Physcs Notes

7/29/2019 Physcs Notes

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One Dimensional Kinematics - Chapter Outline

Lesson 1 : Describing Motion with Words

a.Introduction to the Language of Kinematics

b.Scalars and Vectors

c.Distance and Displacement

d.Speed and Velocity

e.Acceleration

Lesson 2 : Describing Motion with Diagrams

a.Introduction to Diagrams

b.Ticker Tape Diagrams

c.Vector Diagrams

Lesson 3 : Describing Motion with Position vs. Time Graphs

a.The Meaning of Shape for a p-t Graph

b.The Meaning of Slope for a p-t Graph

c.Determining the Slope on a p-t Graph

Lesson 4 : Describing Motion with Velocity vs. Time Graphs

a.The Meaning of Shape for a v-t Graph

b.The Meaning of Slope for a v-t Graph

c.Relating the Shape to the Motion

d.Determining the Slope on a v-t Graph

e.Determining the Area on a v-t Graph

Lesson 5 : Free Fall and the Acceleration of Gravity

a.Introduction to Free Fall

b.The Acceleration of Gravity

c.Representing Free Fall by Graphs

d.How Fast? and How Far?


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e.The Big Misconception

Lesson 6 : Describing Motion with Equations

a.The Kinematic Equations

b.Kinematic Equations and Problem-Solving

c.Kinematic Equations and Free Fall

d.Sample Problems and Solutions

e.Kinematic Equations and Graphs


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Lesson 1

Introduction to the Language of Kinematics

A typical physics course concerns itself with a variety of broad topics. One such topic is

mechanics - the study of the motion of objects. The first six units of The Physics Classroomtutorial will involve an investigation into the physics of motion. As we focus on the language,

principles, and laws that describe and explain the motion of objects, your efforts should

center on internalizing the meaning of the information. Avoid memorizing the information;

and avoid abstracting the information from the physical world that it describes and explains.

Rather, contemplate the information, thinking about its meaning and its applications.

Kinematics is the science of describing the motion of objects using words, diagrams,

numbers, graphs, and equations. Kinematics is a branch of mechanics. The goal of any study

of kinematics is to develop sophisticated mental models that serve to describe (and

ultimately, explain) the motion of real-world objects.

In this lesson, we will investigate the words used to describe the motion of objects. That is,

we will focus on the language of kinematics. The hope is to gain a comfortable foundation

with the language that is used throughout the study of mechanics. We will study such terms

as scalars, vectors, distance, displacement, speed, velocity and acceleration. These wordsare used with regularity to describe the motion of objects. Your goal should be to become

very familiar with their meaning.


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Scalars and Vectors

Physics is a mathematical science. The underlying concepts and principles have a

mathematical basis. Throughout the course of our study of physics, we will encounter avariety of concepts that have a mathematical basis associated with them. While our

emphasis will often be upon the conceptual nature of physics, we will give considerable and

persistent attention to its mathematical aspect.

The motion of objects can be described by words. Even a person without a background in

physics has a collection of words that can be used to describe moving objects. Words and

phrases such as going fast, stopped, slowing down, speeding up, and turning provide a

sufficient vocabulary for describing the motion of objects. In physics, we use these words

and many more. We will be expanding upon this vocabulary list with words such as distance,

displacement, speed, velocity, and acceleration. As we will soon see, these words are

associated with mathematical quantities that have strict definitions. The mathematical

quantities that are used to describe the motion of objects can be divided into two

categories. The quantity is either a vector or a scalar. These two categories can be

distinguished from one another by their distinct definitions:

Scalars are quantities that are fully described by a magnitude (or numerical value) alone.

Vectors are quantities that are fully described by both a magnitude and a direction.

The remainder of this lesson will focus on several examples of vector and scalar quantities

(distance, displacement, speed, velocity, and acceleration). As you proceed through the

lesson, give careful attention to the vector and scalar nature of each quantity. As we

proceed through other units at The Physics Classroom Tutorial and become introduced tonew mathematical quantities, the discussion will often begin by identifying the new quantity

as being either a vector or a scalar.


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Check Your Understanding

1. To test your understanding of this distinction, consider the following quantities listed

below. Categorize each quantity as being either a vector or a scalar. Click the button to seethe answer.

Quantity Category

a. 5 m

b. 30 m/sec, East

c. 5 mi., North

d. 20 degrees Celsius

e. 256 bytes

f. 4000 Calories


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Distance and Displacement

Distance and displacement are two quantities that may seem to mean the same thing yet

have distinctly different definitions and meanings.

Distance is a scalar quantity that refers to "how much ground an object has covered"

during its motion.

Displacement is a vector quantity that refers to "how far out of place an object is"; it is the

object's overall change in position.

To test your understanding of this distinction, consider the motion depicted in the diagram

below. A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2

meters North.

Even though the physics teacher has walked a total distance of 12 meters, her displacement

is 0 meters. During the course of her motion, she has "covered 12 meters of ground"

(distance = 12 m). Yet when she is finished walking, she is not "out of place" - i.e., there is no

displacement for her motion (displacement = 0 m). Displacement, being a vector quantity,

must give attention to direction. The 4 meters east cancels the 4 meters west; and the 2

meters south cancels the 2 meters north. Vector quantities such as displacement are

direction aware. Scalar quantities such as distance are ignorant of direction. In determining

the overall distance traveled by the physics teachers, the various directions of motion can

be ignored.

