UNIT I
INTRODUCTION
THE SCIENCE OF PHYSICS
Physics is a branch of science that involves the study of the physical world: energy, matter, and how they are
related. Everything around you can be described using the tools of physics. Physics uses mathematics as a
powerful language. In Physics, equations are important tools for modeling observations and for making
predictions.
The goal of physics is to use a small number of basic concepts, equations, and assumptions to describe the
physical world. Once the physical world has been described this way, the physics principles involved can be
used to make predictions about a broad range of phenomena. For example, the same physics principles that are
used to describe the interaction between two planets can also be used to describe the motion of a satellite
orbiting the Earth.
Many of the inventions, appliances, tools, and buildings we live with today are made possible by the application
of physics principles. Every time you take a step, catch a ball, open a door, whisper, or check your image in a
mirror, you are unconsciously using your knowledge of physics.
UNITS, STANDARDS AND THE SI SYSTEM
We describe the behavior of physical systems using various quantities that we create for this purpose. However,
there are three quantities, length, mass, and time, that we take as fundamental quantities and we use these three
to create other quantities. The system of units based on these choices is called SI units which stands for System
International.
The base units will be defined as follows:
meter (m): One meter is equal to the path length traveled by light in vacuum during a time
interval of 1/299,792,458 of a second.
kilogram (kg): One kilogram is the mass of a Platinum-Iridium cylinder kept at the International
Bureau of Weights and Measures in Paris.
second (s): One second is the time occupied by 9,192,631,770 vibrations of the light (of a
specified wavelength) emitted by a Cesium-133 atom.
All physical quantities are expressed in terms of base units. For example, the velocity is usually given in units
of m/s. All other units are derived units and may be expressed as a combination of base units. For example: A
Newton is a unit of force: 1 N = 1 kg.m/s2
1
SYSTÈME INTERNATIONAL
The scientific community follows the Systeme International (SI), based on the metric system:
Quantity
Length
Mass
Time
Electric current
Thermodynamic temperature
Amount of substance
Luminous intensity
Unit
meter
kilogram
second
ampere
Kelvin
mole
candela
SI symbol
m
kg
s
A
K
mol
cd
SI PREFIXES
Power Prefix Abbreviation
1012
109
106
103
teragigamegakilo-
T
G
M
k
Power Prefix Abbreviation
10-2
10-3
10-6
10-9
10-12
centimillimicronanopico-
c
m
μ
n
p
MATHEMATICAL NOTATION
Many mathematical symbols will be used throughout this course.
=
denotes equality of two quantities
<
means is less than
>
means greater than
x
(read as “delta x”) indicates the change in the quantity x

represents a sum of several quantities, also called summation
PHYSICS VOCABULARY
A variable refers to a quantity in an equation. Examples: time, velocity, force.
Symbols are used to represent variables: t, v, F
A unit refers to the accepted standard of measurement. For example: time is measured in seconds, velocity is
measured is meters per second, force is measured in Newtons
I. FACTOR-LABEL METHOD FOR CONVERTING UNITS
1 km = 1000 m
1 m = 100 cm
1 m = 1000 mm
1 h = 60 min
1 min = 60 s
1 h = 3600 s
1 kg = 1000 g
1 g = 1000 mg
2
Work out the following conversions using the factor-label method. Show all the steps!
1. Convert 28 km to m.
2. Convert 280 m to km.
3. Convert 28 m to cm.
4. Convert 280 cm to m.
5. Convert 45 kg to g.
6. Convert 450 g to kg.
7. Convert 3 h to min.
8. Convert 35 min to h.
9. Convert 2 h to s.
3
10. Convert 15 km/h to m/s
11. Convert 450 m/s to km/h.
CONVERSION FACTORS
What is the conversion factor to convert km/h to m/s?
What is the conversion factor to convert m/s to km/h?
II. GRAPHING TECHNIQUES
Frequently an investigation will involve finding out how changing one quantity affects the value of another. The
quantity that is deliberately manipulated is called the independent variable. The quantity that changes as a result
of the independent variable is called the dependent variable.
The relationship between the independent and dependent variables may not be obvious from simply looking at
the written data. However, if one quantity is plotted against the other, the resulting graph gives evidence of what
sort of relationship, if any, exists between the variables. When plotting a graph, take the following steps.
1. Identify the independent and dependent variables.
2. Choose your scale carefully. Make your graph as large as possible by spreading out the data on each axis. Let
each space stand for a convenient amount. For example, choosing three spaces equal to ten is not convenient
because each space does not divide evenly into ten. Choosing five spaces equal to ten would be better. Each
axis must show the numbers you have chosen as your scale. However, to avoid a cluttered appearance, you do
not need to number every space.
4
3. All graphs do not go through the origin (0,0). Think about your experiment and decide if the data would
logically include a (0,0) point. For example, if a cart is at rest when you start the timer, then your graph of speed
versus time would go through the origin. If the cart is already in motion when you start the timer, your graph
will not go through the origin.
4. Plot the independent variable on the horizontal (x) axis and the dependent variable on the vertical (y) axis.
Plot each data point. Darken the data points.
Distance versus Time
5. If the data points appear to lie roughly in a straight line,
draw the best straight line you can with a ruler and a
sharp pencil. Have the line go through as many points as
possible with approximately the same number of points
above the line as below. Never connect the dots.
If the points do not form a straight line, draw the best
smooth curve possible.
6. Title your graph. The title should dearly state the
purpose of the graph and include the independent and
dependent variables.
7. Label each axis with the name of the variable and the
unit. Using a ruler, darken the lines representing each axis.
The graph shown on the next page was prepared using
good graphing techniques. Go back and check each of
the items mentioned above.
Graph the following sets of data using proper graphing techniques.
The first column refers to the y-axis and the second column to the x-axis
1.
Volume
(mL)
800
400
200
133
114
100
80
Pressure
(Pa)
100
200
400
600
700
800
1000
What type of curve did
you obtain?
5
2.
Position (m)
0
5
20
45
80
125
Time (s)
0
1
2
3
4
5
What type of curve did
you obtain?
3.
Speed (m/s)
0
20
45
60
84
105
Time (s)
0
1
2
3
4
5
a. What type of curve
did you obtain?
b. What is the equation of the slope of a line?
c. Calculate the slope of the line including appropriate units.
6
III. INTERPRETING GRAPHS
In laboratory investigations, you generally control one variable and measure the effect it has on another variable
while you hold all other factors constant. For example, you might vary the force on a cart and measure its
acceleration while you keep the mass of the cart constant. After the data are collected, you then make a graph of
acceleration versus force, using the techniques for good graphing. The graph gives you a better understanding of
the relationship between the two variables.
There are three relationships that occur frequently in physics:
Graph A: If y varies directly with x, the graph will be a straight line.
Graph B: If y varies inversely with x, the graph will be a hyperbola.
Graph C. If y varies directly with the square of x, the graph is a parabola.
Reading from the graph between data points is called interpolation.
Reading from the graph beyond the limits of your experimentally determined data points is called extrapolation.
Example 1.
Suppose you recorded the following data during a study of the relationship of force and acceleration. Prepare a
graph showing these data.
7
Force
(N)
10
20
30
40
Acceleration
(m/s2)
6.0
12.5
19.0
25.0
a. Describe the relationship between force and acceleration as shown by the graph.
b. What is the slope of the graph? Remember to include units with your slope. (1 N: 1 kg.m/s2)
c. What physical quantity does the slope represent?
d. Write an equation for the line using the variables of the data.
e. What is the value of the force for an acceleration of 15 m/s2?
f. What is the acceleration when the force is 50.0 N?
8
1. Honors Physics –What math review?
CHECK____
PART I. FACTOR-LABEL METHOD FOR CONVERTING UNITS
Carry out the following conversions using the factor-label method. Show all your work!
1. How many seconds are in a month?
2. Convert 3.5 km to cm.
3. Convert 5600 mg to kg.
4. Convert 72 m/s to cm/min
5. Convert 3x108 m/s to km/hr.
9
PART II. GRAPHING TECHNIQUES
Graph the following sets of data using proper graphing techniques.
The first column refers to the y-axis and the second column to the x-axis
1. Plot a graph for the following data recorded for an object falling from rest:
Velocity
(m/s)
3.2
6.3
9.7
12.9
15.9
19.2
22.5
Time
(s)
1
2
3
4
5
6
7
a. What kind of curve did you obtain?
b. What is the relationship between the variables?
c. What do you expect the velocity to be after 4.5 s?
d. How much time is required for the object to attain a speed of 100 m/s?
10
2. Plot a graph showing the relationship between frequency and wavelength of electromagnetic waves:
Frequency
(kHz)
150
200
300
500
600
900
Wavelength
(m)
2000
1500
1000
600
500
333
a. What kind of curve did you obtain?
b. What is the relationship between the variables?
c. Use the graph to determine the wavelength of an electromagnetic wave of frequency 350 kHz
d. Use the graph to determine the frequency of an electromagnetic wave of wavelength 375 m
11
3. In an experiment with electric circuits the following data was recorded. Plot a graph with the data:
Power
(W)
1.0
6.5
16.2
25.8
50.2
72.0
Current
(A)
1.0
2.5
4.0
5.0
7.0
8.5
a. What kind of curve did you obtain?
b. What is the relationship between the variables?
c. Use the graph to determine the power when the current is 3.0 A
d. Use the graph to determine the current when the power is 65 W
12
UNIT II
ONE-DIMENSIONAL KINEMATICS
SCALARS AND VECTORS
A scalar quantity has only magnitude and is completely specified by a number and a unit. Examples are mass (a
stone has a mass of 2 kg), volume (1.5 L), and frequency (60 Hz). Scalar quantities of the same kind are added
by using ordinary arithmetic.
A vector quantity has both magnitude and direction. Examples are displacement (an airplane has flown 200
km to the southwest), velocity (a car is moving at 60 km/h to the north), and force (a person applies an upward
force of 25 N to a package). When vector quantities are added, their directions must be taken into account.
Useful definitions:
Magnitude: Size of the quantity, basically it is the numerical value.
Direction: Alignment or orientation of any position with respect to any other position.
MOTION
An object is in motion if its position changes. The mathematical description of motion is called kinematics.
POSITION, DISTANCE AND DISPLACEMENT
Any description of motion takes place in a coordinate system that allows us to track the position of an object.
One-dimensional motion means that objects are only free to move back and forth along a single line. As a
coordinate system for one-dimensional motion, choose this line to be an x-axis together with a specified origin
and positive and negative directions.
CONSTANT VELOCITY GRAPHS
A. Position (x) – Time (t) Graph
When the velocity is constant the position varies directly with time. This means that the graph must be:
13
2.1 Give a qualitative description of the motion depicted in the following x-versus-t graphs:
a.
x
b.
x
t
t
c.
x
d. x
t
t
2.2 a. Graph the following data and label the points: A, B, C, D, and E
Position versus Time
Time
(s)
A) 0
B) 10
C) 20
D) 25
E) 45
Position
(m)
0
200
200
150
0
14
b. Give a qualitative description of the motion depicted in the following x-versus-t graph for each interval:
c. Calculate the slope of each interval: A-B, B-C, C-D, and D-E. Show all your work.
d. What does the slope represent?
e. What is the meaning of a negative slope?
SUMMARY:
The slope of the position-time graphs yields the average velocity.
15
2.3 Consider the position vs. time graph below for cyclists A and B.
a. Do the cyclists start at the same point? How do you know? If not, which is ahead?
b. At t= 7s, which cyclist is ahead? How do you know?
c. Which cyclist is traveling faster at t = 3s? How do you know?
d. Are their velocities equal at any time? How do you know?
e. What is happening at the intersection of lines A and B?
16
B. Velocity (v) – Time (t) Graph
The velocity-time graph of a constant velocity graph is a line parallel to the x-axis.
2.5 Give a qualitative description of the motion depicted in the following v-versus-t graphs:
a.
v
b.
v
t
t
2.6 a. Graph the following data.
Velocity versus Time
Time
(s)
0
1
2
3
4
5
Velocity
(m/s)
150
150
150
150
150
150
b. Calculate the area under the line.
c. What does the area represent?
SUMMARY:
The area under the velocity-time curve gives the displacement.
17
2. Honors Physics – Love That Homework – 1
CHECK ______
1. Complete the table by writing the position and clock reading of each of the sets below.
18
2. Using a ruler, plot each set of data. Differentiate the line that represents each set with colors.
3. Calculate the SLOPE of each of your best fit lines. Choose two points on each of the best fit lines. Start with
the slope equation and show all your substitutions, with units.
DOT
DIAMOND
TRIANGLE
CIRCLE
19
4. Plot a velocity versus time graph for each of the sets.
5. Describe the motion of each set and include a motion diagram. Be sure to include complete information:
- Where does the motion start? origin? left or right of the origin?
- Object's motion: positive or negative direction?
- Moving fast or slow? For how long?
DOT
DIAMOND
TRIANGLE
CIRCLE
20
DISTANCE AND DISPLACEMENT
Distance is the length between any two points in the path of an object.
Displacement is the length and direction of the change in position measured from the starting point.
The primary difference between the two is that the distance an object travels tells you nothing about the
direction of travel, while displacement tells you precisely how far, and in what direction, from its initial
position an object is located. Distance is a scalar quantity that represents the total length of travel and
displacement is a vector quantity that represents the net length of travel accounting for direction.
2.7 You leave your home and drive 4.83 km North on Preston Rd. to go to the grocery store. After shopping,
you go back home by traveling South on Preston Rd.
a. What distance do you travel during this trip?
b. What is your displacement?
SPEED
If an object takes a time interval t to travel a distance x, then the average speed of the object is given by:
x
v
Units: m/s
t
2.8 A ship steams at an average speed of 30 km/h.
a. What is the speed in m/s?
b. How far in km does it travel in one day?
DATA
EQUATION
SUBSTITUTION
c. How many hours does it take to travel 500 km?
EQUATION
SUBSTITUTION
ANSWER
ANSWER
21
2.9 How long does it take an echo to return to a woman standing 300 m from a cliff? The velocity of sound in
air is about 343 m/s.
DATA
EQUATION
SUBSTITUTION
ANSWER
AVERAGE SPEED AND VELOCITY
Average velocity is the displacement divided by the amount of time it took to undergo that displacement. The
difference between average speed and average velocity is that average speed relates to the distance traveled
while average velocity relates to the displacement.
2.10 A car travels north at 100 km/h for 2 h, slows down to 75 km/h for the next 2 h, and then turns south at 80
km/h for 1 h.
a. How many legs does this trip have?
b. Write down the data per leg:
What is the car’s average speed for the entire journey? Show all your work.
b. What is the car’s average velocity for the entire journey?
22
2.11 A cab driver needs to get an actor to the theater in time for curtain time that is 8 minutes away. The cab
driver speeds off to the theater down a 1000 m long one-way street at a speed of 25 m/s. At the end of the street
the cab driver waits at a traffic light for 1.5 min and then turns north onto a 1700 m long traffic filled avenue on
which he is able to travel at a speed of 10 m/s.
a. How many legs does this trip have?
b. Write down the data per leg:
c. Does the actor arrive before the theater lights dim?
b. Plot a distance versus time graph of the situation.
23
CW. CONSTANT VELOCITY
1. The fastest helicopter, the Westland Lynx, can travel 3.33 km in the forward direction in just 30.0 s. What is
the speed of this helicopter? Express your answer in m/s and km/h. (111 m/s, 399.6 km/h)
2. Jim drives 20.0 km to the East when he realizes he left his wallet at home. He drives 20.0 km West to his
house, takes 5.0 min to find his wallet, and then leaves again. Jim is 40.0 km East of his house exactly 60.0 min
after he left the first time.
a. How many legs does the trip have?
b. Write down the data for each leg of the trip.
a. What is his average speed? (80 km/h)
b. What is his average velocity? (40 km/h, E)
3. A teacher makes a trip to a nearby mall that is located 40 miles from her home. On the trip to the mall she
averages 80 mi/hr but gets a speeding ticket upon her arrival. On the return trip she averages just 40 mi/hr.
a. How many legs does the trip have?
b. Write down the data for each leg of the trip.
c. What was her average speed for the entire trip? (53.3 mi/h)
d. What was her average velocity for the entire trip? (0 mi/h)
24
4. The graph shows the distance versus time for two cars traveling on a straight highway.
a. What can you determine about the relative direction of travel of the cars?
b. At what time do they pass one another?
c. Which car is traveling faster? Explain. (A, why?)
d. What is the speed of each of the cars? (A≈ -70 km/h; B≈ 17 km/h)
5. A bus travels on a northbound street for 20.0 s at a constant velocity of 10.0 m/s. After stopping for 20.0 s, it
travels at a constant velocity of 15.0 m/s for 30.0 s to the next stop, where it remains for 15.0 s. For the next
15.0 s, the bus continues north at 15.0 m/s.
a. How many legs does the trip have?
b. Write down the data for each leg:
25
c. Calculate any additional information needed to construct a x-t graph of the motion of the bus.
d. Calculate the total distance traveled? (875 m)
e. What is the average velocity of the bus for this period? (8.75 m/s)
26
3. Honors Physics – Great, More Tasks
CHECK_____
1. A car travels a distance of 86 km at an average speed of 8 m/s. How many hours were required for the trip?
(2.99 hours)
2. Sound travels at an average speed of 340 m/s. Lightning from a distant thundercloud is seen almost
immediately. If the sound of thunder reaches the ear 3 s later, how far away is the storm? (1020 m)
3. A woman walks for 4 min directly North with an average velocity of 6 km/h; then she walks East at 4 km/h
for 10 min. a. What is her average speed for the trip? (4.57 km/h)
b. What is her average velocity for the trip? Hint: Draw a sketch! (3.33 km/h)
27
4. It is now 10.29 am but when the bell rings at 10.30 am Kate will be late for Physics class. She must get from
the tower to the physics room by hurrying down three different hallways. She runs down the first hallway, a
distance of 35 m at a speed of 3.5 m/s. The second hallway is filled with students and she covers the 48 m
length at an average speed of 1.20 m/s. The final hallway is empty and Kate sprints its 60 m length at a speed of
5.0 m/s.
a. How many legs does her trip have?
b. Write down the data for each leg of the trip.
c. Does Kate make it on time or does she get a detention? Justify your answer by showing ALL your work!
d. Draw a position versus time graph of the situation:
28
ACCELERATION
Acceleration happens when:
 An object's velocity increases
 An object's velocity decreases
 An object changes direction
UNIFORM ACCELERATION GRAPHS
A. Velocity (v) – Time (t) Graph
2.12 a. Graph the following data.
Velocity versus Time
Time
(s)
0
10
20
30
40
50
Velocity
(m/s)
0
200
400
600
800
1000
b. Calculate the slope showing all your work.
c. What does the slope represent?
d. Calculate the total area under the line.
e. What does the area represent?
29
B. Position (x) – Time (t) Graph
2.13 a. Graph the following data.
Position versus Time
Time
(s)
0
1
2
3
4
5
Position
(m)
0
10
40
90
160
250
b. Draw a tangent line to 3 s.
c. Find the slope of the tangent line at 3 s.
d. What does the slope represent?
e. Find the slope of the line between 2 and 4 seconds.
f. What does the slope represent?
30
C. Acceleration (a) – Time (t) Graph
Acceleration versus Time
2.14 a. Graph the following data.
Time
(s)
0
1
2
3
Acceleration
(m/s2)
2.5
2.5
2.5
2.5
b. Calculate the area under the line.
c. What does the area represent?
SUMMARY GRAPHICAL ANALYSIS OF MOTION
Graphical interpretations for motion along a straight line (the x-axis) are as follows:
 the slope of the tangent of an x-versus-t graph gives the instantaneous velocity,
 the slope of the line between two times on a position-time graph gives the average velocity,
 the slope of the v-versus-t graph gives the average acceleration,
 the area under the v-versus-t graph gives the displacement,
 the area under the a-versus-t graph gives the change in velocity.
