x 105 N/C south
19. (G16.25) What is the magnitude and direction of the electric field 30.0 cm
directly above a 33.0 x 10-6 C charge?
Answer: + 3.30 x 106 N/C up
20. (G16.27) An electron is released from rest in a uniform electric field and
accelerates to the north at a rate of 125 m/s2. What is the magnitude and direction
of the electric field?
Answer: 7.12 x 10-10 N/C south
21. (G16.31) Calculate the electric field at one corner of a square 1.00 m on a side if
the other three corners are occupied by 2.80 x 10-6 C charges.
Answer: 3.80 x 106 N/C away from the positive charge
22. (G17.1) How much work is needed to move an -8.6 µC charge from the ground to
a point whose potential is +75 V?
Answer: -6.5 x 10-4 J
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MECHANICS PRACTICE PROBLEM SETS
37
23. (G17.3) How much kinetic energy will an electron gain (in Joules) if it falls
through a potential difference of 21,000 V in a TV picture tube?
Answer: 3.4 x 10-15 J
24. (G17.5) How strong is the electric field between two parallel plates 5.2 mm apart
if the potential difference between them is 220 V?
Answer: 4.2 x 104 V/m
25. (G17.9) The work done by an external force to move a -7.50-µC charge from
point a to point b is 25.0 x 10-4 J. If the charge was started from rest and had 4.82
x 10-4 J of kinetic energy when it reached point b, what must be the potential
difference between point a and point b?
Answer: 269 V
26. (G17.13) What is the electric potential 15.0 cm from a 4.00-µC point charge?
Answer: 2.40 x 105 V
27. (G17.21) Two identical +7.5-µC point charges are initially spaced 5.5 cm from
each other. If they are released at the same instant from rest, how fast will they be
moving when they are very far away from each other? Assume they have identical
masses of 1.0 mg.
Answer: 3.0 x 103 m/s
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MECHANICS PRACTICE PROBLEM SETS
10.2
38
EQUIVALENT RESISTANCE AND CIRCUIT ANALYSIS
Using the formulas for series and parallel circuits, fill in the blanks in the tables
shown opposite each circuit. In the blanks across from Battery under
V: Write the emf of the battery.
I: Write the total current in the circuit.
R: Write the equivalent or total resistance of the entire circuit.
In the blanks across from R I under
V: Write the voltage drop across RI'
I: Write the current flowing through R ,R: Write the resistance of RI'
In the blanks across from R2, R3, - . . , fill in the appropriate numbers under
V, I, and R. (Begin by looking for key information given in the table and work
from there.)
I.rI
V
j
L['
rEP
Battery
R,
R2
R,
-A
A
A
V
R2
1
R
D
2.00D
4.00D
R
Battery
V
12.0 A
RI
V
A
n
]8.0 V
V
V
V
V
(j.00 V
2.00 A
A
A
A
8.00 A
A
D
3.00 n
4.00 n
2.00 n
R2
R3
R.
R.
Rs
R7
I
12.0 V
V
V
Rj
n
n
R(,
n
R7
----.--..-------...------.-.-...-.-...----
~
v ------------'I
B<lttery
RI
R2
R,
R4
Lr-~
I
h-J
T
~
R:'
Battery
Rs
R,
R2
R)
R4
R\
Rh
--
Gamzon & Gregorian-Michaelsen
R
A
3.00 A
4.00 A
A
3.00 A
7.00 A
n
n
n
6.00 n
n
f1
V
I
R
V
V
10.0 V
V
V
V
V
A
A
A
A
\.00 A
5.00 A
A
f1
20.0 D
f1
4.00 f1
D
5.00 n
6.00 D
'
"'--
--------.--------.----
R2
RJ
R,
.-------.---
R,
.J\AA
--'
46.0 V
V
V
V
V
V
--
6/7/16
MECHANICS PRACTICE PROBLEM SETS
39
V
t :
R, f
I
R
Battery
R,
Rz
V
V
V
A
2.00A
A
RJ
V
A
11
4.0011
6.0011
8.00 0
V
I
R
V
V
V
V
A
2.00 A
3.00 A
1.00 A
11
11
12.011
11
V
I
12.0 V
2.00 A
V
V
V
A
A
A
Battery
RI
Rz
i
R.t
R'f
R'f
I
RJ
R,
Battery
RJ
RI
Rz
RJ
LR'
R
n
6.0011
4.0011
15.011
'---
Battery
RI
Rz
R g
RJ
R4
II
R,
I
R,
[
50.0 V
V
25.0 V
5.00 A
2. 00 l
A
10.0 V
V
A
3.00 A
V
----------.------..--.--24.0 V
Battery
RI
V
V
R
I
(!