Now consider another example. The diagram below shows the position of a cross-country

skier at various times. At each of the indicated times, the skier turns around and reverses

the direction of travel. In other words, the skier moves from A to B to C to D.


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Use the diagram to determine the resulting displacement and the distance traveled by the

skier during these three minutes. Then click the button to see the answer.

ANSWER:-

The skier covers a distance of

(180 m + 140 m + 100 m) = 420 m

and has a displacement of 140 m, rightward.

As a final example, consider a football coach pacing back and forth along the sidelines. The

diagram below shows several of coach's positions at various times. At each marked position,

the coach makes a "U-turn" and moves in the opposite direction. In other words, the coach

moves from position A to B to C to D.

What is the coach's resulting displacement and distance of travel? Click the button to see

the answer.


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To understand the distinction between distance and displacement, you must know the

definitions. You must also know that a vector quantity such as displacement is direction-aware and a scalar quantity such as distance is ignorant of direction. When an object

changes its direction of motion, displacement takes this direction change into account;

heading the opposite direction effectively begins to cancel whatever displacement there

once was.


1. What is the displacement of the cross-country team if they begin at the school, run10 miles and finish back at the school?

ANSWER

The displacement of the runners is 0 miles. While they have covered a distance of 10

miles, they are not "out of place" or displaced. They finish where they started.

Round-trip motions always have a displacement of 0

2. What is the distance and the displacement of the race car drivers in the Indy 500?ANSWER

The displacement of the cars is somewhere near 0 miles since they virtually finish

where they started. Yet the successful cars have covered a distance of 500 miles.


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Speed and Velocity

Just as distance and displacement have distinctly different meanings (despite their

similarities), so do speed and velocity. Speed is a scalar quantity that refers to "how fast anobject is moving." Speed can be thought of as the rate at which an object covers distance. A

fast-moving object has a high speed and covers a relatively large distance in a short amount

of time. A slow-moving object has a low speed and covers a relatively small amount of

distance in a short amount of time. An object with no movement at all has a zero speed.

Velocity is a vector quantity that refers to "the rate at which an object changes its position."

Imagine a person moving rapidly - one step forward and one step back - always returning to

the original starting position. While this might result in a frenzy of activity, it would result in

a zero velocity. Because the person always returns to the original position, the motion

would never result in a change in position. Since velocity is defined as the rate at which the

position changes, this motion results in zero velocity. If a person in motion wishes to

maximize their velocity, then that person must make every effort to maximize the amount

that they are displaced from their original position. Every step must go into moving that

person further from where he or she started. For certain, the person should never change

directions and begin to return to the starting position.

Velocity is a vector quantity. As such, velocity is direction aware. When evaluating the

velocity of an object, one must keep track of direction. It would not be enough to say that

an object has a velocity of 55 mi/hr. One must include direction information in order to fully

describe the velocity of the object. For instance, you must describe an object's velocity as

being 55 mi/hr, east. This is one of the essential differences between speed and velocity.

Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity

and is direction aware.

The task of describing the direction of the velocity vector is easy. The direction of the

velocity vector is simply the same as the direction that an object is moving. It would not

matter whether the object is speeding up or slowing down. If an object is moving

rightwards, then its velocity is described as being rightwards. If an object is moving

downwards, then its velocity is described as being downwards. So an airplane moving

towards the west with a speed of 300 mi/hr has a velocity of 300 mi/hr, west. Note that

speed has no direction (it is a scalar) and the velocity at any instant is simply the speed value

with a direction.


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As an object moves, it often undergoes changes in speed. For example, during an average

trip to school, there are many changes in speed. Rather than the speed-o-meter maintaining

a steady reading, the needle constantly moves up and down to reflect the stopping and

starting and the accelerating and decelerating. One instant, the car may be moving at 50

mi/hr and another instant, it might be stopped (i.e., 0 mi/hr). Yet during the trip to school

the person might average 32 mi/hr. The average speed during an entire motion can be

thought of as the average of all speedometer readings. If the speedometer readings could

be collected at 1-second intervals (or 0.1-second intervals or ... ) and then averaged

together, the average speed could be determined. Now that would be a

lot of work. And fortunately, there is a shortcut. Read on.

Calculating Average Speed and Average Velocity

The average speed during the course of a motion is often computed using the following

formula:

In contrast, the average velocity is often computed using this formula

Let's begin implementing our understanding of these formulas with the following problem:

Q: While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8 hours.

What was her average speed?


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To compute her average speed, we simply divide the distance of travel by the time of travel.

That was easy! Lisa Carr averaged a speed of 55 miles per hour. She may not have been

traveling at a constant speed of 55 mi/hr. She undoubtedly, was stopped at some instant in

time (perhaps for a bathroom break or for lunch) and she probably was going 65 mi/hr at

other instants in time. Yet, she averaged a speed of 55 miles per hour. The above formula

represents a shortcut method of determining the average speed of an object.