2.15 Draw qualitative graphs of x-versus-t, v-versus-t, and a-versus-t.
vo = 0
x=0
x
v
a
t
t
t
31
vo = 0
x=0
x
v
a
t
t
t
vo ≠ 0
x=0
Note: the object reaches the top and slides back down.
x
v
a
t
t
t
32
vo = 0
x=0
x
v
a
t
t
t
vo = 0
x=0
x
v
a
t
t
t
ACCELERATION
33
Acceleration is the rate of change of velocity. The change in velocity v is the final velocity vf minus the
initial velocity vo.
The acceleration of an object is given by:
a
v f  vo
Units: m/s2
t
2.16 A car comes to a stop in 6 s from a velocity of 30 m/s. What is its acceleration?
EQUATIONS FOR MOTION UNDER CONSTANT ACCELERATION:
x(
vo  vf
)t
2
v f  vo  at
1
x  vot  at 2
2
v f  vo  2ax
2
2
2.17 An object starts from rest with an acceleration of 10 m/s2.
a. How far does it go in 0.5 s?
b. What is its velocity after 0.5 s?
2.18 A sports car has an acceleration of 3 m/s2. How much distance does it cover while its velocity is increased
from 0 to 10 m/s?
34
2.19 A car has an initial velocity of 10 m/s when it begins to be accelerated at 2.5 m/s2.
a. How long does it take to reach a velocity of 25 m/s?
b. How far does it go during this period?
2.20 Find the acceleration of a car that comes to a stop from a velocity of 60 m/s in a distance of 360 m.
VELOCITY AND ACCELERATION
If the sign of the velocity and the acceleration is the same then the object is speeding up.
If the sign of the velocity and the acceleration is the opposite then the object is slowing down.
CW: VELOCITY AND ACCELERATION
6. With an average acceleration of 16.5 m/s2, Mike started a dragster at rest and reached a speed of 386 km/h.
How much time was needed for the car to reach his final speed? (6.5 s)
35
7. A boat moves slowly inside a marina with a constant speed of 1.50 m/s. As soon as it leaves the marina, it
accelerates at 2.40 m/s2.
a. How fast is the boat moving after accelerating for 5.0 s? (13.5 m/s)
b. Plot a graph of velocity versus time.
v (m/s)
c. Find how far has the boat traveled in this
time by using the graph. Clearly show your
work on the graph. (37.5 m)
t (s)
8. A polar bear running with an initial speed of 4.0 m/s accelerates uniformly for 18 s. What is the bear’s
maximum speed if the bear travels 135 m during the 18 s of acceleration? Give the answer in both m/s and
km/h? (11 m/s, 39.6 km/h)
9. A golf ball at a miniature golf course travels 4.2 m along a carpeted green. When the ball reaches the hole 3.0
s later, its speed is 1.3 m/s. Assuming the ball undergoes constant uniform acceleration, what is the ball’s initial
speed? (1.5 m/s)
36
10. A speedboat uniformly increases its velocity from 25 m/s to the west to 35 m/s to the west. How long does it
take the boat to travel 250 m west while undergoing this acceleration? (8.3 s)
11. A rocket is launched from rest traveling 12.4 m upward in 2.0 s. Find the rocket’s acceleration. (6.2 m/s2)
12. Police find skid marks 60 m long on a highway showing where a car made an emergency stop. The
acceleration was -10 m/s2. Was the car exceeding the 80 km/h speed limit? Justify your answer. (125 km/h, yes)
13. Construct a velocity-time graph using the data that describes a familiar motion of a car traveling during
rush-hour traffic.
Time
(s)
0
1
2
3
4
5
6
Velocity
(m/s)
10
10
10
10
10
5
0
a. Describe the car’s motion from t = 0 s to t = 4 s.
37
b. Describe the car’s motion from t = 4 s to t = 6 s.
c. What is the average acceleration for the first 4 s? (0 m/s2)
d. What is the average acceleration from t = 4 s to t = 6 s? (-5 m/s2)
14. Use the velocity-time graph below to answer the following:
a. What is the acceleration between the points on the graph labeled A and B? (15 m/s2)
b. What is the acceleration between the points on the graph labeled D and E? (125 m/s2)
c. Use the graph to find the total distance that the object travels between points B and C? (3000 m)
38
4. Honors Physics – Fast Task? – 1
CHECK____
While cruising along a dark stretch of highway at 30 m/s (≈ 65 mph), you see, at the fringes of your headlights,
a dead skunk on the highway. It takes you 0.5 s to react, then you apply the brakes and come to a stop 3.5 s
later. Assume the clock starts the instant you see the skunk.
1. Construct a quantitatively accurate v vs t graph to
describe the situation. Clearly label your graph.
2. On the v vs t graph at right, graphically represent the
car’s displacement (mark the area under the curves)
during this incident.
3. Using the graphical representation, determine how
far the car traveled during this incident. Clearly show
your work. (67.5 m)
4. Using the v vs t graph calculate the acceleration of the car once the brakes were applied. Show all your work
for each leg of the trip (-8.57 m/s2)
39
5. Sketch a quantitatively accurate a vs t graph:
6. Two kinds of motion occur in this case. For the first 0.5s, the car is traveling at constant velocity. For the
remainder of the time, the car has an initial velocity and a uniform acceleration.
Using the appropriate equations for each leg of the motion, determine how far the car traveled from the instant
you noticed the hazard until you came to a stop. (67.5 m)
7. Compare your answers to 4 and 6. Explain your answer.
40
5. Honors Physics – Faster Task? – 2
CHECK______
For each problem:
- Sketch a velocity-time graph that represents the situation. Your axes should clearly show starting and
finishing values for velocity and time.
- Solve the problems using either a graphical method or equations but clearly SHOW ALL YOUR WORK!
1. At t = 0 a car has a speed of 30 m/s. At t = 6 s, its speed is 14 m/s. What is its average acceleration during
this time interval? (-2.67 m/s2)
2. A bus moving at 20 m/s (t = 0) slows at a rate of 4 m/s each second.
a) How long does it take the bus to stop? (5 s)
b) How far does it travel while braking? (50 m)
41
3. A physics student skis down a hill, accelerating at a constant 2.0 m/s2. If it takes her 15 s to reach the bottom,
what is the length of the slope? (225 m)
4. A dog runs down his driveway with an initial speed of 5 m/s for 8 s, then uniformly increases
his speed to 10 m/s in 5 s.
a) What was his acceleration during the 2nd part of the motion? (1 m/s2)
b) How long is the driveway? Hint: The motion is not the same all the time! (77.5 m)
42
ACCELERATION DUE TO GRAVITY
All bodies in free fall near the Earth's surface have the same downward acceleration of:
g = 9.8 m/s2
A body falling from rest in a vacuum has a velocity of 9.8 m/s at the end of the first second, 19.6 m/s at the end
of the next second, and so forth. The farther the body falls, the faster it moves. A body in free fall has the same
downward acceleration whether it starts from rest or has an initial velocity in some direction.
The presence of air affects the motion of falling bodies partly through buoyancy and partly through air
resistance. Thus two different objects falling in air from the same height will not, in general, reach the ground at
exactly the same time. Because air resistance increases with velocity, eventually a falling body reaches a
terminal velocity that depends on its mass, size and shape, and it cannot fall any faster than that. Terminal
velocity is a constant velocity.
FREE FALL
When air resistance can be neglected, a falling body has the constant acceleration g, and the equations for
uniformly accelerated motion apply. Just substitute a for g.
Sign Convention for direction of motion:
If the object is thrown downward then: g = 9.8 m/s2
If the object is thrown upward then: g = - 9.8 m/s2
There are a few facts concerning free fall motion that you can use in analyzing situations. These facts can be
deduced from the four equations for motion with constant acceleration.

When an object launched vertically upward reaches the top of its path (its maximum height), its
instantaneous velocity is zero, even though its acceleration continues to be 9.8 m/s2 downward.

An object launched upward from a given height takes an equal amount of time to reach the top of its
path as it takes to fall from the top of its path back to the height from which it was launched.

The velocity an object has at a given height, on its way up, is equal and opposite to the velocity it will
have at that same height on its way back down.
2.21 A stone is dropped from the edge of a cliff.
a. What is its velocity 3 s later?
b. How far does it fall in this time?
43
2.22 A ball is thrown upward from the edge of a cliff with an initial velocity of 6 m/s.
a. How fast is it moving 0.5 s later and in what direction?
b. How fast is it moving 2 s later and in what direction?
2.23 A toy in free fall reaches the ground in 5 s.
a. From what height was it dropped?
b. What is its final velocity?
c. How far did it fall in the last second of its descent?
2.24 A ball is thrown vertically upward with a velocity of 12 m/s.
a. At what height is the ball 2 s later?
b. What is the maximum height the ball reaches?
44
CW: FREE FALL
15. An iPad is accidentally dropped from the edge of a cliff and 6.0 s later hits the bottom.
a. How fast was it going just before it hit? (59 m/s)
b. How high is the cliff? (177 m)
16. A surface probe lands on a highland region of the planet Mercury. A few hours later the ground beneath the
probe gives way and the probe falls, landing below its original position with a velocity of 11.2 m/s downward.
If the free-fall acceleration near Mercury’s surface is 3.70 m/s2 downward, what is the probe’s displacement?
(16.9 m)
17. A ball thrown vertically is caught by the thrower after 5.1 s.
a. Find the initial velocity with which the ball is thrown. (25 m/s)
b. Find the maximum height the ball reaches. (31.9 m)
45
6. Honors Physics – Free- Falling Task!
CHECK____
1. Galileo drops a computer from the top of the tower while
a friend below measures the time to strike the ground below.
What is the height of the tower if the time is 6 s? (176.4 m)
2. A brick is given an initial downward velocity of 6 m/s. What is its final velocity after falling a distance of 40
m? (28.6 m/s)
3. A stone is thrown vertically upward and returns to its starting position in 5 s.
a. What was its initial velocity. (24.5 m/s)
b. How high did it rise? (30.6 m)
46
UNIT III
VECTORS AND TWO-DIMENSIONAL MOTION
TRIGONOMETRY
SOH CAH TOA
sin θ =
opp
hyp
cos θ =
adj
hyp
tan θ =
opp
adj
Solve for all sides and all angles for the following triangles. Your calculator must be in degree mode! Show all
your work.
1.  = 55o and c = 32 m, solve for a and b
2.  = 45o and a = 15 m/s, solve for b and c.
3. b= 17.8 m and  = 65o, solve for a and c.
47
4. a = 250 m and b = 180 m, solve for  and c.
5. a =25 cm and c = 32 cm, solve for b and .
6. b =10.4 cm and c = 65 cm, solve for a and .
(26.2 m, 18.35 m; 15 m/s, 21.2 m/s, 19.6 m, 8.3 m; 35.8o, 308.2 m; 38.6o, 20 cm; 9.2o, 64.2 cm)
A vector is represented by an arrowed line whose length is proportional to the vector quantity and whose
direction indicates the direction of the vector quantity.
Vector Notation:
A
Length of the arrow is proportional to the vectors magnitude.
Direction the arrow points is the direction of the vector.
48
VECTOR COMPONENTS
Working with vector quantities can often be simplified by resolving them into components. In two dimensions
vectors have two components, one corresponding to its extent along the x-axis and the other corresponding to its
extent along the y-axis.
R
+Ry
R
+Rx
R
or
+Ry
+Rx
The quadrant that a vector is in determines the sign of the x and y component vectors.
The direction is specified in degrees measured counterclockwise from the right (East).
Remember the plus and minus signs on you answers.
3.1 Draw and find the components of the vector v (250 km/h, 235o)
3.2. Draw and find the components of the vector x (19.20 m, 35o)
49
CW: VECTORS
1. Label each quantity as being either a scalar (S) or a vector (V).
___ Distance
___ Time
___ Force
___ Velocity
___ Mass
___ Area
___ Displacement
___ Acceleration
2. What are the x and y components of a velocity vector of magnitude 100 km/h and direction of 240°? (-50
km/h, -86.6 km/h)
3. A boy pushes a lawn spreader across a lawn by applying a force of 95 N along the handle that makes an angle
of 60° with the horizontal. What are the horizontal and vertical components of the force? (48 N, 82 N)
VECTOR ADDITION
It is easy to apply trigonometry to find the resultant R of
two vectors A and B that are perpendicular to each other.
The magnitude of the resultant is given by the Pythagorean theorem as: R 
The angle  between R and A may be found from: tan  
A2  B 2
b
b
and  = tan 1
a
a
50
3.3 A man drives 10 km to the north and then 20 km to the east. What are the magnitude and direction of his
displacement from the starting point?
VECTOR ADDITION: COMPONENT METHOD
To add two or more vectors A, B, C,… by the component method, follow this procedure:
1. Resolve the initial vectors into components x and y
2. Add the components in the x direction to give Σx and add the components in the y direction to give Σy .
Example: Σx = Ax + Bx + Cx
3. Calculate the magnitude and direction of the resultant R from its components by using the Pythagorean
y
theorem: R  (x) 2  (y ) 2 and   tan 1
x
3.4 Find the magnitude and direction of the resultant of these forces: A (5 N, 37°) and B (3 N, 180°).
51
3.5 Find the magnitude and direction of the resultant of these forces: A (60 N, 45°), B (20 N, 90°), and
C (40 N, 300°).
CW: VECTORS
4. Bob walks 80 m and then he walks 125 m.
a. What is Bob’s displacement if he walks east both times? (205 m)
b. What is Bob’s displacement if he walks east then west? (- 45 m or 45 m, W)
c. What distance does Bob walk in each case? (205 m)
5. A tiger paces back and forth along the front of its cage, which is 8 m wide. The tiger starts from the right side
of the cage, paces to the left side, then to the right side, and finally to the left.
a. What total distance has the tiger paced? (24 m)
b. What is the tiger’s resultant displacement? (8 m, Left)
52
6. Find the magnitude and direction of the resultant force produced by a vertically upward force of 40 N and a
left horizontal force of 30 N. R (50 N, 127˚) Hint: Draw a sketch!
7. You drive 4.1 km West on a street, then turn North and drive 17.3 km on the highway until you reach an
intersection. Turning on the intersection, you will reach your destination after traveling 1.2 km at an angle of
114.6°. What is your resultant displacement? (18.9 km, 104˚)
53
7. Honors Physics – Vectors are phun, really! –
CHECK____
PART I. TRIGONOMETRY
Solve for all sides and all angles for the following triangles. Show all your work.
Your calculator must be in degree mode!
a.
a.
b.
c.
40 kg
((a) 232.9 m, 606.8 m; (b) 23.6o, 458.3 km; (c) 26.6o, 89.43kg)
54
2. Sketch and calculate the x and y-components of:
a. a displacement of 200 km, at 340
b. a velocity of 40 km/h, at 1200
c. a force of 50 N at 330o
d. an acceleration of 2 m/s2 at 225o
2. A land-rover, on the surface of Mars, moves a distance of 38 m at an angle of 1800. It then turns and moves a
distance of 66 m at an angle of 2700. What is the displacement from the starting position? (76.2 m, 240.10)
55
3. A surveyor starts at the southeast corner of a lot and charts the following displacements:
A = 600 m, N; B = 400 m, W; C = 200 m, S; and D = 100 m, E. What is the net displacement? (500 m, 126.9o)
4. Two forces act on the car in the figure. Force A is 120 N, west and force B is 200 N at 600 N of W. What are
the magnitude and direction of the resultant force on the car? (280 N, 141.8o)
56
PROJECTILE MOTION
An object launched into space without motive
power of its own is called a projectile.
If we neglect air resistance, the only force acting
on a projectile is its weight, which causes its path
to deviate from a straight line.
The projectile has
- a constant horizontal velocity and
- a vertical velocity that changes uniformly under
the influence of gravity.
If an object is projected horizontally, its motion can best be described by considering its horizontal and vertical
motion separately. In the figure we can see that the vertical velocity and position increase with time as those of
a free-falling body. Note that the horizontal distance increases linearly with time, indicating a constant
horizontal velocity.
HORIZONTAL MOTION
Velocity
Distance
Time
VERTICAL MOTION
Velocity
Distance
Time
RESULTANT VELOCITY
57
3.6 A ball is thrown horizontally with a velocity of 12 m/s. How far (x, y coordinates) has the ball fallen in 2 s?
3.7 A rifle with a muzzle velocity of 200 m/s is fired with its barrel horizontally at a height of 1.5 m above the
ground. a. How long is the bullet in the air?
b. How far away from the rifle does the bullet strike the ground?
3.8 A ball is rolled off the edge of a table 1 m high with a horizontal velocity of 1.8 m/s. With what final
velocity does it strike the floor?
58
CW: HORIZONTAL PROJECTION
9. A marble is fired horizontally from a launching device that is 94 cm above the floor. The marble then strikes
the floor 2.35 m from the edge of the table.
a. How long will it take for the marble to reach the floor? (0.44 s)
b. What is the initial velocity of the marble as it leaves the launching device? (5.34 m/s)
c. What will be the horizontal velocity of the marble as it reaches the floor? (5.4 m/s)
d. What will be the vertical velocity of the marble as it reaches the floor? (4.3 m/s)
e. What will be the magnitude of the velocity of the marble as it reaches the floor? (6.8 m/s)
10. A cannon atop a mountain fired a projectile horizontally with a speed of 430 m/s, so that the projectile
landed at a horizontal distance of 4020 m from the cannon. How high would the ridge of the mountain be with
respect to the valley below? (428.3 m)
59
11. In 1977, a helicopter at the heliport atop the 59-story Pan Am building in New York fell over, causing the
rapidly turning rotor blades to splinter. One of these fragments landed about 101 m away, near the corner of
Madison Avenue and 43rd Street. If the fragment moved off the building horizontally with a speed of 14.25
m/s. Find the height of the Pan Am building. (246.16 m)
12. A lunch pail is accidentally kicked off a steel beam on a building under construction with an initial
horizontal speed of 1.50 m/s.
a. How far does the lunch pail fall after it travels 3.50 m horizontally? (26.7 m)
b. If the building is 250 m tall, and the lunch pail is knocked off the top floor, what will be the horizontal
displacement of the lunch pail when it reaches the ground? (10.7 m)
c. What is the magnitude of the velocity of the lunch pail when it reaches the ground? (70 m/s)
60
13. What is the range (horizontal displacement) of an arrow shot horizontally at 85.3 m/s if it is initially 1.50 m
above the ground? (47.2 m)
14. An archer stands 40.0 m from the target. If the arrow is shot horizontally with a velocity of 90.0 m/s, how
far above the bull’s-eye must he aim to compensate for gravity pulling his arrow downward? (0.97 m)
15. A bridge is 176.4 m above a river, If a lead-weighted fishing line is thrown from the bridge with a horizontal
velocity of 22.0 m/s, how far has it moved horizontally when it hits the water? (132 m)
61
8. Honors Physics –Projectiles right! –
CHECK_____
1. A baseball leaves a bat with a horizontal velocity of 20 m/s.
a. In a time of 0.25 s, how far will it have traveled horizontally? (5.0 m)
b. How far has it fallen vertically? (0.306 m)
2. A projectile has an initial horizontal velocity of 40 m/s at the edge of a roof top. Find the horizontal and
vertical components of its velocity after 3 s. (40.0 m/s, 29.4 m/s)
3. An airplane traveling at 70 m/s drops a box of supplies. What horizontal distance will the box travel before
striking the ground 340 m below? (583 m)
62
PROJECTILE MOTION AT AN ANGLE
The more general case of projectile motion occurs when the projectile is fired at an angle as shown in the
figure.