11
n
n
n
R
A
8.00 V
Rz
RJ
I
RJ
V
12
n
A
4.00 A
n
2.00
A
n
--------
-'VVv-
R,
V
r
R,
I
I
R3
R.
Rz
12.0 V
V
R)
24.0 V
A
A
A
A
R,
V
V
R, I
R3
Battery
RI
R2
Rs
nJ
Rs
R
A
- R4
L-J
I
V
Battery
Rz
RJ
R4
Rj
R6
I
11
2.00 n
4.00 n
4.00 n
8.00 n
R
30.0 V
A
6.00 V
3.00 A
V
V
V
8.00 V
V
! 2.00 A
A
1.00 A
A
A
n
n
n
3.00 n
n
11
n
[IT
Gamzon & Gregorian-Michaelsen
6/7/16
MECHANICS PRACTICE PROBLEM SETS
10.3
40
ELECTRICITY REVIEW
1. Two charges are separated by 3.0 cm. Object A has a charge of + 6.0µC while
object B has a charge of +3.0µC. What is the force on object A?
Answer: 180 N away from B
2. A sphere with charge + 6.0µC is located near two other charged spheres. A 3.0µC sphere is located 40.0cm to the right and a + 1.5µC sphere is located 30.0
cm directly underneath. Determine the net force (magnitude and direction) on the
+ 6.0µC charge.
Answer: 1.35 N at 42°
3. A negative charge of 2.0 x 10-8 C experiences a force of 0.060 N to the right in an
electric field. What is the field magnitude and direction?
Answer: 3 x 106 N/C, to the left
4. What is the magnitude and direction of the electric field at a point midway
between a - 8.0µC and a + 6.0µC that are 4.0 cm apart.
Answer: 3.15 x 108 N/C toward negative
5. An electron acquires 3.45 x 10-16 J of kinetic energy when it is accelerated by an
electric field in a computer monitor from plate A to plate B. What is the potential
difference between the plates and which plate has a higher potential?
Answer: 2.15 x 103 V; Plate B is higher
6. Two parallel plates are connected to a 100-V power supply and are separated by
an air gap. How small can the gap be if the air is not to exceed its breakdown
value of E = 3 x 106 V/m?
Answer: 3.3 x 10-5 m
7. An automobile headlight with a resistance of 30 Ω is placed across a 12-V
battery. What is the current through the circuit?
Answer: 0.40 A
8. A bird stands on an electric transmission line carrying 2500 A. The resistance per
meter of the line is 2.5 x 10-5 Ω/m. If the bird’s feet are 4.0 cm apart, what voltage
does the bird feel?
Answer: 0.0025 V
9. What is the diameter of a 1.00-m length of tungsten wire whose resistance is 0.22
Ω? The resistivity of tungsten is 5.6 x 10-8 Ω•m.
Answer: 5.7 x 10-4 m
10. Suppose the resistance in a copper wire with a diameter of 1.02 mm carrying 1.67
A has a resistance of 1.05 Ω at 20°C. What is the resistance at 0°C and 100°C if
the temperature coefficient for copper is 0.00393°C-1? Answer: 0.97 Ω; 1.38 Ω
11. What is the maximum voltage that can be applied to a 2.7-kΩ resistor rated at
0.25 Watts?
Answer: 26 V
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MECHANICS PRACTICE PROBLEM SETS
41
12. A standard 100-W incandescent light bulb has a filament made out of tungsten. At
room temperature (20°C), the tungsten filament has a resistance of 15 Ω. When
the light bulb operating with a wall voltage of 120 V, it gets hot. What is the
temperature of the filament when it is hot? The temperature coefficient of tungsten
is 0.0044°C-1.
Answer: 1975°C
13. Find the equivalent resistance and the current through the battery for each of the
following combination circuits.
a. Req = 27.1 Ω; IB = 4.4 A
10 Ω
25 Ω
30 Ω
15 Ω
40 Ω
120 V
20 Ω
b. Req = 128.8 Ω; IB = 0.93 A
120 V
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MECHANICS PRACTICE PROBLEM SETS
Unit 11
11.1
42
MAGNETISM
MAGNETISM
1. Use the right-hand rule to find the direction of the current in a wire in a magnetic
field that results in the force on the wire shown for each case shown below:
2. A 50 cm long, straight wire conducts 4.0 A of current upward. The wire
experiences a force of 1.0 x 10-2 N when in a magnetic field that is perpendicular
to the wire. What is the magnetic field around the wire?