Average Speed versus Instantaneous Speed

Since a moving object often changes its speed during its motion, it is common to distinguish

between the average speed and the instantaneous speed. The distinction is as follows.

Instantaneous Speed - the speed at any given instant in time.

Average Speed - the average of all instantaneous speeds; found simply by

a distance/time ratio.

You might think of the instantaneous speed as the speed that the speedometer reads at any

given instant in time and the average speed as the average of all the speedometer readings

during the course of the trip. Since the task of averaging speedometer readings would be

quite complicated (and maybe even dangerous), the average speed is more commonly

calculated as the distance/time ratio.

Moving objects don't always travel with erratic and changing speeds. Occasionally, an object

will move at a steady rate with a constant speed. That is, the object will cover the same

distance every regular interval of time. For instance, a cross-country runner might be

running with a constant speed of 6 m/s in a straight line for several minutes. If her speed is

constant, then the distance traveled every second is the same. The runner would cover a

distance of 6 meters every second. If we could measure her position (distance from an

arbitrary starting point) each second, then we would note that the position would be

changing by 6 meters each second. This would be in stark contrast to an object that is

changing its speed. An object with a changing speed would be moving a different distanceeach second. The data tables below depict objects with constant and changing speed.


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Now let's consider the motion of that physics teacher again. The physics teacher walks 4

meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion

lasted for 24 seconds. Determine the average speed and the average velocity.

The physics teacher walked a distance of 12 meters in 24 seconds; thus, her average speed

was 0.50 m/s. However, since her displacement is 0 meters, her average velocity is 0 m/s.Remember that the displacement refers to the change in position and the velocity is based

upon this position change. In this case of the teacher's motion, there is a position change of

0 meters and thus an average velocity of 0 m/s.

Here is another example similar to what was seen before in the discussion of distance and

displacement. The diagram below shows the position of a cross-country skier at various

times. At each of the indicated times, the skier turns around and reverses the direction of

travel. In other words, the skier moves from A to B to C to D.

Use the diagram to determine the average speed and the average velocity of the skier

during these three minutes. When finished, click the button to view the answer.


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ANSWER

The skier has an average speed of

(420 m) / (3 min) = 140 m/min

and an average velocity of

(140 m, right) / (3 min) = 46.7 m/min, right

As a last example, consider a football coach pacing back and forth along the sidelines. The

diagram below shows several of coach's positions at various times. At each marked position,

the coach makes a "U-turn" and moves in the opposite direction. In other words, the coach

moves from position A to B to C to D.

What is the coach's average speed and average velocity? When finished, click the button to

view the answer.

ANSWER

Seymour has an average speed of


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(95 yd) / (10 min) = 9.5 yd/min

and an average velocity of

(55 yd, left) / (10 min) = 5.5 yd/min, left

In conclusion, speed and velocity are kinematic quantities that have distinctly different

definitions. Speed, being a scalar quantity, is the rate at which an object covers distance.

The average speed is the distance (a scalar quantity) per time ratio. Speed is ignorant of

direction. On the other hand, velocity is a vector quantity; it is direction-aware. Velocity is

the rate at which the position changes. The average velocity is the displacement or position

change (a vector quantity) per time ratio.


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Acceleration

The final mathematical quantity discussed in Lesson 1 is acceleration. An often confused

quantity, acceleration has a meaning much different than the meaning associated with it by

sports announcers and other individuals. The definition of acceleration is:

Acceleration is a vector quantity that is defined as the rate at which an object changes its

velocity. An object is accelerating if it is changing its velocity.

Sports announcers will occasionally say that a person is accelerating if he/she

is moving fast. Yet acceleration has nothing to do with going fast. A person

can be moving very fast and still not be accelerating. Acceleration has to do

with changing how fast an object is moving. If an object is not changing its

velocity, then the object is not accelerating. The data at the right are

representative of a northward-moving accelerating object. The velocity is

changing over the course of time. In fact, the velocity is changing by a constant amount - 10

m/s - in each second of time. Anytime an object's velocity is changing, the object is said to

be accelerating; it has an acceleration.

The Meaning of Constant Acceleration

Sometimes an accelerating object will change its velocity by the same amount each second.

As mentioned in the previous paragraph, the data table above show an object changing its

velocity by 10 m/s in each consecutive second. This is referred to as a constant acceleration

since the velocity is changing by a constant amount each second. An object with a constant

acceleration should not be confused with an object with a constant velocity. Don't be

fooled! If an object is changing its velocity -whether by a constant amount or a varying

amount - then it is an accelerating object. And an object with a constant velocity is not

accelerating. The data tables below depict motions of objects with a constant acceleration

and a changing acceleration. Note that each object has a changing velocity.