Problem solution:
1. Upward direction is positive. Acceleration (g) is downward thus negative.
g = ─ 9.8 m/s2
2. Resolve the initial velocity vo into its x and y components:
v0 x  v0 cos and
v0 y  v0 sin 
3. The horizontal and vertical components of its position at any instant is given by:
x  v0 x t
and
y  v0 y t  21 gt 2
4. The horizontal and vertical components of its velocity at any instant are given by:
v x  v0 x
and
v y  v0 y  gt
5. The final position and velocity can then be obtained from their components as a resultant.
63
3.9 A projectile is fired with a velocity of 196 m/s at an angle of 60 with the horizontal.
a. Find the time the projectile is in the air
b. Find the maximum height the projectile reaches
c. Find the final velocity. How does it compare with the initial velocity?
3.10 A projectile is fired at such an angle from the horizontal that the vertical component of its velocity is 49
m/s. The horizontal component of its velocity is 61 m/s.
a. How long does the projectile remain in the air?
b. What horizontal distance does it travel?
c. What is the initial velocity of the projectile?
64
3.11 A shell is fired at an angle of 40° above the horizontal at a velocity of 30 m/s.
a. Find the time required to reach its maximum height
b. Find the horizontal range.
CW: PROJECTILE MOTION AT AN ANGLE
16. A projectile is fired from the ground with an initial velocity of 96 m/s at an angle of 35˚.
a. Find the vertical and horizontal components of the initial velocity of this projectile. (78.6 m/s, 55 m/s)
b. How long will it take this projectile to reach the highest point in its trajectory? (5.6 s)
c. How long will this projectile be in the air? (11.22 s)
d. What will be the velocity of this projectile at the highest point? (78.6 m/s)
e. What will be the final velocity of this projectile as it reaches the ground? (96 m/s)
f. How high will this projectile be at the highest point of its trajectory? (154 m)
g. What will be the range of this projectile? (885 m)
65
17. A discus is released at an angle of 45˚ and a velocity of 24.0 m/s.
a. How long does it stay in the air? (3.5 s)
b. What horizontal distance does it travel? (59.5 m)
18. A golf ball is hit with a velocity of 24.5 m/s at 35.0° above the horizontal.
a. Find the range of the ball. (57.7 m)
b. Find the maximum height of the ball. (10.1 m)
66
9. Honors Physics –Projectiles at an angle! –
CHECK_____
1. Neatly draw and label the velocity vectors (vx and vy) onto the balls below:
2. A stone is given an initial velocity of 20 m/s at an angle of 580. What are its horizontal and vertical
displacements after 3 s? (31.8 m, 6.8 m)
67
3. A baseball leaves the bat with a velocity of 30 m/s at an angle of 300.
a. What are the horizontal and vertical components of its velocity after 3 s? (26 m/s, -14.4 m/s)
b. What is the maximum height. (11.5 m/s)
c. What is the range? (79.5 m/s)
4. A projectile leaves the ground with a velocity of 35 m/s at an angle of 320. What is the maximum height
attained. (17.5 m)
68
UNIT IV
DYNAMICS
FORCE
An object that experiences a push or a pull has a force exerted on it. We will consider the forces exerted ON the
object. The object is called the system. The world around the object that exerts forces on it is called the
environment. Force is a vector quantity and is represented by the symbol F.
CONTACT VERSUS FIELD FORCES
Forces exerted by the environment on a system can be divided into two types:
- Contact Forces: act on an object only by touching it. Examples: tension, friction, normal.
- Field Forces or Long-Range Forces: are exerted without contact. Example: gravitational, magnetic,
electric.
Type of Force
and its Symbol
Applied Force
Normal Force
Friction Force
Description of Force
Direction of Force
An applied force is a force that is applied to an object by
another object or by a person.
If a person is pushing a desk across the room, then there is
applied force acting upon the desk. The applied force is the
force exerted on the desk by the person.
The normal force is the support force exerted upon an object
that is in contact with another stable object. For example, if
a book is resting upon a surface, then the surface is exerting
an upward force upon the book. The normal force is always
perpendicular to the surface
The friction force is the force exerted by a surface as an
object moves across it or makes an effort to move across it.
The friction force opposes the motion of the object. If a
book moves across the surface of a desk, the desk exerts a
friction force in the direction opposite to the motion of the
book.
Tensional Force
Tension is the force that is transmitted through a string,
rope, or wire when it is pulled tight by forces acting at each
end. The tensional force is directed along the wire and pulls
equally on the objects on either end of the wire.
Air Resistance
Force
Air resistance is a special type of frictional force that acts
upon objects as they travel through the air. The force of air
resistance always opposes the motion of the object. This
force will frequently be ignored due to its negligible
magnitude. It is most noticeable for objects that travel at
high speeds (e.g., a skydiver or a downhill skier) or for
objects with large surface areas.
69
The force of gravity is the force with which the earth, moon,
or other massive body attracts an object towards itself. By
Gravitational
definition, this is the weight of the object. All objects upon
Force (also known
earth experience a force of gravity that is directed
as Weight)
"downward" towards the center of the earth. The force of
gravity on an object on earth is always equal to the weight
of the object.
FORCES HAVE AGENTS
Each force has a specific identifiable, immediate cause called agent. You should be able to name the agent of
each force, for example the force of the desk or your hand on your book. The agent can be animate such as a
person, or inanimate such as a desk, floor or a magnet. The agent for the force of gravity is Earth's mass. If you
can't name an agent, the force doesn't exist!
4.1 There are several situations described below. For each situation, decide which forces are present and absent.
1. A block hangs at rest from the ceiling by a piece of rope. Consider the forces acting on the block.
____Gravity
____Tension
____Normal
____Friction
2. A block hangs from the ceiling by a spring. Consider the forces acting on the block when it is at rest (at its
equilibrium position).
____Gravity
____ Spring
____Tension
____Normal
____Friction
3. A ball is shot into the air with a spring-loaded cannon. Consider the forces acting on the ball while it is in the
air.
____Gravity
____Tension
____Normal
____Friction
____Air Resistance
4. A skydiver who hasn't opened his parachute yet is falling. Consider the forces acting on the skydiver.
____Gravity
____Tension
____Normal
____Friction
____Air Resistance
70
5. A block rests on top of a table. Consider only the forces acting on the block.
____Gravity
____Tension
____Normal
____Friction
____Air Resistance
6. A block slides across the top of a table. Consider only the forces acting on the block.
____Gravity
____Tension
____Normal
____Friction
7. The wheels of a car are locked as it skids to a stop while moving across a level highway.
____Gravity
____Tension
____Normal
____Friction
____Air Resistance
8. A block rests on an incline plane without sliding. Consider the forces acting on the block.
____Gravity
____Tension
____Normal
____Friction
9. A block slides across the top of a table. Consider only the forces acting on the table.
____Gravity
____Tension
____Normal
____Friction
10. A car is attached by a cable to a moving truck and is being pulled along a level highway. Consider only the
forces acting on the car.
____Gravity
____Tension
____Normal
____Friction
____Air Resistance
71
11. A person is sitting on a sled and gliding across loosely packed snow along a horizontal surface. Consider
only the forces acting on the person.
____Gravity
____Tension
____Normal
____Friction
____Air Resistance
NOTE: Air Resistance will usually be ignored in most Physics problems.
FREE-BODY-DIAGRAMS
A free-body-diagram (FBD) is a vector diagram that shows all the forces that act on an object whose motion
is being studied.
4.2 Draw a FBD for each situation below following these directions:
- Choose a coordinate system defining the positive direction of motion.
- Replace the object by a dot and locate it in the center of the coordinate system.
- Draw arrows to represent the forces acting on the system.
1. Object lies motionless.
2. Object slides at constant speed without friction
3. Object slows due to kinetic friction.
4. Object slides without friction.
72
5. Static friction prevents sliding.
6. An object is suspended from the ceiling.
7. An object is suspended from the ceiling.
8. The object is motionless.
9. The object is motionless.
10. The object is motionless.
73
11. The object is pulled by a force parallel to the surface.
12. The object is pulled by a force at an angle to the surface..
13. The object is pulled upward at constant speed.
14. The object is pushed by a force applied downward at an angle.
.
15. The object is falling at constant (terminal) velocity.
16. The ball has been punted by a football player.
74
10. Honors Physics – FBD Fiesta! –
CHECK_____
Draw neat, labeled Free-Body-Diagrams for each case:
75
76
2. Draw free-body diagrams (FBD’s) showing all forces acting on each body. Then using each FBD as a guide,
write down the ΣFx and ΣFy expressions,
Object resting on frictionless surface
Object being pulled by a force at an angle.
Object being pulled up the incline by a force
77
HOOKE'S LAW
Whenever a spring is stretched from its equilibrium position and released, it will move back and forth on either
side of the equilibrium position. The force that pulls it back and attempts to restore the spring to equilibrium is
called the restoring force.
F= - k x
Where k (N/m) is a constant called the force constant or spring constant and is a measure of the stiffness of the
particular spring. This equation is known as the spring equation or Hooke’s Law.
4.3 A force of 5 N compresses a spring by 4 cm. Find the force constant of the spring.
CW. HOOKE'S LAW
1. A spring is stretched by 30 mm when a force of 0.40 N is applied to it. Find the spring constant. (13.3 N/m)
2. A child exerts a force of 12 N to shoot a rubber band across the room. If the rubber band has a spring
constant of 180 N/m, what is the rubber band’s displacement? (0.067m)
78
NEWTON'S FIRST LAW OF MOTION
Newton's first law of motion is often stated as:
"An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same
speed and in the same direction unless acted upon by an unbalanced force."
There are two parts to this statement:
- one which predicts the behavior of stationary objects and
- the other which predicts the behavior of moving objects.
The behavior of all objects can be described by saying that objects tend to "keep on doing what they're doing"
(unless acted upon by an unbalanced force). If at rest, they will continue in this same state of rest. If the object
is in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East).
It is the natural tendency of objects to resist changes in their state of motion. This tendency to resist changes in
their state of motion is described as inertia.
Inertia is the resistance an object has to a change in its state of motion.
FORCE UNITS
Force is a vector quantity measured using a standard metric unit known as the Newton (N). One Newton is the
amount of force required to give a 1-kg mass an acceleration of 1 m/s2.
MASS AND WEIGHT
A quantitative measure of inertia is mass. The SI unit of mass is the kilogram (kg).
The weight of a body is the gravitational force with which the Earth attracts the body.
Weight (a vector) is different from mass (a scalar).
The weight of a body varies with its location near the Earth (or other astronomical body), whereas its mass is
the same everywhere in the universe. The weight of a body is the force that causes it to be accelerated
downward with the acceleration of gravity g (9.8 m/s2).
FG = mg
Units: N (Newtons)
4.4 a. What is the weight of 6 kg of potatoes?
b. What is the mass of 6 N of potatoes?
79
CW. MASS AND WEIGHT
3. Refer to the diagrams below to answer questions A-H. Circle the correct answer.
A. The agent of FN is _____ .
a. the bowl
c. friction
b. Earth
d. the shelf
B. The agent of Fg is _______ .
a. the bowl
c. friction
b. Earth
d. the shelf
C. What part of Diagram 2 best represents the bowl in equilibrium?
a. A
c. C
b. B
d. D
D. Which part of Diagram 1 best represents the weight force of the bowl sitting on a shelf?
a. A
c. C
b. B
d. D
E. FN is a symbol that represents the _______ force.
a. friction
c. normal
b. tension
d. weight
F. The magnitude of the net force on the bowl in equilibrium is ______ .
a. FN
c. 0
b. Fg
d. 2 Fg
G. Which of these is true when the bowl is in equilibrium?
a. FN = Fg
c. FN > Fg
b. FN ±Fg
d. FN < Fg
H. Which part of Diagram 2 best represents the bowl if it falls off the shelf?
a. A
c. C
b. B
d. D
80
NET FORCE (ΣF)
An unbalanced force exists whenever all vertical forces (up and down) do not cancel each other and/or all
horizontal forces (left and right) do not cancel each other. Whenever there is an unbalanced force, there is a net
force. The net force is the vector sum of all the forces that act upon the object.
CW: NET FORCE
4. Find the magnitudes of the net force for each case.
Fnet = ______________
Fnet = ______________
Fnet = ______________
Fnet = ______________
FIRST CONDITION FOR EQUILIBRIUM
An object is in translational equilibrium if and only if the vector sum of the forces acting upon it is zero.
Σ Fx = 0
Σ Fy = 0
This is a statement of Newton's First Law of Motion for objects at rest or moving in a straight line at constant
velocity.
4.5 A guy pushes a 2.0 kg broom at constant speed across the floor. The broom handle makes a 50° angle with
the floor. He pushes the broom with a 5.0 N force.
a. Draw a FBD.
81
b. Calculate the normal force.
c. Calculate the frictional force opposing the motion.
4.6 A ball of weight 8.0 N hangs from a cord that is knotted to two other cords, fastened as shown.
a. Draw the free body diagram at the knot.
b. Calculate the tensions T1 and T2.
82
4.7 A 125 N bucket hangs from support cables as shown. What is the tension in the two cables?
28.5
55.0
125 N
4.8 A 441 N crate is being pulled by a rope, up a frictionless inclined plane, which meets the horizontal at an
angle of 35˚.
a. Draw a FBD of the forces acting on the crate.
b. Find the magnitude of the normal force (FN) acting on the crate. (361.2 N)
83
c. Find the magnitude of the tension force (FT) in the rope. (252.9 N)
CW: FIRST LAW
5. The box on the frictionless ramp is held at rest by the tension force. The mass of the box is 20 kg.
a. Draw a neat FBD.
b. What is the tension force? (98 N)
c. What is the normal force? (169.7 N)
84
6. Draw a neat FBD and find the tension in the ropes A and B for the Figure. (A = 1405 N, B = 1140 N)
85
7. A block of weight 50 N hangs from a cord that is knotted to two other cords, A and B fastened to the ceiling.
B makes an angle of 60˚ with the ceiling and A forms a 30° angle
a. Draw the free body diagram at the knot.
b. Calculate the tensions A and B. (25 N, 43.3 N)
86
8. A man pulls a 50 kg box at constant speed across the floor. He applies a 200 N force at an angle of 30°.
a. Draw a neat FBD.
b. Calculate the frictional force opposing the motion? (173.2 N)
c. Calculate the normal force. (390 N)
87
11. Honors Physics – Equilibrium is N1L
CHECK_____
1. A woman weighs 800 N on earth. When she walks on the moon, she weighs only 133 N.
a. What is the acceleration due to gravity on the moon? (1.63 m/s2)
b. What is her mass on the moon? (81.6 kg)
2. Three identical bricks are strung together with cords and hung from
a scale that reads a total of 24 N.
a. What is the tension in the cord that supports the lowest brick? (8 N)
Draw a neat FBD!
b. What is the tension in the cord between the middle brick and the top brick? (16 N) Draw a neat FBD!
88
2. Find the tension in each cable supporting the 80 N SuperKitKat burglar. (95.4 N, 124.5 N)
o
40
37.0
A
B
3. A 50.51 kg traffic light hangs from two cables at the angles shown. Calculate the tensions in the two cables.
(443 N, 110 N)
78
33
89
4.
A teacher pulls a 50 kg desk with a 200 N force acting at 30° angle above the horizontal.
The desk does not budge.
a. Draw a neat FBD for the desk.
b. Determine the value of the frictional force. (173.2 N)
c. Determine the value of the normal force. (390 N)
6. Suppose in the diagram above, that the teacher is pushing down at a 30° angle with 200 N of force. The
desk still does not move. a. Draw a force diagram for the desk.
b. Determine the value of the frictional force. (173.2 N)
c. Determine the value of the normal force. (590 N)
90
5. A 70-N block rests on a 300 inclined plane. a. Draw a neat, labeled FBD.
b. Determine the normal force. (60.6 N)
c. Determine the friction force that keeps the block from sliding. (35 N)
6. What push P directed up the plane will cause the block in the previous problem to move up the plane with
constant speed if the force of friction is 18 N? Draw a neat, labeled, FBD. (53 N)
91
TORQUE
Torque is a measure of a force's ability to rotate an object. It depends on the magnitude of the applied force and
on the length of the moment arm d, according to the following equation:
τ=Fd
Units: N.m
d is measured perpendicular to the line of action of the force F
Sign Convention:
Torque will be positive if F tends to produce counterclockwise rotation.
Torque will be negative if F tends to produce clockwise rotation.
ROTATIONAL EQUILIBRIUM
An object is in rotational equilibrium when the sum of the forces and torques acting on it is zero.
First Condition of Equilibrium: Σ Fx = 0 and Σ Fy = 0
Second Condition of Equilibrium: Σ τ = 0
(translational equilibrium)
(rotational equilibrium)
By choosing the axis of rotation at the point of application of an unknown force, problems may be simplified.
CENTER OF MASS
The terms "center of mass" and "center of gravity" are used synonymously to represent the average position of
all the mass that makes up the object. For example, a symmetrical object, such as a ball, has its center of mass at
its geometrical center.
4.9 A uniform beam of negligible weight is held up by two supports A and B. Given the distances and forces
listed find the forces exerted by the supports.
92
4.10 A 300 N girl and a 400 N boy stand on a 16 m platform supported by posts A and B as shown. The
platform itself weighs 200 N. What are the forces exerted by the supports on the platform?
CW: EQUILIBRIUM
9a. What is the resultant torque about point A in the figure. (90 Nm CCW)
15 N
4m
30 N
3m
2m
A
20 N
93
b. Find the resultant torque if the axis is moved to the left end of the bar. (120 Nm CW)
10. Assume that the weight of the bar is negligible. Find forces A and F. (26.7 N, 106.7 N)
94
12. Honors Physics – Torque
CHECK_____
1. Weights of 100, 200, and 500 N are placed on a light board resting on two supports as shown in the figure.
What are the forces exerted by the supports? (375 N, 425 N)
A
2m
B
3m
3m
2m
Axis
100 N
200 N
500 N
2. A 4-m pole is supported at each end by hunters carrying an 800-N deer which is hung at a point 1.5 m from
the left end. Draw a neat FBD and calculate the upward forces required by each hunter. (500 N, 300 N)
95
3. A bridge whose total weight is 4500 N is 20 m long and supported at each end. Find the forces exerted at
each end when a 1600-N tractor is located 8 m from the left end. (2890 N, 3210 N)
B
A
8m
2m
10 m
Axis
1600 N
4500 N
4. An 8-m steel metal beam weighs 2400 N and is supported 3 m from the right end.
a. If a 9000-N weight is placed on the right end, what force must be exerted at the left
end to balance the system? (4920 N)
4m
A
1m
2400 N
F
3m
9000 N
b. What is the magnitude of the support force F? (16,320 N)
96
NEWTON'S THIRD LAW OF MOTION
According to Newton's third law of motion, when one object exerts a force on another object, the second object
exerts on the first an equal force in opposite direction.
The third law of motion applies to two different forces on two different objects: the action force one object
exerts on the other, and the equal but opposite reaction force the second object exerts on the first. Action and
reaction forces never balance out because they act on different objects.
4.11 On each side of the arrows for the action-reaction pair write a statement of the third law.
CW: THIRD LAW
11. Complete each statement of the third law and add the reaction arrow for the following situations
97
12. Nellie Newton holds an apple weighing 1 N at rest on the palm of her hand.
a. Show the force vectors that act on the apple on the diagram below.
b. To say the weight of the apple is 1 N is to say that a downward gravitational force of 1 N is exerted on the
apple by
(the earth)
(her hand)
c. Nellie’s hand supports the apple with normal force FN, which acts in a direction opposite to FG. We can say
that FN
(equals FG) (has the same magnitude as FG)
d. Since the apple is at rest, the net force on the apple is: (zero) (nonzero).
e. Since FN is equal and opposite to FG, we (can) (cannot) say that FN and FG comprise an action- reaction pair.