Answer: 0.005 T
3. A magnetic field of 1.0 x 10-4 T at 30° North of West acts on a 1.0 m wire that
carries a current of 15 A to the west. What is the magnitude and direction of the
magnetic force on the wire? Answer: 7.5 x 10-4 N into the page
4. A wire with a length of 2.7 m and a mass of 0.89 kg is in a region of space with a
magnetic field of 0.72 T. What is the minimum current needed to levitate the
wire?
Answer: 4.5 A
5. You want to produce a magnetic field with a magnitude of 5.50 x 10-4 T at a
distance of 0.040 m from a long, straight wire.
a. What current is required to produce this field?
Answer: 110 A
b. What is the magnitude of the field located at a distance of 0.080 m and
0.160 m? Answer: 2.75 x 10-4 T; 1.38 x 10-4
6. Two hikers are reading a compass under an overhead transmission line that is 5.50
m above the grand and carries a current of 800 A in a horizontal direction from
north to south. What is the magnitude and direction of the magnetic field at a
point directly under the conductor?
Answer: 2.91 x 10-5 east
7. Two, straight, parallel, 1.0 m superconducting cables 4.5 mm apart carry equal
currents of 15,000 A in opposite directions. What is the force on each wire and
what type of force is it?
Answer: 10,000 N repulsive
8. Two long, parallel wires that are each 1.20 meters long are separated by a distance
of 2.50 cm. The force each wire exerts on the other is 4.80 x 10-5 N. The wires
attract each other. The current in one wire is 0.600 A. What is the current in the
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MECHANICS PRACTICE PROBLEM SETS
43
second wire and is it going in the same direction or the opposite direction as the
first current? Answer: 8.33 A, same direction
9. An electron moving at right angles to a 0.10-T magnetic field experiences an
acceleration of 6.00 x 1015 m/s2. Ignoring the effects of gravity, what is the
electron’s speed?
Answer: 3.42 x 105 m/s
10. A particle with a charge of 6.40 x 10-19 C travels in a circular orbit with a radius
4.68 mm due to the force exerted on it by a magnetic field with magnitude of 1.65
T that is perpendicular to its orbit. What is the magnitude of the linear momentum
(ρ) of the particle?
Answer: 4.94 x 10-21 kg•m/s
11. A deuteron, which is the nucleus of an isotope of hydrogen, has a mass of 3.34 x
10-27 kg and a charge of +e. The deuteron travels in a circular path with a radius of
6.96 mm in a magnetic field with a magnitude of 2.50 T.
a. What is the speed of the deuteron?
Answer: 8.35 x 105 m/s
b. What time is required for it to make half of a revolution?
Answer: 2.62 x 10-8 s
12. An ion with a mass of m and a charge of +2e leaves a velocity selector moving at
a speed of 400 km/s. It then moves in a half circle in a magnetic field of 60-mT
that is perpendicular to the plane of its motion. At the end of this trip it is
detected. If the radius of the circle is 13.9 cm, what is the mass of the ion?
Answer: 6.67 x 10-27 m/s
The following problems are from Giancoli Chapter 20.
13. (G1) What is the force per meter on a wire carrying a 9.80-A current when
perpendicular to a 0.80 T magnetic field?
Answer: 7.8 N/m
a. What if the angle between the wire and field is 45.0°? Answer: 5.5 N/m
14. (G3) How much current is flowing in a wire 4.20 m long if the maximum force on
it is 0.900 N when placed in a uniform 0.0800-T field? Answer: 2.68 A
15. (G9) Alpha particles of charge q = +2e and a mass m = 6.6 x 10-27 kg are emitted
from a radioactive source at a speed of 1.67 x 107 m/s. What magnetic field
strength would be required to bend these into a circular path of radius r = 0.25 m?
Answer: 1.3 T
16. (G13) A proton moves in a circular path perpendicular to a 1.15 T magnetic field.
The radius of its path is 8.40 mm. Calculate the energy of the proton.
Answer: 7.16 x 10-16 J
17. (G19) Jumper cables used to start a stalled vehicle often carry a 15-A current.
How strong is the magnetic field 15 cm away?
Answer: 2.0 x 10-5 T
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MECHANICS PRACTICE PROBLEM SETS
44
18. (G21) What is the magnitude and direction of the force between two parallel
wires 45 m long and 6.0 cm apart, each carrying 35 A in the same direction?
Answer: 0.18 N attraction
19. (G27) A stream of protons passes a given point in space at a rate of 109 protons.
What magnetic field do they produce 2.0 m from the beam?
Answer: 1.67 x 10-17 T
Gamzon & Gregorian-Michaelsen
6/7/16