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Since accelerating objects are constantly changing their velocity, one can say that the

distance traveled/time is not a constant value. A falling object for instance usually

accelerates as it falls. If we were to observe the motion of a free-falling object (free fall

motion will be discussed in detail later), we would observe that the object averages a

velocity of approximately 5 m/s in the first second, approximately 15 m/s in the second

second, approximately 25 m/s in the third second, approximately 35 m/s in the fourth

second, etc. Our free-falling object would be constantly accelerating. Given these average

velocity values during each consecutive 1-second time interval, we could say that the object

would fall 5 meters in the first second, 15 meters in the second second (for a total distance

of 20 meters), 25 meters in the third second (for a total distance of 45 meters), 35 meters in

the fourth second (for a total distance of 80 meters after four seconds). These numbers are

summarized in the table below.

Time Interval Ave. Velocity During Time Interval Distance Traveled During Time Interval

Total Distance Traveled from 0s to End of Time Interval

0 - 1 s ~ 5 m/s ~ 5 m ~ 5 m

1 -2 s ~ 15 m/s ~ 15 m ~ 20 m

2 - 3 s ~ 25 m/s ~ 25 m ~ 45 m

3 - 4 s ~ 35 m/s ~ 35 m ~ 80 m

Note: The ~ symbol as used here means approximately.


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This discussion illustrates that a free-falling object that is accelerating at a constant rate will

cover different distances in each consecutive second. Further analysis of the first and last

columns of the data above reveal that there is a square relationship between the total

distance traveled and the time of travel for an object starting from rest and moving with a

constant acceleration. The total distance traveled is directly proportional to the square ofthe time. As such, if an object travels for twice the time, it will cover four times (2^2) the

distance; the total distance traveled after two seconds is four times the total distance

traveled after one second. If an object travels for three times the time, then it will cover

nine times (3^2) the distance; the distance traveled after three seconds is nine times the

distance traveled after one second. Finally, if an object travels for four times the time, then

it will cover 16 times (4^2) the distance; the distance traveled after four seconds is 16 times

the distance traveled after one second. For objects with a constant acceleration, the

distance of travel is directly proportional to the square of the time of travel.

Calculating the Average Acceleration

The average acceleration (a) of any object over a given interval of time (t) can be calculated

using the equation

This equation can be used to calculate the acceleration of the object whose motion is

depicted by the velocity-time data table above. The velocity-time data in the table shows

that the object has an acceleration of 10 m/s/s. The calculation is shown below.

Acceleration values are expressed in units of velocity/time. Typical acceleration units

include the following:

m/s/s

mi/hr/s

km/hr/s

m/s2


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These units may seem a little awkward to a beginning physics student. Yet they are very

reasonable units when you begin to consider the definition and equation for acceleration.

The reason for the units becomes obvious upon examination of the acceleration equation.

Since acceleration is a velocity change over a time, the units on acceleration are velocity

units divided by time units - thus (m/s)/s or (mi/hr)/s. The (m/s)/s unit can be

mathematically simplified to m/s2.

The Direction of the Acceleration Vector

Since acceleration is a vector quantity, it has a direction associated with it. The direction of

the acceleration vector depends on two things:

whether the object is speeding up or slowing down

whether the object is moving in the + or - direction

The general RULE OF THUMB is:

If an object is slowing down, then its acceleration is in the opposite direction of its motion.

This RULE OF THUMB can be applied to determine whether the sign of the acceleration of

an object is positive or negative, right or left, up or down, etc.

Consider the two data tables below. In each case, the acceleration of

the object is in the positive direction. In Example A, the object is

moving in the positive direction (i.e., has a positive velocity) and is

speeding up. When an object is speeding up, the acceleration is in

the same direction as the velocity. Thus, this object has a positive

acceleration. In Example B, the object is moving in the negative

direction (i.e., has a negative velocity) and is slowing down.

According to our RULE OF THUMB, when an object is slowing down, the acceleration is in

the opposite direction as the velocity. Thus, this object also has a positive acceleration.


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This same RULE OF THUMB can be applied to the motion of the objects represented in the

two data tables below. In each case, the acceleration of the object is in the negative

direction. In Example C, the object is moving in the positive direction (i.e., has a positive

velocity) and is slowing down. According to our RULE OF THUMB, when an object is slowing

down, the acceleration is in the apposite direction as the velocity. Thus, this object has a

negative acceleration. In Example D, the object is moving in the negative direction (i.e., has

a negative velocity) and is speeding up. When an object is speeding up, the acceleration is in

the same direction as the velocity. Thus, this object also has a negative acceleration.

Observe the use of positive and negative as used in the discussion above (Examples A - D). In

physics, the use of positive and negative always has a physical meaning. It is more than a

mere mathematical symbol. As used here to describe the velocity and the acceleration of a

moving object, positive and negative describe a direction. Both velocity and acceleration are

vector quantities and a full description of the quantity demands the use of a directional

adjective. North, south, east, west, right, left, up and down are all directional adjectives.

Physics often borrows from mathematics and uses the + and - symbols as directional

adjectives. Consistent with the mathematical convention used on number lines and graphs,

positive often means to the right or up and negative often means to the left or down. So to

say that an object has a negative acceleration as in Examples C and D is to simply say that its

acceleration is to the left or down (or in whatever direction has been defined as negative).

Negative accelerations do not refer acceleration values that are less than 0. An acceleration

of -2 m/s/s is an acceleration with a magnitude of 2 m/s/s that is directed in the negativedirection.