The reason is because action and reaction always
(act on the same object)
(act on different objects),
and here we see FN and FG
(both acting on the apple)
(acting on different objects).
f. In accord with the rule, “If ACTION is A acting on B, then REACTION is B acting on A,” if we say action
is the earth pulling down on the apple, reaction is
(the apple pulling up on the earth)
(FN, Nellie’s hand pushing up on the apple).
g. To repeat for emphasis, we see that FN and FG are equal and opposite to each other
(and comprise an action-reaction pair)
(but do not comprise an action-reaction pair).
h. Another pair of forces is FN [shown] and the downward force of the apple against Nellie’s
hand [not shown].
This force pair
(is)
(isn’t)
an action-reaction pair.
i. Once the apple leaves Nellie’s hand, FN is
(zero)
(still twice the magnitude of FG),
and the net force on the apple is
(zero)
(only FG)
(still FG - FN, which is a negative force).
98
SECOND LAW OF MOTION
If there is a net force acting on an object, the object will have an acceleration and the object's velocity will
change. How much acceleration will be produced by a given force?
Newton's Second Law states that for a particular force, the acceleration of an object is directly proportional to
the net force and inversely proportional to the mass of the object. The direction of the force is the same as that
of the acceleration. In equation form:
a
F
m
or
F = ma
UNITS: Newton (N)
A Newton is that net force which, when applied to a 1-kg mass, gives it an acceleration of 1 m/s2.
The second law of motion is the key to understanding the behavior of moving bodies since it links cause (force)
and effect (acceleration) in a definite way.
4.12 A force of 80 N gives an object of unknown mass an acceleration of 20 m/s2. What is its mass?
4.13 Danny the skater, total mass 25 kg, is propelled by rocket power.
a. Draw a FBD of the skater:
b. Write the net force equation (∑F)
c. Calculate Danny’s acceleration.
99
c. Assume that there is a 50 N force of air resistant. Redo the FBD including this force.
d. Write the net force equation (ΣF)
e. Calculate Danny’s acceleration.
4.14 A box of mass m hangs by a string from the ceiling of an elevator as shown.
a. Draw a FBD of the forces acting on the box.
m
b. What is the tension in the string when the elevator accelerates upward?
c. Calculate the tension in the string if the mass of the box is 5.0 kg and the upward acceleration of the elevator
is 1.4 m/s2?
100
4.15 A 20 kg mass is allowed to accelerate down a frictionless 15° ramp. a. Draw the FBD.
b. Determine the value of the x-component of the gravitational force.
c. What is the acceleration of the block down the ramp?
CW: NET FORCE
13. Calculate the acceleration of a 113-kg cart when the net force on it is 5 N. (0.044 m/s2)
14. Calculate the horizontal force needed to make a 895 g hockey puck accelerate at 135 m/s2. (120.8 N)
101
15. A 1.5 kg bucket at rest is suspended from a string.
a. Draw a simple sketch and a neat, labeled FBD of the forces acting on the mass.
b. Write the net force equation (ΣF)
c. What is the net force on the bucket? (0 N)
d. What is the net force if the mass moves down at constant speed? (0 N) Explain your answer.
16. The 1.5 kg bucket is lifted with a tension of 45 N in order to accelerate it.
a. Draw a neat, labeled FBD of the forces acting on the bucket.
b. Write the net force equation (ΣF)
c. What is the acceleration on the bucket? (20 m/s2)
102
17. The 1.5 kg bucket is now being lowered with a tension of 3.5 N in order to accelerate it.
a. Draw a neat, labeled FBD of the forces acting on the bucket.
b. Write the net force equation (ΣF)
c. What is the acceleration on the bucket? (7.67 m/s2)
18. A crate with a mass of 125 kg is accelerated upward at 0.45 m/s2 by means of a cable.
a. Calculate the weight of the crate. (1250 N)
b. Draw a neat, labeled FBD of the forces acting on the crate.
c. Write the net force equation (ΣF)
d. Find the tension force. (1306.25 N)
For the following problems:
- Draw a neat, labeled FBD
- Write the equation for the net force: ΣF
- Solve for the unknown
103
19. A woman exerts 100 N of force to lift a laundry basket weighing 75 N. Calculate the upward acceleration on
the basket. (3.33 m/s2)
20. A person whose mass is 75 kg is lowered with a rope down the side of a cliff. The rope being used can
support a maximum of 595 N. In order to prevent the rope from breaking, at what minimum rate must the
person accelerate downward? (2 m/s2)
21. An applied 25 N force pushes on a 5.0 kg block resting on a frictionless horizontal surface. The force is
directed downwards at an angle of 20°. a. Draw the FBD.
20°
104
b. Determine the x-component of the applied force. (23.5 N)
b. Find the normal force. (57.5 N)
c. What is the acceleration of the block? (4.7 m/s2)
22. A 70.0 kg box is pulled by a 400 N force at an angle of 30° to the horizontal. The force of friction is 75 N.
a. Draw the FBD for the box.
b. What is the acceleration of the box? (3.88 m/s2)
105
23. A block is pulled by a rope as shown in the diagram below. Assume that the ramp is frictionless.
a. Draw the force diagram for the block on the ramp.
b. What is the x-component of the force of gravity acting on the block on the ramp? (84.3 N)
c. What is the acceleration of the block? (4.38 m/s2)
APPARENT WEIGHT
The actual weight of a body is the gravitational force that acts on it. The body's apparent weight is the force
the body exerts on whatever it rests on. Apparent weight can be thought of as the reading on a scale a body is
placed on.
4.16 A 80-kg man is standing on a scale inside of an elevator. For each case:
- Draw the free-body diagram.
- Find the net force.
- Calculate the force of the scale of the man.
a. At rest or traveling at constant velocity
106
b. Going up with an acceleration of 1.5 m/s2.
c. Going down with an acceleration of 1.5 m/s2.
d. After too much playing with the elevator the cable snaps!
CW: APPARENT WEIGHT
24. An 800-N woman stands on a scale in an elevator. What does the scale read when the elevator is
a. Ascending at a constant velocity of 3 m/s, (800 N)
b. Ascending at a constant acceleration of 0.8 m/s2, (864 N)
c. Descending at a constant acceleration of 0.8 m/s2, (736 N)
107
13. Honors Physics – Newton’s Laws – 2
CHECK____
1. It is determined that a resultant force of 60 N will give a wagon an acceleration of 10 m/s2. What force is
required to give the wagon an acceleration of only 2 m/s2? (12 N)
For the following problems:
- Draw a neat, labeled FBD
- Write the equation for the net force: ΣF
- Solve for the unknown
2. What horizontal pull is required to drag a 6-kg sled with an acceleration of 4 m/s2 if a friction force of 20 N
opposes the motion? (44 N)
108
3. A 10-kg mass is lifted upward by a light cable. What is the tension in the cable if the acceleration is
a. zero, (98 N)
b. 6 m/s2 upward, (158 N)
c. 6 m/s2 downward? (38 N)
4. An 800-kg elevator is lifted vertically by a strong rope. Find the acceleration of the elevator if the rope
tension is 9000 N? (1.45 m/s2)
109
5. In the figure an unknown mass slides down the 300 inclined plane. What is the acceleration in the absence of
friction? (4.90 m/s2)
6. An elevator is moving downward with an acceleration of 2.0 m/s2.
a. What is the force exerted by the elevator floor on a 75 kg person. (585 N)
b. The elevator is now moving upward with an acceleration of 2.0 m/s2 what is the force exerted by the elevator
floor on a 75 kg person. (885 N)
110
PULLEY SYSTEMS
4.17 The figure shows a 5-kg block 1 that hangs from a string that passes over a frictionless pulley and is joined
at its other end to a 12-kg block 2 that lies on a frictionless table.
This arrangement is called a modified Atwood’s machine.
a. Draw a FBD of the system. Use a highlighter to trace the direction of motion.
2
1
b. Write the net force equation (ΣF) for the system (i.e. along the direction of motion).
c. Find the acceleration of the system.
d. Find the tension in the string.
111
4.18 The figure shows the same two blocks, 1 and 2, from the previous problem but now suspended by a string
on either side of a frictionless pulley.
a. Draw a FBD of the system. Use a highlighter to trace the direction of motion.
1
2
b. Write the net force equation (ΣF) for the system (i.e. along the direction of motion).
c. Find the acceleration of the system.
d. Find the tension in the string.
112
CW: PULLEYS.
25. A 47 N box is pulled along a frictionless horizontal surface by a 25 N weight hanging from a cord on a
frictionless pulley.
a. Draw a sketch of the situation.
b. Find the mass of the box and label it m1. Find the mass of the hanging weight and label it m2.
c. Draw a neat, labeled FBD of the forces acting on each object. Use a highlighter to trace the direction of
motion.
d. Write the net force equation (ΣF)
e. Find the acceleration of the system. (3.4 m/s2)
f. Find the tension in the cord. (16.3 N)
113
26. The same two objects on the previous problem are now hanging side to side over a frictionless pulley.
a. Draw a sketch of the situation.
b. Draw a neat, labeled FBD of the forces acting on each object. Use a highlighter to trace the direction of
motion. (Hint: which side is heavier?)
c. Write the net force equation (ΣF)
d. Find the acceleration of the system. (3.0 m/s2)
e. Find the tension in the cord. (32.6 N)
114
27. A 4-kg block 1 that hangs from a string that passes over a frictionless pulley and is joined at its other end to
a 10-kg block 2 that lies on a frictionless table.
a. What is the acceleration of the blocks? Show ALL your work: FBD, ΣF (2.8 m/s2)
b. What is the tension, FT, in the rope? (28.0 N)
115
28. The same two blocks, 1 and 2, from the previous problem are now suspended by a string on either side of a
frictionless pulley. Show ALL your work: FBD, ΣF
a. What is the acceleration of blocks? (4.2 m/s2)
b. What is the tension, FT, in the rope? (56 N)
116
14. Honors Physics – Newton’s Laws Ain’t Over – 3
CHECK_____
For the following problems:
- Draw a neat, labeled FBD
- Write the equation for the net force: ΣF
- Solve for the unknown
1a. What is the acceleration of the system? (Friction is negligible) (9.8 m/s2)
2 kg
6 kg
T
80 N
b. What is the tension T in the connecting cord? (19.6 N)
117
2a. What is the acceleration of the system? (5.88 m/s2)
b. What is the tension in the connecting cord for the arrangement shown? (23.5 N)
3a. Assume that the masses m1 = 2 kg and m2 = 8 kg are connected by a cord that passes over a light frictionless
pulley as in the figure. What is the acceleration of the system? (5.88 m/s2)
b. What is the tension in the cord? (39.3 N)
118
FRICTIONAL FORCE
When two surfaces are in direct contact and one surface either slides or attempts to slide across the other, a
force, called friction, that opposes the motion (or attempted motion) is generated between the surfaces.
The origin of friction is based on the microscopic structure of the surfaces involved.
Static friction occurs between surfaces at rest relative to each other. When an increasing force is applied to a
book resting on a table, for instance, the force of static friction at first increases as well to prevent motion. In a
given situation, static friction has a certain maximum value called starting friction. When the force applied to
the book is greater than the starting friction, the book begins to move across the table.
The kinetic friction (or sliding friction) that occurs afterward is usually less than the starting friction, so less
force is needed to keep the book moving than to start it moving.
COEFFICIENT OF FRICTION
The frictional force between two surfaces depends on the normal (perpendicular) force FN pressing them
together and on the nature of the surfaces. The latter factor is expressed quantitatively in the coefficient of
friction  (mu) whose value depends on the materials in contact. The frictional force is experimentally found to
be:
Ff = FN 
4.19 A force of 300 N is sufficient to keep a 100-kg wooden crate moving at constant velocity across a wooden
floor. What is the coefficient of kinetic friction?
4.20 A force of 1000 N is applied to a 1200-kg car. If the coefficient of friction is 0.04, what is the car’s
acceleration?
119
4.21 A car whose brakes are locked skids to a stop in 70 m from an initial velocity of 80 km/h. Find the
coefficient of kinetic friction.
CW: FRICTION
29. A 2.0-kg brick has a sliding coefficient of friction of 0.38. What force must be applied to the brick for it to
move at a constant velocity? (7.4 N)
30. A truck moving at 100 km/h carries a steel girder that rests on its wooden floor. What is the minimum time
in which the truck can come to a stop without the girder moving forward? The coefficient of static friction
between steel and wood is 0.5. (5.65 s)
120
31. A force of 20 N accelerates a 9.0-kg wagon at 2.0 m/s2 along the sidewalk.
a. How large is the frictional force? (2 N)
b. What is the coefficient of friction? (0.02)
32. A 70 kg box is pulled by a 400 N force at an angle of 30 with the horizontal. If μ = 0.5.
a. Draw a free body diagram.
b. Find: the x and y components of the applied force. (346.4 N, 200 N)
c. Find the normal force. (486 N)
121
d. Find the frictional force. (243 N)
e. Find the acceleration of the box. (1.47 m/s2)
33. A rope is being used to pull a sled along a horizontal surface at a constant speed with an applied force of
125 N. The sled, including the load, has a mass of 28.0 kg. The angle between the rope and the horizontal is 23˚.
a. Complete the FBD.
b. Calculate the normal force acting on this sled as it is pulled to the right at a constant speed? (225.6 N)
d. Find the frictional force acting on the sled as it is pulled to the right at a constant speed? (115.06 N)
122
34. A sled with a mass of 125 kg is sitting on an icy horizontal surface. A rope is attached to the front end of the
sled such that the angle between the rope and the horizontal is 33˚ and a force of 375 N is applied to the rope.
As a result the sled moves along the horizontal surface with a uniform acceleration.
a. Complete the free body diagram showing all the forces acting on the sled.
b. What is the magnitude of the normal force acting on the sled? (1020.76 N)
c. If the coefficient of friction between the sled and the icy horizontal surface is 0.21, what is the frictional force
acting on this sled? (214.35 N)
d. Find the acceleration of the system. (0.8 m/s2)
123
15. Honors Physics – Mu? Me?
CHECK_____
1. Assume surfaces where s = 0.7 and k = 0.4. a. What horizontal force is needed to just start a 50-N block
moving along a wooden floor? (35 N)
b. What force will move it at constant speed? (20 N)
2. A 60-N toolbox is dragged horizontally at constant speed by a rope making an angle of 350 with the floor.
The tension in the rope is 40 N.
a. Draw a neat FBD
b. Determine the magnitude of the normal force. (37.1 N)
124
c. Determine the magnitude of the friction force. (32.8 N)
c. What is the coefficient of kinetic friction? (0.884)
3. A horizontal force of 100 N pulls an 8-kg cabinet across a level floor. Find the acceleration of the cabinet if
k = 0.2. (10.5 m/s2)
4. An unknown mass slides down the 300 inclined plane shown below. Determine the acceleration if k = 0.2
(3.2 m/s2)
125
CIRCULAR MOTION
Suppose that you were driving a car with the steering wheel turned in such a way that your car followed the
path of a perfect circle with a constant radius. The speedometer maintained a constant reading of 10 mi/hr. In
this situation, the motion of your car would be described to be experiencing uniform circular motion. Uniform
circular motion is the motion of an object in a circle with a constant or uniform speed.
When moving in a circle, an object travels a distance around the perimeter of the circle. So if your car were to
move in a circle with a constant speed of 5 m/s, then the car would travel 5 meters along the perimeter of the
circle in each second of time. The distance of one complete cycle (or revolution) around the perimeter of a
2 r
circle is known as the circumference 2 r, therefore the speed in m/s is given by: v 
T
where the period T is the time to complete one full rotation or revolution and is given in seconds. The
frequency is the number of rotations or revolutions per unit time, the unit is called Hertz (Hz).
1
1
(s)
and
f  (Hz)
T
T
f
4.22 A car travels 125 rpm (revolutions per minute) around a track that has radius of 30 meters
a. Find the period.
b. Find the velocity
Objects moving in uniform circular motion will have a constant speed. But does this mean that they will have a
constant velocity? No! Although the magnitude of the velocity of a body in uniform circular motion is
constant, its direction changes continually. The best word that can be used to describe the direction of the
velocity vector is the word tangential. The direction of the velocity vector at any instant is in the direction of a
tangent line drawn to the circle at the object's location. (A tangent line is a line that touches the circle at one
point but does not intersect it.)
We learned that if an object changes its direction, then the object is accelerating. In the case of circular motion,
the acceleration is called centripetal acceleration. The term centripetal means that the acceleration is always
directed towards the center of the circle.
126
v2
The centripetal acceleration is given by: ac 
r
Units: m/s2.
The acceleration is perpendicular to the velocity.
4.23 A car moving at 10 m/s takes a turn around a circle with a radius of 25.0 m. Determine its centripetal
acceleration.
CENTRIPETAL FORCE
The inward force that must be applied to keep an object moving in a circle is called centripetal force. Without
centripetal force, circular motion cannot occur!
From Newton's second law of motion: F = ma, substituting the centripetal acceleration we obtain:
mv 2
Units: Newtons (N).
Fc 
r
The centripetal force is not a 'special' kind of force. The centripetal force is provided by the force that keeps the
object in a circle, this is called the centripetal force requirement.
4.24 a. A car makes a turn, what force is required to keep it in circular motion?
b. A bucket of water is tied to a string and spun in a circle, what force is required to keep it in circular motion?
c. The moon orbits the Earth, what force is required to keep it in circular motion?
127
4.25 A 1000 kg car rounds a turn of radius 30 m at a velocity of 9 m/s.
a. What provides the centripetal force?
b. Calculate the force.
4.26 How much centripetal force is required to keep a 45 kg skater moving in a circle 12 m in diameter at a
velocity of 3 m/s? What provides the centripetal force?
4.27 The maximum force of friction a road can exert on the tires of a 1500 kg car is 8500 N. What is the
maximum velocity at which the car can round a turn of radius 120 m?
4.28 A car is traveling at 45 km/h on a level road where the coefficient of friction between tires and road is 0.8.
Find the minimum turn radius of the car.
128
CW: CENTRIPETAL FORCE
35. A physics student swings a small rubber ball attached to a string over her head in a horizontal, circular path.
The piece of string is 50 cm long and the ball makes 75 complete circles each minute.
a. What is the period? (0.8 s)
b. Calculate the velocity. (3.92 m/s)
c. Find the ball’s centripetal acceleration? (30.7 m/s2)
35. A model electric train moves along a circular track. The train has a speed of 0.35 m/s and has a centripetal
acceleration of 0.29 m/s2. What is the diameter of the track? (84 cm)
36. The radius of the earth’s orbit about the sun is about 1.5x1011 m. The mass of the earth is 5.98x1024 kg and
it takes the earth 365 days to go around the sun.
a. Find the period of the earth's orbit. (3.15x107 s)
b. Find the velocity of the Earth.(2.99x104 m/s)
129
c. Calculate the centripetal acceleration of the earth. (5.96x10-3m/s2)
d. How much centripetal force is acting on the earth? (3.56x1022N)
37. A small asteroid with a mass of 2.05x108 kg is pulled into a circular orbit around Earth. The distance from
the asteroid to Earth’s center is 7378 km. If the gravitational force needed to keep the asteroid in orbit has a
magnitude of 3x109 N, what is the asteroid’s speed? (1.04x104 m/s)
38. A 60.0-kg speed skater with a velocity of 18.0 m/s comes into a curve of 20 m radius. How much friction
must be exerted between to negotiate the curve? (972 N)
39. A1250 kg automobile with a speed of 48 km/h follows a circular road that has a radius of
35 m. The pavement is wet and oily, so the coefficient of kinetic friction between the car’s tires and the
pavement is only 0.5. a. How large is the force needed to maintain the car’s circular motion? (6349.2 N)
b. How large is the available frictional force? (6125 N)
c. Is the available frictional force large enough to maintain the automobile’s circular motion?