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To test your understanding of the concept of acceleration, consider the following problems

and the corresponding solutions. Use the equation for acceleration to determine theacceleration for the following two motions.

ANSWER A

a = 2 m/s/s

Use a = (vf - vi) / t and pick any two points.

a = (8 m/s - 0 m/s) / (4 s)

a = (8 m/s) / (4 s)

a = 2 m/s/s

ANSWER B

a = -2 m/s/s

Use a = (vf-vi) / t and pick any two points.

a = (0 m/s - 8 m/s) / (4 s)


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a = (-8 m/s) / (4 s)

a = -2 m/s/s

Lesson 2

Introduction to Diagrams

Throughout the course, there will be a persistent appeal to your ability to represent physicalconcepts in a visual manner. You will quickly notice that this effort to provide visual

representation of physical concepts permeates much of the discussion in The Physics

Classroom Tutorial. The world that we study in physics is a physical world - a world that we

can see. And if we can see it, we certainly ought to visualize it. And if we seek to understand

it, then that understanding ought to involve visual representations. So as you continue your

pursuit of physics understanding, always be mindful of your ability (or lack of ability) to

visually represent it. Monitor your study and learning habits, asking if your knowledge has

become abstracted to a series of vocabulary words that have (at least in your own mind) no

relation to the physical world which it seeks to describe. Your understanding of physicsshould be intimately tied to the physical world as demonstrated by your visual images.

Like the study of all of physics, our study of 1-dimensional kinematics will be concerned with

the multiple means by which the motion of objects can be represented. Such means include

the use of words, the use of graphs, the use of numbers, the use of equations, and the use

of diagrams. Lesson 2 focuses on the use of diagrams to describe motion. The two most

commonly used types of diagrams used to describe the motion of objects are:

ticker tape diagrams

vector diagrams

Begin cultivating your visualization skills early in the course. Spend some time on the rest of

Lesson 2, seeking to connect the visuals and graphics with the words and the physical

reality. And as you proceed through the remainder of the unit 1 lessons, continue to make

these same connections.


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Ticker Tape Diagrams

A common way of analyzing the motion of objects in physics labs is to perform a ticker tape

analysis. A long tape is attached to a moving object and threaded through a device that

places a tick upon the tape at regular intervals of time - say every 0.10 second. As the object

moves, it drags the tape through the "ticker," thus leaving a trail of dots. The trail of dots

provides a history of the object's motion and therefore a representation of the object's

motion.

The distance between dots on a ticker tape represents the object's position change during

that time interval. A large distance between dots indicates that the object was moving fast

during that time interval. A small distance between dots means the object was moving slow

during that time interval. Ticker tapes for a fast- and slow-moving object are depicted

below.

The analysis of a ticker tape diagram will also reveal if the object is moving with a constant

velocity or accelerating. A changing distance between dots indicates a changing velocity and

thus an acceleration. A constant distance between dots represents a constant velocity and

therefore no acceleration. Ticker tapes for objects moving with a constant velocity and with

an accelerated motion are shown below.


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And so ticker tape diagrams provide one more means of representing various features of

the motion of objects.

Vector Diagrams

Vector diagrams are diagrams that depict the direction and relative magnitude of a vector

quantity by a vector arrow. Vector diagrams can be used to describe the velocity of a

moving object during its motion. For example, a vector diagram could be used to represent

the motion of a car moving down the road.

In a vector diagram, the magnitude of a vector quantity is represented by the size of the

vector arrow. If the size of the arrow in each consecutive frame of the vector diagram is the

same, then the magnitude of that vector is constant. The diagrams below depict the velocityof a car during its motion. In the top diagram, the size of the velocity vector is constant, so

the diagram is depicting a motion of constant velocity. In the bottom diagram, the size of

the velocity vector is increasing, so the diagram is depicting a motion with increasing

velocity - i.e., an acceleration.


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Vector diagrams can be used to represent any vector quantity. In future studies, vector

diagrams will be used to represent a variety of physical quantities such as acceleration,

force, and momentum. Be familiar with the concept of using a vector arrow to represent thedirection and relative size of a quantity. It will become a very important representation of an

object's motion as we proceed further in our studies of the physics of motion.


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Lesson 5

Free Fall and the Acceleration of Gravity

Introduction to Free Fall

A free falling object is an object that is falling under the sole influence of gravity. Any object

that is being acted upon only be the force of gravity is said to be in a state of free fall. There

are two important motion characteristics that are true of free-falling objects:

Free-falling objects do not encounter air resistance.

All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s

(often approximated as 10 m/s/s for back-of-the-envelope calculations)

Because free-falling objects are accelerating downwards at a rate of 9.8 m/s/s, a

ticker tape trace or dot diagram of its motion would depict an acceleration. The dot

diagram at the right depicts the acceleration of a free-falling object. The position of

the object at regular time intervals - say, every 0.1 second - is shown. The fact that

the distance that the object travels every interval of time is increasing is a sure sign

that the ball is speeding up as it falls downward. Recall from an earlier lesson, that

if an object travels downward and speeds up, then its acceleration is downward.

Free-fall acceleration is often witnessed in a physics classroom by means of an ever-popular

strobe light demonstration. The room is darkened and a jug full of water is connected by a

tube to a medicine dropper. The dropper drips water and the strobe illuminates the falling

droplets at a regular rate - say once every 0.2 seconds. Instead of seeing a stream of water

free-falling from the medicine dropper, several consecutive drops with increasing separation

distance are seen. The pattern of drops resembles the dot diagram shown in the graphic at

the right.


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The Acceleration of Gravity

It was learned in the previous part of this lesson that a free-falling object is an object that is

falling under the sole influence of gravity. A free-falling object has an acceleration of 9.8

m/s/s, downward (on Earth). This numerical value for the acceleration of a free-falling

object is such an important value that it is given a special name. It is known as the

acceleration of gravity - the acceleration for any object moving under the sole influence of

gravity. A matter of fact, this quantity known as the acceleration of gravity is such an

important quantity that physicists have a special symbol to denote it - the symbol g. The

numerical value for the acceleration of gravity is most accurately known as 9.8 m/s/s. There

are slight variations in this numerical value (to the second decimal place) that are

dependent primarily upon on altitude. We will occasionally use the approximated value of

10 m/s/s in The Physics Classroom Tutorial in order to reduce the complexity of the many

mathematical tasks that we will perform with this number. By so doing, we will be able tobetter focus on the conceptual nature of physics without too much of a sacrifice in

numerical accuracy.

g = 9.8 m/s/s, downward

( ~ 10 m/s/s, downward)

Recall from an earlier lesson that acceleration is the rate at which an object changes its

velocity. It is the ratio of velocity change to time between any two points in an object's path.

To accelerate at 9.8 m/s/s means to change the velocity by 9.8 m/s each second.

If the velocity and time for a free-falling object being dropped from a position of rest were

tabulated, then one would note the following pattern.

Time (s) Velocity (m/s)

0 0

1 - 9.8

2 - 19.6


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3 - 29.4

4 - 39.2

5 - 49.0

Observe that the velocity-time data above reveal that the object's velocity is changing by 9.8

m/s each consecutive second. That is, the free-falling object has an acceleration of

approximately 9.8 m/s/s.

Another way to represent this acceleration of 9.8 m/s/s is to

add numbers to our dot diagram that we saw earlier in this

lesson. The velocity of the ball is seen to increase as depicted

in the diagram at the right. (NOTE: The diagram is not drawnto scale - in two seconds, the object would drop considerably

further than the distance from shoulder to toes.)


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Representing Free Fall by Graphs

Early in Lesson 1 it was mentioned that there are a variety of means of describing the

motion of objects. One such means of describing the motion of objects is through the use of

graphs - position versus time and velocity vs. time graphs. In this part of Lesson 5, the

motion of a free-falling motion will be represented using these two basic types of graphs.

A position versus time graph for a free-falling object is shown below.

Observe that the line on the graph curves. As learned earlier, a curved line on a position

versus time graph signifies an accelerated motion. Since a free-falling object is undergoing

an acceleration (g = 9.8 m/s/s), it would be expected that its position-time graph would be

curved. A further look at the position-time graph reveals that the object starts with a small

velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs.

time graph is the velocity of the object (as learned in Lesson 3), the small initial slope

indicates a small initial velocity and the large final slope indicates a large final velocity.

Finally, the negative slope of the line indicates a negative (i.e., downward) velocity.

A velocity versus time graph for a free-falling object is shown below.


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Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal

line on a velocity versus time graph signifies an accelerated motion. Since a free-falling

object is undergoing an acceleration (g = 9,8 m/s/s, downward), it would be expected that

its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals

that the object starts with a zero velocity (as read from the graph) and finishes with a large,negative velocity; that is, the object is moving in the negative direction and speeding up. An

object that is moving in the negative direction and speeding up is said to have a negative

acceleration (if necessary, review the vector nature of acceleration). Since the slope of any

velocity versus time graph is the acceleration of the object (as learned in Lesson 4), the

constant, negative slope indicates a constant, negative acceleration. This analysis of the

slope on the graph is consistent with the motion of a free-falling object - an object moving

with a constant acceleration of 9.8 m/s/s in the downward direction.

How Fast? and How Far?

Free-falling objects are in a state of acceleration. Specifically,

they are accelerating at a rate of 9.8 m/s/s. This is to say that

the velocity of a free-falling object is changing by 9.8 m/s

every second. If dropped from a position of rest, the object

will be traveling 9.8 m/s (approximately 10 m/s) at the end of

the first second, 19.6 m/s (approximately 20 m/s) at the end

of the second second, 29.4 m/s (approximately 30 m/s) at theend of the third second, etc. Thus, the velocity of a free-falling

object that has been dropped from a position of rest is

dependent upon the time that it has fallen. The formula for

determining the velocity of a falling object after a time of t

seconds is

vf = g * t

where g is the acceleration of gravity. The value for g on Earth is 9.8 m/s/s. The above

equation can be used to calculate the velocity of the object after any given amount of time

when dropped from rest. Example calculations for the velocity of a free-falling object after

six and eight seconds are shown below.