130
16. Honors Physics – Going 'round in circles-1
CHECK_____
1. A ball is attached to the end of a 1.5 m string and it swings in a circle with a constant speed of 8 m/s.
a. What is the centripetal acceleration? (42.7 m/s2)
b. What are the period and frequency of rotation for the ball? (1.18 s, 0.85 Hz)
2. A 1500-kg car moves at a constant speed of 22 m/s along a circular track with a centripetal acceleration of
6 m/s2. a. What is the radius of the track? (80.7 m)
b. What is the centripetal force on the car? (9000 N). What provides the centripetal force?
131
3. A 3-kg rock, attached to a 2-m cord, swings in a horizontal circle so that it makes one revolution in 0.3 s.
What is the centripetal force on the rock? What provides the centripetal force? (2631.9 N)
4. On a rainy day the coefficient of static friction between tires and the roadway is only 0.4. What is the
maximum speed at which a car can negotiate a turn of radius 80 m? (17.7 m)
5. A 20-kg child sits 3 m from the center of a rotating platform. If s = 0.4, what is the maximum number of
revolutions per minute that can be achieved without slipping? (10.9 rpm)
132
MOTION IN A VERTICAL CIRCLE
When a body moves in a vertical circle at the end of
a string, the tension FT in the string varies with the
body's position. The centripetal force Fc on the body
at any point is the vector sum of FT and the
component of the body's weight toward the center of
the circle.
At the highest point in the circular turn the
centripetal force is given by:
At the lowest point in the loop the centripetal force is:
4.29 A string 0.5 m long is used to whirl a 1-kg stone in a vertical circle at a velocity of 5 m/s.
a. Calculate the tension in the string when the stone is at the top of the circle.
b. Calculate the tension in the string when the stone is at the bottom of the circle.
133
CW: VERTICAL CIRCLES
40. Spin Out is a carnival ride consisting of a large open cylinder. Riders stand inside with their backs against
the cylinder wall. As the ride starts to spin, the floor drops down. a. Draw a FBD of the situation.
b. What is the centripetal force acting on you in the ride if the radius of the cylinder is 3.5 m, your mass is 50 kg
and you are traveling at 5 m/s? (357 N)
c. What provides the centripetal force?
d. Calculate and name the force keeps you from sliding downward. (490 N)
41. A certain cord will break if its tension exceeds 250 N. The cord is attached to a 0.5 kg ball.
a. The ball is swung in a horizontal circle at the end of a 0.8 m cord, how fast can the ball move before the cord
breaks? (20 m/s)
b. If the ball swings with a speed of 25 m/s, what is the shortest length of cord that can be used? (1.25 m)
134
17. Honors Physics – Going 'round in circles-2
CHECK_____
1. A 3-kg ball swings in a vertical circle at the end of an 8-m cord. When it reaches the top of its path, its
velocity is 16 m/s. Draw a neat FBD.a. What is the tension in the cord? (66.6 N)
b. If the tension at the bottom is 50 N, what is the ball’s speed? Draw a neat FBD. (9.75 m/s)
135
2. A 2.0-kg object is attached to a 1.5 m long string and swung in a vertical circle at a constant speed of 12 m/s.
a. Draw a FBD and find the tension in the string when the object is at the bottom of its path. (211.6 N)
b. Draw a FBD and find the tension in the string when the object is at the top of its path? (172.4 N)
136
UNIVERSAL GRAVITATION
Newton proposed that the force that causes objects to fall to the earth exists between all other bodies, even the
sun and planets. He found that the gravitational force varies inversely with the square of the distance between
two objects. This is also known as the inverse square law.
The Law of Universal Gravitation states that: "Every object in the universe attracts every other object in the
universe with a force that varies directly with the product of their masses and inversely with the square of the
distance between the centers of the two masses."
FG 
Gm1m2
r2
Units: Newtons (N)
Cavendish determined the first reasonably accurate numerical value for G more than one hundred years after
Newton’s Law was published. The value of the constant is: G = 6.67 x10-11 N.m2/kg2
4.30 The mass of an electron is 9.1 x10-31 kg. The mass of proton is 1.7x10-27 kg. They are about 1.0 x10-10 m
apart in a hydrogen atom. What gravitational force exists between the proton and the electron of a hydrogen
atom?
CW: GRAVITATION
42. Two satellites of equal mass are put into orbit 30 m apart. The gravitational force between them is 2.0 x10-7
N. a. What is the mass of each satellite? (1642.7 kg)
b. What is the initial acceleration given to each satellite by the gravitational force? (1.22 x10-10 m/s2)
137
43. Two large spheres are suspended close to each other. Their centers are 4.0 m apart. One sphere weighs 980
N. The other sphere has a weight of 196 N. What is the gravitational force between them? (8.3 x10-9 N)
44. If the centers of Earth (ME = 6x1024 kg) and the moon are 3.9 x108 m apart, the gravitational force between
them is about 1.9 x1020 N. What is the approximate mass of the moon? (7.2 x1022 kg)
45. A satellite is placed in a circular orbit with a radius of 1.0 x107 m and a period of 9.9 x103 s. Calculate the
mass of Earth. (6.0 x1024 kg)
GRAVITATIONAL ACCELERATION
We can use Newton’s Law of Universal Gravitation to find the acceleration due to gravity of an object of mass,
m. The weight on an object on Earth's surface is equal to the gravitational force between that object and Earth:
GmM
mg 
the masses 'cancel out' so:
r2
g
GM
r2
Units: m/s2
138
4.31 The radius of the earth is about 6.38x106 m and its mass is 5.98x1024 kg. Calculate the value of the
acceleration due to gravity on the surface of the earth.
CW: GRAVITY
46. If you weigh 637 N on Earth’s surface, how much would you weigh on the planet Mars? (Mars has a mass
of 6.37x1023 kg and a radius of 3.43x106 m.) (234.6 N)
47. What would be the value of g, acceleration of gravity, if
a. Earth’s mass was double its actual value, but its radius remained the same? (19.6 m/s2)
b. If the radius was doubled, but the mass remained the same? (2.45 m/s2)
c. If both the mass and radius were doubled? (4.9 m/s2)
48. On the surface of the moon, a 91 kg physics teacher weighs only 145.6 N. What is the value of the moon's
gravitational field at its surface? (1.60 m/s2)
139
18. Honors Physics – Gravity!
CHECK_____
1. Jupiter’s largest moon, Ganymede, is also the eight largest known object in the solar system. The
gravitational force between Ganymede and Jupiter is 1.636x1022 N. Given that Jupiter’s mass is 1.9x1027 kg and
the distance between their centers of mass is 1.071x106 km, calculate Ganymede’s mass. (1.47x1023 kg)
2. On a distant planet, the acceleration due to gravity is 5.00 m/s2. and the radius of the planet is roughly 4560
m. Use the law of gravitation to estimate the mass of this planet. (1.56x1018 kg)
3. The mass of the earth is about 81 times the mass of the moon. If the radius of the earth is 4 times that of the
moon, what is the acceleration due to gravity on the moon? (1.94 m/s2)
140
UNIT V
CONSERVATION LAWS
ENERGY
Energy - a conserved, substance-like quantity with the capability to produce change.
The idea of energy is an invention that proves very useful. Energy can be moved around and stored in a variety
of ways, but the energy itself is unchanged. Energy is universal and it does not come in different "kinds" or
exist in different "forms."
There are many mechanisms for energy storage such as elastic Eel, kinetic Ek, gravitational potential Eg,
and chemical potential Echem, where the energy can be easily retrieved. Also, energy can be stored in the
random motion of molecules Ethermal, or the wave motion of molecules Esonic, where the stored energy is
very difficult to retrieve. Friction within a system often causes energy to be irretrievably stored in Ethermal and
Esonic which we lump together as energy dissipated, Ediss or Einternal. A numerical amount of energy can be
calculated for each storage mechanism.
As energy is transferred from one method of storage to another, the total amount of energy stays constant
(energy is conserved).
When examining energy transfers, it is helpful to choose what methods of energy storage are in our system and
what methods are outside our system. Generally, the smallest system that contains all the needed ways of
storing energy is the easiest. Transferring energy from one storage method to another or transferring energy into
or out of a system is the process of "working" (your textbook simply calls it "work.")
Money analogy:
We will define "the system" as the personal and institutional places where you keep your money. You can store
your money in a number of ways, in a checking account, savings account, cash in a piggy bank, or a stock
mutual fund. As you transfer money from cash and savings to checking, the amount of money stays the same (is
conserved) even though the money is now stored somewhere else. Some transfers cost you money, such as
using a debit card or getting a cash advance. In this case the money is transferred out of your account and into
the bank's account. The money still exists, you just can't have it anymore. (This is like dissipated energy.)
ENERGY TRANSFER
Energy can be transferred in or out of a physical system in three ways:
1. Working: energy is transferred by forces that cause displacements.
2. Heating – temperature is a measure of the average kinetic energy of the molecules of a substance.
Temperature differences between a system and its surroundings cause energy to be transferred from the warmer
object to the cooler object.
3. Electromagnetic Radiation (such as visible light, microwaves, ultraviolet light and infrared light) can
transfer energy. Matter loses kinetic energy as it emits electromagnetic radiation and gains kinetic energy when
absorbing electromagnetic radiation.
BAR GRAPHS AND ENERGY FLOW DIAGRAMS
Energy can be stored and it can be transferred from one storage mechanism to another. This can be represented
with an energy conservation bar graph diagram.
141
Steps in constructing a bar graph/energy flow representation:
1. Identify the system.
2. Identify the initial energy storage modes, and represent them with bar graphs.
3. Identify the resulting final energies with final bar graphs.
4. Identify the energy transfer(s) that occur across the system boundary to cause the changes and represent the
transfer with quantified arrows pointing into or out of the energy flow diagram.
In summary, you will use bar graphs to represent the Initial and Final energies, and the energy flow diagram to
represent the During processes. The difference in the Initial and Final energies is the change in internal
energy (or dissipated energy), ∆E, since ∆E = Ef - Ei.
Example of Bar Graph/Energy Flow Diagram
A person pushes a box from a 0 m position up a ramp to a stop.
v=0
Initial
System = box + surface of ramp + earth
Final
v=0
y=0
Energy Flow
Diagram
Initial
EK Eg Eel
EK
Final
Eg Eel Eint
W
0
0
Analysis:
1. Assuming the box starts at a 0 reference point, it has no initial energy.
2. Energy is transferred to the system via the external force provided by the person. This is defined as working.
The working arrow is 5 blocks long.
3. At the final point, the energy transferred by working done has been stored as the gravitational potential
energy, Eg, and some has been dissipated due to friction, Eint.
Note that Eg and Eint add up to 5 blocks, in agreement with the Conservation of Energy.
5.1 For each situation shown below:
a. Show your choice of system in the energy flow diagram, unless it is specified for you.
Always include the earth in your system!
b. Sketch an energy bar graph for the initial situation.
c. Then complete the analysis by showing energy transfers and the final energy bar graph.
142
1. A car on a roller coaster track, launched by a huge spring, makes it to the top of the loop.
Friction? NO
2. The same car is launched by the spring, but it is only half way up the loop.
Friction? NO
3. A car moving up a hill coasts to a stop at the top.
Friction? YES
143
4. A load of bricks, resting on a compressed spring, is launched into the air.
Friction? NO
5. A bungee jumper falls off the platform and reaches the limit of stretch of the cord.
Friction? YES
144
19. Honors Physics – Energy Stuff
CHECK_____
For each situation shown below:
a. What is the system? Always include the earth in your system.
b. Sketch an energy bar graph for the initial situation.
c. Then complete the analysis by showing energy transfers and the final energy bar graph.
1. A person pushes a stalled car to get it to the service station.
Friction? YES
2. A crate, starting at rest, is propelled up a hill by a tightly coiled spring.
Friction? YES
145
3. Superman, stopping a speeding locomotive, is pushed backwards a few meters in the process.
Friction? YES
4. Create your own situation and construct corresponding energy bar graphs and flow diagram.
System = ______________________
146
WORK
Work is a measure of the amount of change that a force produces when it acts on a body. The change may be in
the velocity of the body, in its position, in its size or shape, and so forth. Work is a scalar quantity; no direction
is associated with it.
The work done by a force acting on a body is equal to the product of the force and the distance through which
the force acts, if the F and x are in the same direction. Thus:
W=Fx
If F and x are not parallel but F is at the angle  with respect to x, then
W = F x cos 
When F is perpendicular to x,  = 90 and cos 90 = 0. No work is done in this case.
In SI units, the unit of work is the joule (J). 1 J = 1 N.m
5.2 A boy pulls a wagon with a force of 45 N by means of a rope that makes an angle of 40° with the ground.
How much work does he do in moving the wagon 50 m?
5.3 A 185 kg refrigerator is loaded into a moving van by pushing it up a 10.0 m ramp at an angle of 11. How
much work is done?
CW: WORK
1. During a tug-of-war, Team A does 2.20 x 105 J of work in pulling Team B a distance of 8.0 m. What force
was Team A exerting? (2.75 x104 N)
147
2. A lawn mower is pushed with a force of 88.0 N at an angle of 41° with the horizontal. How much work is
done in pushing the mower 1.2 km in mowing the yard? (7.9 x104 J)
3. A 17.0-kg crate is to be pulled a distance of 20.0 m, requiring 1210 J of work being done. If the job is done
by attaching a rope and pulling with a force of 75.0 N, at what angle is the rope held? (36.2˚)
POWER
Power is the rate at which work is done by a force. Thus P 
W
t
When a constant force F does work on a body that is moving at the constant velocity v, if
F is parallel to v the power involved is:
P
W Fx

 Fv
t
t
Unit: watts (W)
5.4 A man uses a horizontal force of 200 N to push a crate up a ramp 8 m long that is 20 above the horizontal.
If it takes 12 s to push the crate up the ramp, what is his power output?
5.5 The four engines of a DC-8 airplane develop a total of 22 MW when its velocity is 240 m/s. How much
force do the engines exert?
148
CW: POWER
4. An elevator lifts a total mass of 1100 kg, a distance of 40.0 m in 12.5 s. How much power does the elevator
demonstrate? (3.45 x104 W)
5. A 120-kg lawn tractor goes up a 21° incline of 12.0 m in 2.5 s. What is the power of the tractor? (2.02x103 W)
6. The engines of the Queen Mary could deliver 174 MW to propel the massive ship. How long does it take for
the engines to do 7.31x1010 J of work on the ship? (7 min)
KINETIC ENERGY
The energy a body has by virtue of its motion is called kinetic energy. If the body's mass is m and its velocity is
v, its kinetic energy is:
1
KE  mv 2
2
Units: Joules (J)
POTENTIAL ENERGY
The gravitational potential energy of a body of mass m at a height h above a given reference level is
PE = mgh
Units: Joules (J)
149
5.6 What velocity does a 1-kg object have when its kinetic energy is 2.5 J?
5.7 A 5.0 kg stone is slid up a frictionless ramp that has an incline of 25°. How long is the ramp if the
gravitational potential energy associated with the stone is 240 J?
WORK-ENERGY THEOREM
The net work done on a body (by the net force) equals the change in energy of that body:
W = ΔKE = ΔPE
5.8 A 10-g bullet has a velocity of 600 m/s when it leaves the barrel of a rifle. If the barrel is 60 cm long, find
the average force on the bullet while it is in the barrel.
ELASTIC POTENTIAL ENERGY
Elastic potential energy is associated with elastic materials.
The elastic potential energy is given by:
PEe = ½ kx2
Units: Joules (J)
k is the spring constant or force constant that measures the stiffness of a spring in N/m.
x is the displacement given in meters.
150
5.9 A dart of mass 0.100 kg is pressed against the spring of a toy dart gun. The spring (k = 250 N/m) is
compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its normal length,
what speed does the dart acquire?
PROBLEM SOLVING STRATEGY
For each situation first ask these questions:
1. Is the object at a height?
If yes, it has gravitational potential energy
2. Is the object moving?
If yes, it has kinetic energy
3. Is the object in contact with a spring?
If yes, it has elastic potential energy
4. Is a force applied through a distance?
If yes, it work has been done.
CW: ENERGY
7. A walrus swimming at a speed of 35 km/h has a mass of 900 kg, what is its kinetic energy? (4.25x104 J)
8. A force of 30.0 N pushes a 1.5-kg cart, initially at rest, a distance of 2.8 m along a frictionless surface.
a. Find the work done on the cart. (84 J)
b. What is its change in kinetic energy? (84 J)
c. What is the cart’s final velocity? (10.6 m/s)
151
9. A bike and rider, 82.0-kg combined mass, are traveling at 4.2 m/s. A force of -140 N is applied by the brakes.
What braking distance is needed to stop the bike? (5.17 m)
10. A 0.18 kg ball is placed on a compressed spring on the floor. The spring exerts an average force of 2.8 N
through a distance of 15 cm as it shoots the ball upward. How high will the ball travel above the release spring?
(0.24 m)
11. A highway guardrail is designed so that it can be distorted as much as 5 cm when struck by a car. What is
the minimum spring constant of the guardrail if it is to withstand the impact of a car with 1.09x104 J of energy?
(8.72x106 N/m)
152
12. A 70.0 kg stuntman jumps from a bridge that is 50 m above the water. A bungee cord with an unstretched
length of 15.0 m is attached to the stuntman, so that he breaks his fall 12.0 m above the water’s surface. If the
total potential energy associated with the stuntman and cord is 3.43 x104 J, what is the spring constant of the
cord? (98.56 N/m)
13. A 51-kg bungee jumper steps off a bridge situated 321 m above a river. The bungee's cord spring constant is
32 N/m, the cord's relaxed length is 104 m, and its length is 179 m when the jumper stops falling. What is the
total potential energy of the jumper at the end of the fall? (1.62x105 J)
153
20. Honors Physics – Work on This - 1
CHECK_____
1. A push of 120 N is applied along the handle of a lawn mower producing a horizontal displacement of 14 m.
If the handle makes an angle of 300 with the ground, what work was done by the 120-N force? (1455 J)
2. A sled is dragged a distance of 12.0 m by a rope under constant tension. The task requires 1200 J of work. If
the angle the rope make with the ground is 44.4o, what is the tension in the rope? (140 N)
FT
12 m

3. A 17 kg crate is pulled a distance of 20 m by exerting 1210 J of work. If the job is done by attaching a rope
and pulling with a force of 75 N, at what angle is the rope held? (36.2o)
154
4. A girl pulls a wagon along a level path for a distance of 15 m. The handle of the wagon makes an angle of
20o with the horizontal. The girl exerts a force of 35 N on the handle. If the friction force is 24 N, what is the
resultant work on the wagon? (133.5 J). Hint: Find the net force first!
5. Identical boxes of mass 10 kg are moving at the same initial velocity to the right on a flat surface. The same
magnitude force, F, is applied to each box for the distance, d, shown. Rank these situations in order of the work
done on the box by F while the box moves the indicated distance to the right.
Greatest 1_______ 2________ 3________ 4________ 5________ 6________ Least
EXPLAIN:
155
6. A 40-kg mass is lifted through a distance of 20 m in a time of 3 s. Find average power. (2613.3 W)
7. What is the change in kinetic energy when a 50.0 g ball hits the pavement with a velocity of 16 m/s and
rebounds with a velocity of 10 m/s? (-3.90 J)
8. What average force is needed to increase the velocity of a 2.0 kg object from 5 m/s to 12 m/s over a distance
of 8 m? (14.9 N)
156
CONSERVATION OF MECHANICAL ENERGY
Mechanical energy is the sum of kinetic energy and all forms of potential energy.
ME = KE + PE
5.10 A 150 kg roller coaster car starts from the top of the track at 37 m of height.
a. What is the energy present at the top of the hill?
b. What is the energy present at point A?
c. Find the speed of the car at point A.
d. What is the energy present at point B?
e. Find the speed of the car at point B (8 m).
f. What is the energy present at point C?
g. Find the speed of the car at point C (5 m).
157
CW: CONSERVATION OF ENERGY
14. An average force of 8.2 N is used to pull a 0.40 kg rock, stretching a sling shot 43 cm. The rock is shot
downward from a bridge 18 m above a stream. What will be the velocity of the rock just before it enters the
water? (19 m/s)
16. A 100.0 g arrow is pulled back 30 cm against a bowstring. If the spring constant of the bowstring is 1250
N/m, at what speed will the arrow leave the bow?
(33.5 m/s)
17. A 50 kg circus performer jumps from a platform into a safety net below. The net, which has a force constant
of 3.4 x104 N/m, is stretched by 0.65 m. If the unstretched net is positioned 1 m above the ground, what is the
height of the platform? (15 m)
158
21. Honors Physics – Work is Still a 4 Letter Word
CHECK____
1. At a particular instant a mortar shell has a velocity of 60 m/s. If its potential energy at that point is one-half
of its kinetic energy, what is its height above the earth? (91.8 m)
2. A 4-kg hammer is lifted to a height of 10 m and dropped. What are the potential and kinetic energies of the
hammer when it has fallen to a point 4 m from the earth? (157 J, 235 J)
3. A spring with a spring constant value of 125 N/m is compressed 12.2 cm by pushing on it with a 215 g block.
When the block is released, what velocity will the block have when it leaves the spring? (2.94 m/s)
159
4. An 8-kg block in the figure has an initial downward velocity of 7 m/s.
a. Find the velocity when it reaches point B? (21 m/s)
7 m/s
C
20 m
B
8m
b. What is the velocity at point C? (16.9 m/s)
160
LINEAR MOMENTUM
The linear momentum (usually called simply momentum) of a body of mass m and velocity v is the product of
m and v:
p = mv
Units: kg.m/s
The greater the momentum of a body, the greater its tendency to continue in motion.
IMPULSE
A force F that acts on a body during time t provides the body with an impulse of Ft
Impulse = F t
Units: N.s
When a force acts on a body to produce a change in its momentum, the momentum change m v is equal to the
impulse provided by the force. Thus
F t = m v = m (vf - vo)
Impulse = momentum change
5.11 Find the momentum of a 100 kg ostrich running at 4.5 km/h
5.12 A 2500-kg truck crashes into a wall at 40 km/h and comes to a stop in 0.5 s. What is the average force on
the truck?
5.13 A DC-9 airplane has a mass of 50,000 kg and a cruising velocity of 70 km/h. Its engines develop a total
thrust of 70,000 N. If air resistance, change in altitude, and fuel consumption are ignored, how long does it take
the plane to reach its cruising velocity, starting from rest?
161
CW: MOMENTUM
18. Jim strikes a 0.058-kg golf ball with a force of 272 N and gives it a velocity of 62.0 m/s. How long was the
club in contact with the ball? (0.013 s)
19. A force of 186 N acts on a 7.3-kg bowling ball for 0.40 s.
a. What is the bowling ball’s change in momentum? (74.4 N.s)
b. What is its change in velocity? (10.19 m/s)
20. A 0.24-kg volleyball approaches Tina with a velocity of 3.8 m/s. Tina bumps the ball, giving it a velocity of
2.4 m/s. Find the force that she apply if the interaction time between her hands and the ball is 0.025 s? (-59.5 N)
21. A 0.145-kg baseball is pitched at 42 m/s. The batter hits it horizontally to the pitcher at
58 m/s. a. Find the change in momentum of the ball. (-14.5 kg.m/s)
162
b. If the ball and bat were in contact 4.6 x 10-4 s, what would be the average force while they touched?
(-3.15 x104 N)
22. In running a ballistics test at the police department an officer fires a 6 g bullet at 350 m/s into a container
that stops it in 0.30 m. What average force stops the bullet? (-1.24x10-3 N)
23. A net force of 10 N to the right pushes a 3.0 kg book across a table. If the force acts on the book for 5.0 s,
what is the book’s final velocity? Assume the book to be initially at rest. (16.7 m/s, right)
24. How much time would it take for a 0.17 kg ice hockey puck to decrease its speed by 9.0 m/s if the
coefficient of kinetic friction between the ice and the puck is 0.05? (18.4 s)
163
22. Honors Physics – Momentum – 1
CHECK____
1. A 0.5-kg wrench is dropped from a height of 10 m. Find the momentum just before it strikes the floor.
(7 kg.m/s)
2. . A 2500-kg truck traveling at 40 km/h strikes a brick wall and comes to a stop in 0.2 s.
a. What is the change in momentum? (-27,800 kg m/s)
b. What is the impulse? (-27,800 kg m/s)
c. What is the average force on the wall during the crash? (+ 139,000 N)
164
CONSERVATION OF LINEAR MOMENTUM
According to the law of conservation of linear momentum, when the vector sum of the external forces that act
on a system of bodies equals zero, the total linear momentum of the system remains constant no matter what
momentum changes occur within the system.
Although interactions within the system may change the distribution of the total momentum among the various
bodies in the system, the total momentum does not change. Such interactions can give rise to two general
classes of events:
a. explosions, in which an original single body flies apart into separate bodies, and
b. collisions, in which two or more bodies collide and either stick together or move
apart, in each case with a redistribution of the original linear momentum.
For two objects interacting with one another, the conservation of momentum can be expressed as:
m1v1  m2 v2  m1v1'  m2 v2'
v1 and v2 are initial velocities, v1' and v2' are final velocities
COLLISIONS
Momentum is also conserved in collisions. If a moving billiard ball strikes a stationary one, the two move off in
such a way that the vector sum of their momentum is the same as the initial momentum of the first ball. This is
true even if the balls move off in different directions.
A perfectly elastic collision is one in which the bodies involved move apart in such a way that kinetic energy
as well as momentum is conserved.
In a perfectly inelastic collision, the bodies stick together and the kinetic energy loss is the maximum possible
consistent with momentum conservation. Most collisions are intermediate between these two extremes.
5.14 A 5-kg ball moving at 6 m/s strikes a 3-kg ball initially at rest. The 5-kg ball continues moving in the same
direction at 2 m/s. Find the velocity and direction of the 3-kg ball.
165
5.15 An unoccupied 1200-kg car has coasted down a hill and is moving along a level road at 10 m/s. A 6000-kg
truck moving in the opposite direction collides head-on with it. What was the truck’s velocity if both vehicles
come to a stop after the collision?
5.16 A 50-kg boy at rest on roller skates catches a 0.6-kg ball moving toward him at 30 m/s. How fast does he
move backward as a result?
5.17 A child jumps from a moving sled with a speed of 2.2 m/s and in the direction opposite the sled’s motion.
The sled continues to move in the forward direction, but with a new speed of 5.5 m/s. If the child has a mass of
38 kg and the sled has a mass 68 kg, what is the initial velocity of the sled?
166
5.18 A 1200-kg car traveling at 10 m/s overtakes a 1000-kg car traveling at 8 m/s and collides with it. a. If the
two cars stick together, what is their final velocity?
b. How much kinetic energy is lost?
CW: COLLISIONS
25. In the game of marbles, a shooter is a large marble about 2 cm in diameter that is used to knock smaller
marbles out of the ring. Suppose a shooter with a speed of 0.80 m/s hits a 4.8 g marble that is at rest in the ring.
The shooter continues forward with a speed of 0.51 m/s while the smaller marble moves forward with a speed
of 1.33 m/s. What is the mass of the shooter? (22 g)
167
26. The moon’s orbital speed around Earth is 3680 km/h. Suppose the moon suffers a perfectly elastic collision
with a comet whose mass is 50% that of the moon. After the collision, the moon moves with a speed of - 440
km/h, while the comet moves away from the moon at 5740 km/h. What is the comet’s speed before the
collision? (-2500 km/h)
27. Two astronauts are working in outer space. They have equal masses and accidentally run into each other.
The first astronaut moves 5 m/s to the right before the collision and 2 m/s to the left afterwards. If the second
astronaut moves 5 m/s to the right after the perfectly elastic collision, what was the second astronaut’s initial
velocity? (2 m/s, left)
168
28. A 50 g projectile is launched with a horizontal velocity of 647 m/s from a 4.65 kg launcher moving in the
same direction at 2 m/s. What is the velocity of the launcher after the projectile is launched? (- 4.94 m/s)
29. A swimmer with a mass of 58 kg and a velocity of 1.6 m/s to the north climbs onto a 142 kg raft. The
combined velocity of the swimmer and raft is 0.32 m/s to the north. What is the raft’s velocity before the
swimmer reaches it? (0.2028 m/s, South)
169
30. A twig floating in a small pond is initially at rest. On the twig is a snail, which begins moving along the
length of the twig with a speed of 1.2 cm/s. The twig moves in the opposite direction with a speed of 0.40 cm/s.
If the snail’s mass is 2.5 g, what is the mass of twig? (7.5 g)
31. An arrow is fired into a small target at rest on a frictionless surface. The arrow’s mass is 20 g and the
target’s mass is 2.50 kg. If the speed of the arrow and target combined is 0.67 m/s, what is the arrow’s initial
speed? (84.4 m/s)
170
23. Honors Physics – Mo' momentum - 2
CHECK______
1. A 4.2 kg ball traveling to the left collides with a 5.7 kg ball traveling to the right at 3.1 m/s. Find the velocity
of the first ball after the collision. (-1.7 m/s)
Before:
v1 = 5.5 m/s
v2 = -3.1 m/s
After:
v2’ = 2.2 m/s
2. The mass of the toy truck in Fig. 9-8 is three times that of the car, and they are tied together against a
compressed spring. When the string breaks, the car moves to the left at 6 m/s. What was the velocity imparted
to the truck? (2 m/s, right)
3m
v2
6 m/s m
v1
171
3. A 70-kg person standing on a frictionless surface throws a football forward with a velocity of 12 m/s. If the
person moves backward at 34 cm/s, what was the mass of the football?(1.98 kg)
4. A 60-g firecracker explodes sending a 45-g piece to the left and another to the right with a velocity of 40 m/s.
What is the velocity of the 45-g piece? (13.3 m/s, left)
5. A 6-kg bowling ball collides head on with a 1.8 kg pin. The pin moves forward at 3 m/s and the ball slows to
1.6 m/s. What was the initial velocity of the bowling ball? (2.5 m/s)
172
6. A 60 kg man on a lake of ice catches a 2 kg ball. The ball and man each move at 8 cm/s after the ball is
caught. a. What was the velocity of the ball before it was caught? (2.48 m/s)
b. What energy was dissipated in the process? (5.95 J)
7. The block in has a mass of 1.5 kg. How high will it rise if a 40-g projectile with an initial velocity of 80 m/s
embeds itself into the block? (22 cm)
h
173
UNIT VI
SIMPLE HARMONIC MOTION
Simple harmonic motion (SHM) is periodic motion that occurs when the restoring force on a body displaced
from an equilibrium position is proportional to the displacement and opposite in direction. The amplitude A of a
body undergoing SHM is the maximum value of its displacement on either side of the equilibrium position.
GRAPH OF SHM
The graph shown depicts the up and down oscillation of the mass at the end of a spring. One complete cycle is
from a to b, or from c to d, or from e to f. The time taken for one cycle is T, the period.
6.1 For the motion shown in the figure:
a. What is the amplitude?____________
b. What is the period?____________
c. What is the frequency? ________________
MASS-SPRING SYSTEM
The period T of a body of mass m attached to a spring of force constant k
m
Units: s
T  2
k
The frequency f of a body undergoing SHM is the number of cycles per second it executes, so that
f=1
T
Units: hertz (1 Hz = 1 cycle/s)
174
6.2 A box is placed on a spring scale whose spring constant is 362 N/m. If the scale’s platform oscillates with a
frequency of 1.20 Hz, what is the mass of the box?
THE SIMPLE PENDULUM
The simple pendulum very nearly undergoes SHM if its angle of swing is not too large. The period of vibration
for a pendulum of length l at a location where the acceleration is g is given by:
L
g
T  2
Units: s
THE SIMPLE PENDULUM
A simple pendulum has its entire mass concentrated at the end of a string. It undergoes SHM provided that the
arc through which it travels is only a few degrees. The period of a simple pendulum of length L is given by:
T  2
L
g
where g is the acceleration due to gravity
Units: s
6.3 Find the frequency of a simple pendulum 20 cm long.
6.4 A pendulum 1.00 m long oscillates 30.0 times per minute in a certain location. What is the value of g there?
175
CW: SHM
1. A vertical spring 60 mm long resting on a table is compressed by 5.0 mm when a 200 g mass is placed on it.
If the spring is pressed down and released, what is the period of the oscillation? (0.14 s)
2. A 50 g mass vibrates in SHM at the end of a spring. The amplitude of the motion is
12 cm, and the period is 1.70 s. Find:
a. The frequency, (0.588 Hz)
b. The spring constant, (0.68 N/m)
3. A 20.0 kg sphere oscillates on a spring 42.7 times each minute, what is the spring constant? (397.1 N/m)
4. The hummingbird makes a humming sound with its wings, which beat with a frequency of 90 Hz. Suppose a
mass is attached to a spring with a spring constant of 2.50x102 N/m. How large is the mass if its oscillation
frequency is 3.00x10-2 times that of a hummingbird’s wings? (0.86 kg)
176
5. The 400-g piston in a compressor oscillates up and down through a total distance of 80 mm. Calculate the
maximum force on the piston when it goes through 10 cycles/s. (63.2 N)
6. Calculate the period and frequency of a 6.200 m long pendulum in Oslo, Norway, where g = 9.819 m/s2.
(4.993 s, 0.2003 Hz)
7. On Mars, a simple pendulum with a length of 65.0 cm would have a period of 2.62 s. Calculate the
acceleration of gravity on Mars. (3.74 m/s2)
177
24. Honors Physics – Harmonium Stuff
CHECK____
1. When a mass of 200-g is hung from a spring, the spring is displaced downward a distance of 1.5 cm. What is
the spring constant k? (130.7 N/m)
2. A car and it passengers have a total mass of 1600 kg. The frame of the car is supported by four springs, each
having a spring constant of 20,000 N/m. Find the frequency of vibration of the car when it drives over a bump
in the road. (1.125 Hz)
3. A student constructs a pendulum of length 3 m and determines that it makes 50 vibrations in 2 min and 54 s.
What is the acceleration due to gravity at this student’s location? (9.78 m/s2)
178
UNIT VII
WAVES AND SOUND
A wave is, in general, a disturbance that moves through a medium. (An exception is an electromagnetic wave,
which can travel through a vacuum. Examples are light and radio waves.) A wave carries energy, but there is no
transport of matter. In a periodic wave, pulses of the same kind follow one another in regular succession.
In a transverse wave, the particles of the medium move back and forth perpendicular to the direction of the
wave. Waves that travel down a stretched string when one end is shaken are transverse.
In a longitudinal wave, the particles of the medium move back and forth in the same direction as the wave.
Waves that travel down a coil spring when one end is pulled out and released are longitudinal. Sound waves are
also longitudinal.
Water waves are a combination of longitudinal and transverse waves.
WAVE PROPERTIES
The period T of a wave is the time required for one complete wave to pass a given point. The frequency f is the
number of waves that pass that point per second.
The amplitude A of a wave is the maximum displacement of the particles of the medium through which the
wave passes on either side of their equilibrium positions. In a transverse wave, the amplitude is half the distance
between the top of a crest and the bottom of a trough.
The wavelength λ (lambda) of a periodic wave is the distance between adjacent wave crests. Frequency and
wavelength are related to wave velocity by:
v = f λ or v 

T
Units: m/s
The speed of a mechanical wave is constant in a given medium. The amplitude of a wave does not affect its
wavelength, frequency or velocity.
179
When a wave travels from one medium to another, the wave is both reflected and transmitted.
When a wave passes into a new medium, its speed changes. The wave must have the same frequency in the new
medium as in the old medium. Thus, the wavelength adjusts.
7.1 Water waves in a small tank are 6.0 cm long. They pass a given point at the rate of 4.8 waves per second.
a. What is the speed of the water waves?
b. What is the period of the waves?
7.2 Microwaves are electromagnetic waves that travel through space at a speed of 3x108 m/s. Most microwave
ovens operate at a frequency of 2450 MHz.
a. What is the period of these microwaves?
b. How long is the wavelength of these microwaves?
180
7.3 A sound wave is directed toward a vertical cliff 680 m from the source. A reflected wave is detected 4.0 s
after the wave is produced.
a. What is the speed of sound in air?
b. The sound has a frequency of 500 Hz. What is its wavelength?
c. What is the period of the wave?
ELECTROMAGNETIC WAVES
Electromagnetic waves consist of oscillating electric and magnetic fields with different wavelengths. The wave
speed equation is:
c = f λ where c is the speed of light (3 x108 m/s)
CW: WAVES
1. What is the frequency of violet light of wavelength 4.10x10-7 m? (7.32 x1014 Hz)
2. The local hit music radio station can be found at 103.6 on the FM dial. This means that they broadcast at a
frequency of 103.6x106 (103.6 Megahertz). What is the wavelength of these radio waves? (2.894 m)
181
WAVES ON A STRING
The velocity of a wave depends on the properties of the medium through which it travels. For a transverse wave
on a string, the velocity is given by the following equation:
v
FT
m/ L
Units: m/s
FT is tension in the string and m/L is the mass per unit length of the string (linear density)
7.4 A metal wire of mass 500.0-g and length 50.0-cm is under a tension of 80 N. What is the speed of a
transverse wave in the wire?
CW: WAVES
3. What tension is needed to produce a wave speed of 12 m/s in a 900 gram string that is 2 m long? (64.8 N)
4. A string 200 cm long has a mass of 500 g. What string tension is required to produce a wave speed of 120
cm/s? (0.360 N)
182
BEHAVIOR OF WAVES
Reflection: Waves bounce off a surface:
Refraction: Waves bend when they pass through a
boundary:
Diffraction: Waves spread out (bend) when they pass
through a small opening or move around a barrier.
Interference: When two or more waves meet, their displacements add. This process is called superposition.
The effect of two or more waves traveling through a medium is called interference. Waves can produce
constructive or destructive interference.
Constructive Interference occurs when two waves combine to produce a wave of greater amplitude.
Destructive Interference occurs when two waves combine to produce a wave with smaller amplitude than
either the original amplitudes.
After two waves pass through one another, they return to their original shapes.
183
STANDING WAVES IN STRINGS
At certain frequencies standing waves can be produced in which the waves seem to be standing still rather than
traveling. This means that the string is vibrating as a whole. This is called resonance and the frequencies at
which standing waves occur are called resonant frequencies or harmonics.
As shown in the diagram below, a standing wave is produced by the superposition of two periodic waves
having identical frequencies and amplitudes which are traveling in opposite directions. In stringed musical
instruments, the standing wave is produced by waves reflecting off a fixed end and interfering with oncoming
waves as they travel back through the medium.
The points of destructive interference (no vibration) are called nodes, and points of constructive interference
(maximum amplitude of vibration) are called antinodes.
SOUND WAVES
Sound is a longitudinal wave produced by a vibration and which travels away from the source through solids,
liquids or gases, but not through a vacuum. The speed of sound depends on the medium. As a result, sound
waves generally travel faster through solids than through gases.
Pitch usually refers to the frequency of a sound wave and is measured in Hz. The range of human hearing
(audible range) is from 20 Hz to 20,000 Hz. Above 20,000 Hz waves are ultrasonic and below 20 Hz waves are
infrasonic.