Example Calculations:

At t = 6 s


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vf = (9.8 m/s2) * (6 s) = 58.8 m/s

At t = 8 s

vf = (9.8 m/s2) * (8 s) = 78.4 m/s

The distance that a free-falling object has fallen from a position of rest is also dependent

upon the time of fall. This distance can be computed by use of a formula; the distance fallen

after a time of t seconds is given by the formula.

d = 0.5 * g * t2

where g is the acceleration of gravity (9.8 m/s/s on Earth). Example calculations for the

distance fallen by a free-falling object after one and two seconds are shown below.

Example Calculations:

At t = 1 s

d = (0.5) * (9.8 m/s2) * (1 s)2 = 4.9 m

At t = 2 s

d = (0.5) * (9.8 m/s2) * (2 s)2 = 19.6 m

At t = 5 s

d = (0.5) * (9.8 m/s2) * (5 s)2 = 123 m

(rounded from 122.5 m)

The diagram below (not drawn to scale) shows the results of several distance calculations

for a free-falling object dropped from a position of rest.


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The Big Misconception

Earlier in this lesson, it was stated that the acceleration of a free-falling object (on earth) is

9.8 m/s/s. This value (known as the acceleration of gravity) is the same for all free-falling

objects regardless of how long they have been falling, or whether they were initially

dropped from rest or thrown up into the air. Yet the questions are often asked "doesn't a

more massive object accelerate at a greater rate than a less massive object?" "Wouldn't an

elephant free-fall faster than a mouse?" This question is a reasonable inquiry that is

probably based in part upon personal observations made of falling objects in the physical

world. After all, nearly everyone has observed the difference in the rate of fall of a single

piece of paper (or similar object) and a textbook. The two objects clearly travel to the

ground at different rates - with the more massive book falling faster.

The answer to the question (doesn't a more massive object accelerate at a greater rate than

a less massive object?) is absolutely not! That is, absolutely not if we are considering the

specific type of falling motion known as free-fall. Free-fall is the motion of objects that move

under the sole influence of gravity; free-falling objects do not encounter air resistance.

More massive objects will only fall faster if there is an appreciable amount of air resistance

present.

The actual explanation of why all objects accelerate at the same rate involves the concepts

of force and mass. The details will be discussed in Unit 2 of The Physics Classroom. At that

time, you will learn that the acceleration of an object is directly proportional to force and

inversely proportional to mass. Increasing force tends to increase acceleration while

increasing mass tends to decrease acceleration. Thus, the greater force on more massive

objects is offset by the inverse influence of greater mass. Subsequently, all objects free fall

at the same rate of acceleration, regardless of their mass.


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Newton's Laws - Chapter Outline

Lesson 1: Newton's First Law of Motion

a.Newton's First Law

b.Inertia and Mass

c.State of Motion

d.Balanced and Unbalanced Forces

Lesson 2: Force and Its Representation

a.The Meaning of Force

b.Types of Forces

c.Drawing Free-Body Diagrams

d.Determining the Net Force

Lesson 3 : Newton's Second Law of Motion

a.Newton's Second Law

b.The Big Misconception

c.Finding Acceleration

d.Finding Individual Forces

e.Free Fall and Air Resistance

f.Double Trouble (a.k.a., Two Body Problems)

Lesson 4 : Newton's Third Law of Motion

a.Newton's Third Law

b.Identifying Action and Reaction Force Pairs


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Lesson 1

Newton's First Law of Motion

Newton's First Law

In a previous chapter of study, the variety of ways by which motion can be described

(words, graphs, diagrams, numbers, etc.) was discussed. In this unit (Newton's Laws of

Motion), the ways in which motion can be explained will be discussed. Isaac Newton (a 17th

century scientist) put forth a variety of laws that explain why objects move (or don't move)

as they do. These three laws have become known as Newton's three laws of motion. The

focus of Lesson 1 is Newton's first law of motion - sometimes referred to as the law of

inertia.

Newton's first law of motion is often stated as

An object at rest stays at rest and an object in motion stays 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 that predicts the behavior of stationary objects

and the other that predicts the behavior of moving objects. The two parts are summarized

in the following diagram.

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


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in this same state of rest. If in motion with an eastward velocity of

5 m/s, they will continue in this same state of motion (5 m/s,

East). If in motion with a leftward velocity of 2 m/s, they will

continue in this same state of motion (2 m/s, left). The state of

motion of an object is maintained as long as the object is not acted upon by an unbalancedforce. All objects resist changes in their state of motion - they tend to "keep on doing what

they're doing."

Suppose that you filled a baking dish to the rim with water and walked around an oval track

making an attempt to complete a lap in the least amount of time. The water would have a

tendency to spill from the container during specific locations on the track. In general thewater spilled when:

the container was at rest and you attempted to move it

the container was in motion and you attempted to stop it

the container was moving in one direction and you attempted to change its direction.

The water spills whenever the state of motion of the container is changed. The waterresisted this change in its own state of motion. The

water tended to "keep on doing what it was doing."