184
DOPPLER EFFECT
When a source of sound waves and a listener approach one another, the pitch of the sound is increased as
compared to the frequency heard if they remain at rest. If the source and the listener recede from one another,
the frequency is decreased.
v  vL
Units: Hz
f L  fS
v  vS
fL: frequency of the listener, fS: frequency of the source
v: velocity of sound, vL: velocity of listener, vS: velocity of source
Sign Convention: velocity: (+) approaching and (-) receding
7.5 A train moving at 20 m/s emits a sound at a frequency of 400 Hz. The speed of sound is 340 m/s
a. Find the frequency of the sound heard when the train is moving toward a stationary observer.
b. What is the frequency heard when the train is moving away from the observer at this speed?
7.6 A stationary source of sound has a frequency of 800 Hz on a day when the speed of sound is 340 m/s. What
frequency is heard by a person who is moving from the source at 30 m/s?
185
CW: DOPPLER EFFECT
6. An auto with a siren having a frequency of 365 Hz is approaching a stationary detector at a speed of 75 km/h.
What will the frequency of the siren appear to be if sound travels at 345 m/s? (388.4 Hz)
7. The whistle of a train has a frequency of 440 Hz. The train is moving at 35 m/s toward a person standing on
the platform. If sound travels at 345 m/s, what is the wavelength of the sound heard by the person? (0.7 m)
186
25. Honors Physics – Waves is Wavy - 1
CHECK____
1. A person on a pier counts the slaps of a wave as the crests hit a post. If 80 slaps are heard in one minute and a
particular crest travels a distance of 8 m in 4 s, what is the length of a single wave? (1.5 m)
2. A transverse wave is shown. It has a frequency of 12 Hz. (12 cm, 28cm, 0.0833 s, 3.36 m/s)
Find the amplitude: ______________, wavelength: ______________
Period:
Speed of the wave:
187
3. A longitudinal wave is shown. It has a frequency of 8 Hz. (12 cm, 28cm, 0.125 s, 2.24 m/s)
Find the amplitude: ______________, wavelength: ______________
Period:
Speed of the wave:
4. A tension of 400 N causes a 300-g wire of length 1.6 m to vibrate with a frequency of 40 Hz. What is the
wavelength of the transverse waves? (1.15 m)
5. A truck traveling at 24 m/s overtakes a car traveling at 10 m/s in the same direction. The trucker blows a
600-Hz horn. If the speed of sound is 340 m/s, what frequency is heard by the car driver? (663.9 Hz)
188
UNIT VIII
GEOMETRIC OPTICS
REFLECTION OF LIGHT
The law of reflection states that: "The angle of incidence θi is equal to the angle of reflection θr."
Both angles are measured with respect to the normal N.
N
incidence θi θr reflection
Light reflection from a smooth surface is called regular (or specular) reflection. Light reflection from a rough
or irregular surface is called diffuse reflection.
FLAT MIRRORS
A flat mirror reflects light rays in the same order as they approach it. Flat mirrors are made from pieces of
plate glass that have been coated on the back with a reflecting material like silver or aluminum. The image is
the same size of the object and the same distance behind the mirror as the object is in front of the mirror.
Notice that the images formed by a flat mirror are, in truth, reflections of real objects. The images themselves
are not real because no light passes through them. These images which appear to the eye to be formed by rays of
light but which in truth do not exist are called virtual images.
On the other hand real images are formed when rays of light actually intersect at a single point.
189
CURVED MIRRORS
A curved mirror is a mirror that may be thought of as a portion of a reflecting sphere.
A curved mirror has a geometric center or vertex V.
The center of curvature C is equal to the radius R.
The focal length f of the mirror is half the radius:
f 
1
R
2
The focal point is F.
If the inside of the spherical surface is the reflecting surface, the mirror is said to be concave or converging. If
the outside portion is the reflecting surface, the mirror is convex or diverging.
CONCAVE MIRROR
CONVEX MIRROR
IMAGES FORMED BY CURVED SPHERICAL MIRRORS
The best method of understanding the formation of images by mirrors is through geometrical optics or ray
tracing.
The three principal rays are:
Ray 1. A ray parallel to the mirror axis passes through the focal point of a concave mirror or seems to come
from the focal point of a convex mirror.
Ray 2. A ray that passes through the focal point of a concave mirror or proceeds toward the focal point of a
convex mirror is reflected parallel to the mirror axis.
Ray 3. A ray that proceeds along a radius of the mirror is reflected back along its original path.
190
IMAGE CHARACTERISTICS
There are three characteristics of the image to be determined:
1. Size: is the image greater (enlarged), smaller (reduced) or the same size as the object?
2. Position: is the image inverted (upside down) or upright?
3. Nature: Is the image on the same side as the object (real image) or is it on the other side of the mirror (virtual
image)?
8.1 RAY DIAGRAMS STEP-BY-STEP
Concave Mirror
1. Pick a point on the top of the object and draw one incident ray traveling parallel towards the mirror.
2. Reflect that ray through the focal point F. This is Ray 1.
3. Draw one incident ray traveling through F towards the mirror.
4. Reflect that ray parallel to the principal axis. This is Ray 2.
5. Where the two rays intersect, mark the image of the top of the object with a dot and complete the image.
6. Write down the three characteristics of the image.
Convex Mirror
1. Pick a point on the top of the object and draw one incident ray traveling parallel towards the mirror.
2. Reflect that ray as it seems to come from the focal point F 'inside' the mirror. This is Ray 1.
3. Draw one incident ray traveling through C 'inside' the mirror. This is Ray 3.
4. Where the two rays intersect, mark the image of the top of the object with a dot and complete the image.
5. Write down the three characteristics of the image.
8.2 a. Find the images formed by the following mirrors using the Ray Tracing method.
b. Write the characteristics of each image:
 real or virtual
 larger, smaller or same size as object
 inverted or upright
191
192
CW: MIRRORS
1. a. Find the images formed by the following converging mirrors using the Ray Tracing method.
b. Write the characteristics of each image: real or virtual, larger, smaller or same size as object and upright or
inverted.
193
2. a. Find the images formed by the following diverging mirrors using the Ray Tracing method.
b. Write the characteristics of each image: real or virtual, larger, smaller or same size as object and upright or
inverted.
194
THE MIRROR EQUATION
The mirror equation can be used to locate the image:
1 1 1
 
d o di
f
di 
do f
do  f
M
hi
d
 i
ho
do
The ratio M is called the magnification.
SIGN CONVENTION
R
f
di
hi
radius of curvature
focal length
image distance
image size
+ for converging, - for diverging
+ for converging, - for diverging
+ for real images, - for virtual images
+ if upright, - if inverted
8.3 An object stands 18 cm from a concave mirror. An image forms 36 cm from the mirror.
What is the focal length of the mirror?
195
8.4 A 3.5 cm candle is placed 6 cm away from a concave mirror with a focal length of 24 cm.
a. Where does the image of the candle form?
b. Is it virtual or real?
c. What is the height of the image?
d. Is it inverted or upright?
8.5 When you hold a convex mirror 21 cm from your eye; your image forms 7 cm behind the mirror.
a. What is the magnification of the image?
b. What are the mirror’s focal length and the radius of curvature?
CW: MIRROR EQUATION
3. A ray of light strikes a mirror at an angle of 62° with the horizontal.
a. Draw a sketch showing the incident ray, reflected ray and angle of incidence.
b. What is the angle of reflection? (28°)
c. Why isn't the answer 62°?
196
4. A store clerk uses a diverging mirror with a radius of curvature of 1 m to keep an eye on shoplifters. How tall
would the image of the man 1.85 m in height be if he were standing 5 m in front of the mirror? (0.17 m)
5. A spherical concave mirror has a radius of curvature of 30 cm. An object that is 8.0cm tall is placed 20 cm in
front of the mirror.
a. What is the location at which the image is formed? (60 cm)
b. Is the image real or virtual?
c. What is the height of the image? (-24 cm)
d. Is it inverted or upright?
6. An object 5 cm in height is placed 10 cm in front of a spherical concave mirror with a 15 cm focal length.
a. What is the position at which the image is formed? (-30 cm)
b. Is the image real or virtual?
c. What is the height of the image? (15 cm)
d. Is it inverted or upright?
197
27. Honors Physics – Optical Stuff - 1
CHECK:____
1. A light bulb 3 cm high is placed 20 cm in front of a concave mirror with a radius of curvature of 15 cm.
Determine the nature, size, and location of the image formed. (12.0 cm, -1.80 cm; Real, Inverted)
2. A spherical concave mirror has a focal length of 20 cm. What are the nature size and location of the image
formed when a 6 cm tall object is located 15 cm from this mirror? (-60.0 cm, 24.0 cm, Virtual, Upright)
198
3. An 8-cm pencil is placed 10 cm from a diverging mirror of radius 30 cm. Determine the nature, size, and
location of the image formed. (-6.0 cm, 4.8 cm, Virtual, Upright)
4a. Find the magnification of an object if it is located 10 cm from a mirror and its image is upright and seems to
be located 40 cm behind the mirror? (4)
b. Calculate the focal length. (13.33 cm)
c. Is this mirror diverging or converging? Explain your answer.
199
5. Construct the image on the drawings below via ray tracing. Write the three characteristics of each image.
f
C
C
f
f
200
REFRACTION
The bending of a ray of light as it passes from one medium to another is called refraction.
The speed of light c in a material is generally less than the free-space velocity of 3 x108 m/s. In water light
travels about three-fourths of its velocity in air. Light travels about two-thirds as fast in glass. The ratio of the
velocity c of light in a vacuum to the velocity v of light in a particular medium is called the index of refraction
n for that material.
c
c = 3x108 m/s
n
v
Indices of Refraction for Yellow Light
Medium
vacuum
air
water
ethanol
n
1.00
1.00
1.33
1.36
Medium
crown glass
quartz
flint glass
diamond
n
1.52
1.54
1.61
2.42
8.6 The speed of light in a plastic is 2 x108 m/s. What is the index of refraction of the plastic?
8.7 Find the velocity of yellow light in a diamond whose refractive index is 2.42.
SNELL’S LAW
n1 sin θ1 = n2 sin θ2
 n1 sin 1 

 n2 
2  sin 1 
n1 = index of refraction of the incident medium and θ1 = incident angle
n2 = index of refraction of the second medium and θ2 = refracted angle
8.8 Light is incident upon a piece of crown glass at an angle of 45. What is the angle of refraction?
201
8.9 A ray of light travels from air into liquid. The ray is incident upon the liquid at an angle of 30. The angle of
refraction is 22.
a. What is the index of refraction of the liquid?
b. Look at the Table of values. What might the liquid be?
CW: REFRACTION
10. Light travels through an unknown transparent substance at a speed of 2.15x108 m/s. What is the index of
refraction of the substance? (1.39)
11. A ray of light traveling in air enters a transparent substance with an angle of incidence of 40° and it is
refracted at an angle of 25°. How fast does the light travel through the substance? (1.97 x108 m/s)
12. A ray of light traveling in air enters a square container made of crown glass and filled with water. It strikes
the container at an angle of incidence of 45°. What is the angle of refraction as the light ray goes from the glass
wall into the water? (32˚)
202
13. A light beam in air (n = 1.00) makes an angle of 45° with the normal to a glass window pane (n= 1.44).
a. What is the angle to the normal of the light beam after it is refracted into the glass? (29°)
b. What is the angle to the normal of the light beam after it exits the other side of the glass into air? (45°)
THIN LENSES
Lenses are an essential part of telescopes, eyeglasses, cameras, microscopes and other optical instruments. A
lens is usually made of glass, or transparent plastic.
The two main types of lenses are convex and concave lenses. The focal length (f) of a lens depends on its shape
and its index of refraction.
A converging (convex) lens is thick in the center and thin at the edges.
A diverging (concave) lens is thin in the center and thick at the edges.
203
IMAGE FORMATION BY LENSES
The three principal rays are:
Ray 1. A ray parallel to the axis passes through the second focal point F2 of a converging lens or appears to
come from the first focal point F1 of a diverging lens.
Ray 2. A ray which passes through the first focal point F1 of a converging lens or proceeds toward the second
focal point F2 of a diverging lens is refracted parallel to the lens axis.
Ray 3. A ray through the geometrical center of a lens will not be deviated.
A real image is always formed on the side of the lens opposite to the object. A virtual image will appear to be
on the same side of the lens as the object.
Principal Rays for Convex Lenses
Principal Rays for Concave Lenses
8.10 a. Find the images formed by the following lenses using the Ray Tracing method.
b. Write the characteristics of each image: real or virtual, larger, smaller or same size as object and upright or
inverted.
204
205
CW: LENSES
14 a. Find the images formed by the following converging lenses using the Ray Tracing method.
b. Write the characteristics of each image: real or virtual, larger, smaller or same size as object and upright or
inverted.
15 a. Find the images formed by the following diverging lenses using the Ray Tracing method.
206
b. Write the characteristics of each image: real or virtual, larger, smaller or same size as object and upright or
inverted.
THE LENS EQUATION
The lens equation can be used to locate the image:
1 1 1
 
d o di
f
di 
do f
do  f
M
hi
d
 i
ho
do
SIGN CONVENTION
R
f
di
hi
radius of curvature
focal length
image distance
image size
+ for converging, - for diverging
+ for converging, - for diverging
+ for real images, - for virtual images
+ if upright, - if inverted
8.11 a. Find the location of the image of a 5 cm tall object located 30 cm from a convex lens of 10 cm focal
length.
b. Is the image real or virtual?
c. What is the height of the image?
207
d. Is the image inverted or upright?
8.12 A 1.25 cm tall object is 4 cm from a concave lens of 6 cm focal length.
a. Locate the image formed and say if it is real or virtual.
b. Find the size of the image and say if it is inverted or upright
CW: LENS EQUATION
16. A converging lens of focal length 25 cm is used to form an image of an object located 1.0 m from the lens.
The object has a height of 4.0 cm.
a. What is the position and nature at which the image is formed? (33.3 cm, real)
b. What is the height of the image? (-1.3 cm, inverted)
17. A diverging lens has a focal length of -30 cm. An object of height 10 cm is located 30 cm from the lens.
a. What is the position and nature at which the image is formed? (-15 cm, virtual)
208
b. What is the height of the image? (5 cm, upright)
18. If an object is 10.0 cm from a converging lens that has a focal length of 5 cm, how far from the lens will the
image be? (10 cm)
19. The focal length of a lens in a box camera is 10.0 cm. The fixed distance between the lens and the film is
11.0 cm. If an object is clearly focused on the film, how far must the object be from the lens? (110 cm)
TOTAL INTERNAL REFLECTION
The incident angle that causes the refracted ray to lie right along the boundary of the substance (i.e. at 90o) is
unique to the substance and is known as critical angle of the substance:
n
sin  c  r
ni
For incident angles greater than θc, there is no refracted ray at all, and all of the light is reflected. This effect is
called total internal reflection.
8.13 Find the critical angle for an air-crown glass boundary.
209
28. Honors Physics – Optical Stuff - 2
CHECK:____
1. The speed of light through a certain medium is 1.6 x 108 m/s in a transparent medium, what is the index of
refraction in that medium? (1.87)
2. Light passes from water (n = 1.33) to air. The beam emerges into air at an angle of 320 with the horizontal
water surface? What is the angle of incidence inside the water? (39.6o)
3. Light in air is incident at 600 and is refracted into an unknown medium at an angle of 400. What is the index
of refraction for the unknown medium? (1.35)
4. If the critical angle of incidence for a liquid to air surface is 460, what is the index of refraction for the liquid?
(1.39)
210
5. Construct the image on the drawings below via ray tracing. Write the three characteristics of each image.
f
f
f
f
f
f
211
f
f
6. A 7-cm tall pencil is placed 35 cm from a thin converging lens of focal length 25 cm. What are the nature,
size, and location of the image formed? (+87.5 cm, Real; -17.5 cm, Inverted)
7. A virtual, upright image appears to be located 40 cm in front of a lens of focal length 15 cm. What is the
object distance? (10.9 cm)
8. An object located 30 cm from a lens produces a real, inverted image located a distance of 60 cm on the
opposite side of the lens. What is the focal length of the lens? (+20 cm)
212
UNIT IX
ELECTROSTATICS
HISTORY OF ATOMIC STRUCTURE
The search for the atom began as a philosophical question. The natural philosophers of ancient Greece began
the search for the atom by asking such questions as: What is stuff composed of? What is the structure of
material objects? Is there a basic unit from which all objects are made? As early as 400 B.C., some Greek
philosophers proposed that matter is made of indivisible building blocks known as atomos. (Atomos in Greek
means indivisible.) To these early Greeks, matter could not be continuously broken down and divided
indefinitely. This indivisible building block of which all matter was composed became known as the atom.
From the 1600s to the present century, the search for the atom became an experimental pursuit. Several
scientists are notable; among them are Robert Boyle, John Dalton, J.J. Thomson, Ernest Rutherford, and Neils
Bohr.
The conclusions regarding atomic structure are:



All material objects are composed of atoms. There are different kinds of atoms known as elements;
these elements can combine to form compounds. Different compounds have distinctly different
properties. Material objects are composed of atoms and molecules of these elements and compounds,
thus providing different materials with different electrical properties.
An atom consists of a nucleus and a vast region of space outside the nucleus. Electrons are present in
the region of space outside the nucleus. They are negatively charged and weakly bound to the atom.
Electrons are often removed from and added to an atom by normal everyday occurrences.
The nucleus of the atom contains positively charged protons and neutral neutrons. These protons and
neutrons are not removable by usual everyday methods. It would require some form of high-energy
nuclear occurrence to disturb the nucleus and subsequently dislodge its positively-charged protons.
ELECTRIC CHARGE
Electric charge, like mass, is one of the basic properties of certain of the elementary particles of which all
matter is composed.
There are two kinds of electrical charges, positive (+) and negative (-).
Positively-Charged Negatively-Charged Uncharged
Possesses more
Possesses more
Equal numbers of
protons than electrons electrons than protons protons and electrons
An electrically neutral object is an object that has a balance of protons and electrons. In contrast, a charged
object has an imbalance of protons and electrons.
CHARGE INTERACTIONS
Any charged object can exert an electric force upon other objects - both charged and uncharged objects.
Electric forces are forces-at-a-distance.
213
Like charges repel and unlike charges attract.
Case I.
Case II.
Any charged object - whether positively-charged or negatively-charged - will have an attractive interaction
with a neutral object. Positively-charged objects and neutral objects attract each other; and negatively-charged
objects and neutral objects attract each other.
9.1 On two occasions, the following charge interactions between balloons A, B and C are observed. In each
case, it is known that balloon B is charged negatively. Based on these observations, what can you conclusively
confirm about the charge on balloon A and C for each situation.
214
9.2 Upon entering the room, you observe two balloons suspended from the ceiling. You notice that instead of
hanging straight down vertically, the balloons seems to be repelling
each other. You can
conclusively say ...
a. both balloons have a negative charge.
b. both balloons have a positive charge.
c. one balloon is charge positively and the other negatively.
d. both balloons are charged with the same type of charge.
Explain your answer.
9.3. Chris is investigating the charge on several objects and makes the following findings.
Object C
Object D
Object E
Object F
attracts B
repels C
attracts D
attracts A
repels F
He knows that object A is negatively charged and object B is electrically neutral. What can he conclude about
the charge on objects C, D, E, and F? Explain.
9.5. Balloons X , Y and Z are suspended from strings as shown at the right. Negatively-charged balloon X
attracts balloon Y and balloon Y attracts balloon Z.
Balloon Z _____. Circle all that apply.
a. may be positively-charged
b. may be negatively-charged
c. may be neutral
d. must be positively-charged
e. must be negatively-charged
f. must be neutral
CHARGE AS A QUANTITY
The unit of charge is the coulomb (C). The charge of the proton is +1.6 x10-19 C, and the charge of the electron
is -1.6 x10-19 C
9.2 An object has - 1 C of charge. Calculate the number of electrons it contains.
215
COULOMB'S LAW
Electrical charges exert a force on other electrical charges. This electrostatic force is directly proportional to the
product of the charges and inversely proportional to the square of their distance of separation. Coulomb’s Law
describes the electrostatic force between two charged objects.
F
k q1q2
r2
k = 9 x109 Nm2/C2
F is the electrostatic force in Newtons (N). It is either attractive or repulsive; q is the magnitude of each charge
in Coulombs (C) and r is the distance of separation in meters (m)
Prefixes:
1 C (1 micro Coulomb)= 1 x10-6 C
1 nC (1 nano Coulomb) = 1x10-9 C
9.3 What is the magnitude and nature of the force on a charge of +4 nC that is 5 cm from a charge of +50 nC?
9.4 Two charges, one of +5 x10-7 C and the other of -2 x10-7 C attract each other with a force of 100 N. How far
apart are they?
9.5 A charge q1 of +1 C is placed halfway between a charge q2 of +5 C and a charge q3 of +3 C that are 20
cm apart. Find the resultant force on the +1 C charge. Draw a sketch.
a. What are the interactions of the + 1 μC charge?
216
b. Write down the data paying attention to the distances between the charges and the units!
c. Draw the FBD of the interactions mentioned above. Clearly label the forces.
d. Calculate the forces.
e. Find the net force: magnitude and direction.
9.6 A +4 C charge lies 2 m to the left of a –5 C charge. A –6 C charge lies 4 m to its right. What is the
resultant force on the center charge (- 5C)? Draw a sketch.
a. What are the interactions of the - 5 μC charge?
217
b. Write down the data paying attention to the distances between the charges and the units!
c. Draw the FBD of the interactions mentioned above. Clearly label the forces.
d. Calculate the forces.
e. Find the net force: magnitude and direction.
PROBLEM-SOLVING STRATEGY
- Find the interaction between the charges: is it attractive or repulsive?
- Draw a neat FBD showing the direction of the forces
- Apply Coulomb's Law to find the electric forces between two charges. Do NOT include the sign of the
charges in the equation!
- Write down your answer indicating the magnitude of the force in Newtons and whether the force is attractive
or repulsive. If three forces are interacting, indicate the direction of the net force.
218
CW: COULOMB'S LAW
1. What is the magnitude and nature of the electric force between a pair of electrons 1.0 m apart? (2.3x10-28 N,
repulsive)
2. A metal sphere has a net charge of -4 C. How many excess electrons does the metal sphere contain?
(2.5 x1019electrons)
3. If the force between a pair of electrons is 1.0 N, how far are they from each other? (1.5x10-14 m)
4. Two charged objects are separated by a distance of 0.9 m. One carries a charge of 18 μC and the force
between them is 2.7 N. What is the charge on the second object? (13.5 μC)
5. A pair of electrically charged coins suspended from insulating threads, a certain distance from each other.
There is a specific amount of electrostatic force between them.
a. If the charge on one coin were halved, what would happen to the force between them?
b. If the charge on both coins were doubled, what would happen to the force between them?
219
c. If the distance between the coins were tripled, what would happen to the force between them?
d. If the charge on each object were doubled and the distance between them were doubled, what would happen
to the force between them?
6. Three charges are on a straight line as shown below. What is the resultant force on the - 5 μC charge?
q1
- 5 μC
8 cm
q2
+30 μC
13 cm
q3
+72 μC
a. What are the interactions of the - 5 μC charge?
b. Write down the data paying attention to the distances between the charges and the units!
c. Draw the FBD of the interactions mentioned above. Clearly label the forces.
d. Calculate the forces. (210.9 N, 73.5 N)
220
e. Find the net force: magnitude and direction. (284.4 N, to the right)
7. Three charges are on a straight line as shown below. What is the resultant force on the
+ 30 μC charge?
q1
- 5 μC
8 cm
q2
+30 μC
13 cm
q3
+72 μC
a. What are the interactions of the + 30 μC charge?
b. Write down the data paying attention to the distances between the charges and the units
c. Draw the FBD of the interactions mentioned above. Clearly label the forces.
d. Calculate the forces paying attention to the distances between the charges. (210.9 N, 1150.3 N)
e. Find the net force: magnitude and direction. (1361.2 N, to the left)
221
8. Three charges are on a straight line as shown below. What is the resultant force on the
+ 72 μC charge?
q1
- 5 μC
8 cm
q2
+30 μC
13 cm
q3
+72 μC
a. What are the interactions of the + 72 μC charge?
b. Write down the data paying attention to the distances between the charges and the units
c. Draw the FBD of the interactions mentioned above. Clearly label the forces.
d. Calculate the forces paying attention to the distances between the charges. (73.5 N, 1150.3 N)
e. Find the net force: magnitude and direction. (1076.8 N, to the right)
222
29. Honors Physics – Be shocked!
CHECK______
1. An alpha particle consists of two protons (qe = 1.6 x 10-19 C) and two neutrons (no charge). What is the
repulsive force between two alpha particles separated by 2x10-9 m? (2.30x10-10 N)
2. What is the separation of two -4 C charges if the force of repulsion between them is 200 N? (26.8 mm)
3. Two identical charges separated by 30 mm experience a repulsive force of 980 N. What is the magnitude of
each charge? (9.9 C)
223
4. A +60 C charge is placed 60 mm to the left of a +20 C charge. A -35 C charge is placed midway
between the two charges.
a. Draw a sketch
b. What are the interactions of the - 35 μC charge?
b. Write down the data paying attention to the distances between the charges and the units
c. Draw the FBD of the interactions mentioned above. Clearly label the forces.
d. Calculate the forces paying attention to the distances between the charges. (21,000 N; 7000 N)
e. Find the net force: magnitude and direction. (14,000 N, right)
224
UNIT X
ELECTRICITY
ELECTRIC CURRENT
A flow of charge from one place to another constitutes an electric current. An electric circuit is a closed path
in which an electric current carries energy from a source (such as a battery or generator), to a load (such as a
motor or a lamp). In such a circuit, electric current is assumed to go from the positive terminal of the battery (or
generator) through the circuit and back to the negative terminal of the battery. The direction of a current is
conventionally considered to be that in which positive charge would have to move to produce the same effects
as the actual current.
A conductor is a substance through which charge can flow easily, and an insulator is one through which
charge can flow only with great difficulty. Metals, many liquids, and plasmas (gases whose molecules are
charged) are conductors; nonmetallic solids, certain liquids, and gases whose molecules are electrically neutral
are insulators. A number of substances, called semiconductors, are intermediate in their ability to conduct
charge.
If an amount of charge q passes a given point in a conductor in the time interval t, the current in the conductor
is
q
I
Unit: the ampere (A), where 1A = 1 C/s
t
10.1 A wire carries a current of 1.5 A. How many electrons pass any point in the wire each second?
OHM’S LAW
For a current to exist in a conductor, there must be a potential difference between its ends, just as a difference in
height between source and outlet is necessary for a river current to exist.
In the case of a metallic conductor, the current is proportional to the applied potential difference.
This relationship is known as Ohm’s law and is expressed in the form:
I
V
R
I is the current in amperes (A),
V is the voltage or potential difference in volts (V)
R is the resistance in ohms (Ω).
225
10.2 Find the current in a 200 Ω resistor when the potential difference across it is 40 V.
10.3 An electric water heater draws 10 A of current from a 240 V power line. What is its resistance?
CW: OHM'S LAW
1. How many electrons per second must pass a given point in order to have a current of 2.0 A?
(1.25x1019 electrons/s)
2. The battery of a car delivers 50 A for 2 s to start a car.
a. How many coulombs of charge move through the starter during this 2 s? (100 C)
b. How many electrons make up this amount of charge? (6.25x1020 electrons)
3. A car’s starter motor draws 50 A. How much charge flows if the motor runs for 0.75 s? (37.5 C)
226
4. How long does it take for 52 C to pass through a wire carrying a current of 8.0A? (6.5 s)
5. A current of 4 A flows when a resistor is connected across a 12-V battery Find the resistance of this resistor.
(3 Ω)
ELECTRIC POWER
The rate at which work is done to maintain an electric current is given by the product of the current I and the
potential difference V:
P = VI
Units: Watts (W)
The electric energy transferred to a resistor in a time interval t (in seconds) is converted to thermal energy (heat)
then:
E = Pt
Units: Joules (J)
10.4 A 6 V battery delivers 0.5 A of current to an electric motor connected across its terminals.
a. What is the power rating of the motor?
b. How much energy does the motor use in 5.0 min?
10.5 A heater has a resistance of 10 . It operates on 120-V. What is the energy supplied by the heater in 10 s?
227
10.6 A 100 W light bulb is 20% efficient. That means that 20% of the electric energy is converted to light
energy.
a. How much energy does the light bulb convert into light each minute it is in operation?
b. How much heat does the light bulb produce each minute?
10.7 a. How much energy does a 60 W bulb use in half an hour?
b. If the light bulb is 12% efficient, how much heat does it generate during the half hour?
CW: POWER
6. A motor connected to 120V draws a current of 10 A.
a. What power is being consumed? (1200 W)
b. How much energy does the motor use in 8 h of operation? (3.5x107 J))
228
7. A 75 W light bulb is used in a lamp connected to a 120 V power source.
a. What current flows through the bulb? (0.625 A)
b. If the light bulb is 75% efficient, how much energy per minute is given of as light and as heat?
(3375 J and 1125 J)
THE KILOWATT-HOUR
The kilowatt-hour (kWh) is an energy unit equal to the energy delivered by a source whose power is 1 kW in 1
h of operation.
Energy in kWh =
Pt
1000
Note: Time (hours)
10.8 An old color TV set draws 2 A when operated on 120 V.
a. How much power does the set use?
b. If the set is operated for an average of 7 hours per day, what energy in kWh does it consume per month (30
days)?
c. At $0.080 per kWh, what is the cost of operating the set per month?
229
10.9 A digital clock has an operating resistance of 12,000  and is plugged into a 115 V outlet.
a. How much current does it draw?
b. How much power does it use?
c. If the owner of the clock pays $0.09 per kWh, what does it cost to operate the clock for 30 days?
CW: ENERGY CONSUMPTION
9. A 1500-W hair dryer connected to a 120-V outlet runs for 3 mm.
a. How much current does the hair dryer draw? (12.5 A)
b. How much energy does it use? (270,000 J)
c. How many kWh of energy were used? (0.075 kWh)
230
10. A household appliance was inadvertently left on for an entire week (7 days) while a family was away on
vacation. A current of 5 A was drawn from the 120 V line. The cost of electricity in the area is 5 cents per kWh.
a. How much power is needed by the appliance? (600 W)
b. How much energy was used? (3.63 x108 J)
c. How much did it cost to run the appliance? ($5; 5)
11. A girl blows her hair dry every morning using a 1200 W electric hair dryer. On the average, it takes her 5
minutes each time she uses it. The cost of electricity in her area is 5 cents per kWh. What is the annual cost for
running the hair dryer? ($1.83)
231
30. Honors Physics – Electric Stuff
CHECK_____
1. Determine the current through a 5- resistor that has a 40-V drop in potential across it? (8 A)
2. A hot plate has an internal resistance of 22.0 . It operates on 120 V household AC electricity.
a. How much current did it draw? (5.45 A)
b. How much power did it develop? (654 W)
c. If it operated for 15 minutes, how much heat did it develop? (5.88x105 J)
d. If a kWh costs 12 cents, how much did it cost to run the thing? ($0.019)
232
3. Assume that the cost of energy in a home is 8 cents per kilowatt-hour. A family goes on a two-week vacation
leaving a single 80-W light bulb burning. What is the cost? ($2.15)
4. A light bulb heats at the rate of 250 W when the voltage is 120 V. What is its resistance? (57.6 )
5. A power line has a total resistance of 4 k. What is the power loss through the wire if the current is reduced
to 6.0 mA? (0.144 W)
233
ELECTRIC CIRCUITS
The following symbols are used in electric circuits:
An ammeter measures current and it must be connected so that all of the electrons flow through it. Such a
connection is called a series connection.
A voltmeter measures the potential difference across a circuit element. One voltmeter terminal is connected to
one side of the element. The other terminal is connected to the other side. This connection is called a parallel
connection.
10.10 Draw a circuit diagram to include a 60-V battery, an ammeter, a voltmeter and a resistance of 12.5 .
Indicate the ammeter reading.
RESISTORS IN SERIES
- In a series circuit, the current is the same at all points along the wire.
IT = I1 = I2 = I3
- An equivalent resistance is the resistance of a single resistor that could replace all the resistors in a circuit.
The single resistor would have the same current through it as the resistors it replaced.
RE= R1 + R2 + R3
- In a series circuit, the sum of the voltage drops equal the voltage drop across the entire circuit.
VT = V1 + V2 + V3
10.11 A 5  resistor and a 10  resistor are connected in series and placed across a 45 V potential difference.
Draw the circuit.
234
a. What is the equivalent resistance of the circuit?
b. What is the current through the circuit?
c. What is the voltage drop across each resistor?
d. What is the total voltage of the circuit?
10.12 A 10  resistor, a 15  resistor and a 5  resistor are connected in series and placed across a 90-V
potential difference. Draw the circuit.
a. What is the equivalent resistance of the circuit?
b. What is the current through the circuit?
c. What is the voltage drop across each resistor?
d. What is the total voltage of the circuit?
235
CW: SERIES CIRCUITS
11. Three 10 Ω resistors are connected in series to a 12-V battery.
a. Draw a diagram of the circuit.
b. What is the equivalent resistance of the circuit?(30 Ω)
c. What is the total current in the circuit? (0.4 A)
d. What is the potential drop across each resistor? (4 V)
e. What is the current in each resistor? (0.4 A each)
f. What is the power dissipated by each resistor? (1.6 W)
236
PARALLEL CIRCUITS
- In a parallel circuit, each resistor provides a new path for electrons to flow. The total current is the sum of the
currents through each resistor.
IT = I1 + I2 + I3
- The equivalent resistance of a parallel circuit decreases as each new resistor is added.
1
1
1
1
 

RE R1 R2 R3
- The voltage drop across each branch is equal to the voltage of the source.
VT = V1 = V2 = V3
10.13 A 6  resistor, a 30 resistor and a 20 resistor are connected in parallel across a 90-V potential difference.
Draw the circuit.
a. Find the equivalent resistance of the circuit.
b. Find the current in the entire circuit.
c. Find the current through each branch of the circuit.
237
10.14 A 120  resistor, a 60  resistor and a 40  resistor are connected in parallel and placed across a 120-V
potential difference. Draw a diagram of the circuit.
a. Find the equivalent resistance of the circuit.
b. Find the current in the entire circuit.
c. Find the current through each branch of the circuit.
CW: PARALLEL CIRCUITS
12. Three 10 Ω resistors are connected in parallel to a 12 V battery.
a. Draw a diagram of the circuit.
238
b. What is the equivalent resistance of the circuit? (3.33 Ω)
c. What is the total current in the circuit? (3.6 A)
d. What is the voltage across each resistor? (12 V)
e. What is the current in each resistor? (1.2 A)
f. What is the power dissipated by each resistor? (14.4 W)
SERIES-PARALLEL CIRCUITS
The current in a complex circuit can be found by first calculating the effective resistance of the parallel circuits.
Then the effective resistance of the parallel circuits and all the series resistances can be combined into one total
effective resistance and the total current can be determined.
239
10.15 A 30 Ω and a 20 Ω resistor are connected in parallel. This arrangement is connected in series with an 8 Ω
resistor. The voltage in the circuit is 60 V. Draw the circuit.
a. Find the equivalent resistance of the parallel portion of the circuit.
b. What is the equivalent resistance of the entire circuit?
c. What is the current in the entire circuit?
d. What is the voltage drop across the 8  resistor?
e. What is the voltage drop across the parallel portion of the circuit?
f. What is the current in each branch of the parallel portion of the circuit?
240
10.16 Two resistors (60  and 45 ) are connected in parallel. This parallel arrangement is connected in series
with a 30  resistor. The entire circuit is then placed across a 120-V potential difference. Draw a diagram of
the circuit.
a. Find the equivalent resistance of the parallel portion of the circuit.
b. What is the equivalent resistance of the entire circuit?
c. What is the current in the entire circuit?
d. What is the voltage drop across the 30  resistor?
e. What is the voltage drop across the parallel portion of the circuit?
f. What is the current in each branch of the parallel portion of the circuit?
241
CW: SERIES-PARALLEL CIRCUITS
13. A 25 Ω and a 15 Ω resistor are connected in parallel. This parallel arrangement is connected in series with
an 8 Ω resistor. The voltage in the circuit is 60 V. Draw the circuit.
a. Find the equivalent resistance of the circuit. (17.4 Ω)
b. Find the total current of the circuit. (3.4 A)
14. A hair dryer with a resistance of 12 Ω and a lamp with a resistance of 125 Ω are connected in parallel to a
125 V source through a 1.5 Ω resistor in series. Draw the circuit.
a. Find the equivalent resistance (12.4 Ω)
b. Find the total current (10 A)
c. Find the voltage through the 1.5 Ω (15. V)
242
d. Find the current through the lamp and through the hair dryer. (0.88 A, 9.16 A)
15. In the circuit the potential difference, V, across the circuit is 12 V. The values of the four resistors are as
follows: R1 = 2 Ω, R2 = 3 Ω, R3 = 4 Ω and R4 = 6 Ω
Determine:
a. The equivalent resistance of the circuit (3.33 Ω)
b. The total current of the circuit (3.6 A)
c. The current through each resistor (2.4 A, 2.4 A, 1.2 A, 1.2 A)
d. The potential difference across each resistor (4.8 V, 7.2 V, 4.8 V, 7.2 V)
e. The power dissipated by each resistor (11.5 W, 17.3 W, 5.8 W, 8.6 W)
243
16. In the circuit the potential difference, V, across the circuit is 12 V. The values of the four resistors are as
follows: R1 = 2 Ω, R2 = 3 Ω, R3 = 4 Ω and R4 = 6 Ω
Determine:
a. The equivalent resistance of the circuit (3.33 Ω)
b. The total current of the circuit (3.6 A)
c. The potential difference across each resistor (7.2 V, 4.8 V, 4.8 V, 4.8 V)
d. The current through each resistor (3.6 A, 1.6 A, 1.2 A, 0.8 A)
e. The power dissipated by each resistor (25.9 W, 7.7 W, 5.8 W, 3.8 W)
244
31.Honors Physics – Not too Short Circuits
CHECK_____
1. A 15- resistor is connected in parallel with a 30- resistor and a 30-V source. What is the total current is
delivered? (3 A)
2. Three 3- resistors are connected in parallel. This combination is then placed in series with another 3-
resistor. What is the equivalent resistance? (4 
4. A 9- resistor is connected in series with two parallel resistors of 6 and 12 . What is the potential
difference if the total current from the battery is 4 A? (52 V)
9
6
12 
245
4a. Draw a circuit which has a 12  resistor in series with three resistors that are in parallel with each other: a
25 , 35 , and 45  resistor. The voltage source is a 9.0 V battery.
b. What is the current for this circuit? (0.40 A)
c. What is the voltage across the 12  resistor? (4.8 V)
d. What is the voltage across the 35  resistor? (4.2 V)
d. What is the current through each of the resistors in the parallel branch? (0.168 A, 0.12 A, 0.11 A)
246
5. For the following circuit calculate:
85.0 
35.0 
115 
R1
R2
R3
R4
V
6.00 V
25.0 
a. the total resistance of the circuit (45.4 
b. the total current in the circuit, (0.29 A)
c. the power developed in the circuit? (3.8 W)
247
248
249