The container was moved from rest to a high speed at

the starting line; the water remained at rest and

spilled onto the table. The container was stopped near

the finish line; the water kept moving and spilled over container's leading edge. The

container was forced to move in a different direction to make it around a curve; the water

kept moving in the same direction and spilled over its edge. The behavior of the water

during the lap around the track can be explained by Newton's first law of motion.

Everyday Applications of Newton's First Law

There are many applications of Newton's first law of motion. Consider some of your

experiences in an automobile. Have you ever observed the behavior of coffee in a coffee

cup filled to the rim while starting a car from rest or while bringing a car to rest from a stateof motion? Coffee "keeps on doing what it is doing." When you accelerate a car from rest,


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the road provides an unbalanced force on the spinning wheels to push the car forward; yet

the coffee (that was at rest) wants to stay at rest. While the car accelerates forward, the

coffee remains in the same position; subsequently, the car accelerates out from under the

coffee and the coffee spills in your lap. On the other hand, when braking from a state of

motion the coffee continues forward with the same speed and in the same direction,ultimately hitting the windshield or the dash. Coffee in motion stays in motion.

Have you ever experienced inertia (resisting changes in your state of motion) in an

automobile while it is braking to a stop? The force of the road on the locked wheels

provides the unbalanced force to change the car's

state of motion, yet there is no unbalanced force to

change your own state of motion. Thus, you

continue in motion, sliding along the seat in forward

motion. A person in motion stays in motion with the

same speed and in the same direction ... unless

acted upon by the unbalanced force of a seat belt.

Yes! Seat belts are used to provide safety for

passengers whose motion is governed by Newton's laws. The seat belt provides the

unbalanced force that brings you from a state of motion to a state of rest. Perhaps you

could speculate what would occur when no seat belt is used.

There are many more applications of Newton's first law of motion. Several applications arelisted below. Perhaps you could think about the law of inertia and provide explanations for

each application.

Blood rushes from your head to your feet while quickly stopping when

riding on a descending elevator.

The head of a hammer can be tightened onto the wooden handle by

banging the bottom of the handle against a hard surface.

A brick is painlessly broken over the hand ofa physics teacher by

slamming it with a hammer. (CAUTION: do not attempt this at home!)

To dislodge ketchup from the bottom of a ketchup bottle, it is often

turned upside down and thrusted downward at high speeds and then

abruptly halted.

Headrests are placed in cars to prevent whiplash injuries during rear-end collisions.


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While riding a skateboard (or wagon or bicycle), you fly forward off the board when hitting

a curb or rock or other object that abruptly halts the motion of the skateboard.

Inertia and Mass

Newton's first law of motion states that "An object at rest stays at rest and an object inmotion stays in motion with the same speed and in the same direction unless acted upon by

an unbalanced force." Objects tend to "keep on doing what they're doing." In fact, 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: the resistance an object has to a change in its state

of motion.

Newton's conception of inertia stood in direct opposition to more popular conceptions

about motion. The dominant thought prior to Newton's day was that it was the natural

tendency of objects to come to a rest position. Moving objects, so it was believed, would

eventually stop moving; a force was necessary to keep an object moving. But if left to itself,

a moving object would eventually come to rest and an object at rest would stay at rest;

thus, the idea that dominated people's thinking for nearly 2000 years prior to Newton was

that it was the natural tendency of all objects to assume a rest position.

Galileo and the Concept of Inertia

Galileo, a premier scientist in the seventeenth century, developed the concept of inertia.

Galileo reasoned that moving objects eventually stop because of a force called friction. In

experiments using a pair of inclined planes facing each other, Galileo observed that a ball

would roll down one plane and up the opposite plane to approximately the same height. If

smoother planes were used, the ball would roll up the opposite plane even closer to the

original height. Galileo reasoned that any difference between initial and final heights was

due to the presence of friction. Galileo postulated that if friction could be entirely

eliminated, then the ball would reach exactly the same height.


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Galileo further observed that regardless of the angle at which the planes were oriented, the

final height was almost always equal to the initial height. If the slope of the opposite incline

were reduced, then the ball would roll a further distance in order to reach that original

height.

Galileo's reasoning continued - if the opposite incline were elevated at nearly a 0-degree

angle, then the ball would roll almost forever in an effort to reach the original height. And if

the opposing incline was not even inclined at all (that is, if it were oriented along the

horizontal), then ... an object in motion would continue in motion... .


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A common physics demonstration relies on this principle that the more massive the object,

the more that object resist changes in its state of motion. The demonstration goes as

follows: several massive books are placed upon a teacher's head. A wooden board is placed

on top of the books and a hammer is used to drive a nail into the board. Due to the large

mass of the books, the force of the hammer is sufficiently resisted (inertia). This isdemonstrated by the fact that the teacher does not feel the hammer blow. (Of course, this

story may explain many of the observations that you previously have made concerning your

"weird physics teacher.") A common variation of this demonstration involves breaking a

brick over the teacher's hand using the swift blow of a hammer. The massive bricks resist

the force and the hand is not hurt. (CAUTION: do not try these demonstrations at home.)

State of Motion

Inertia is the tendency of an object to resist changes in its state of motion. But what is

meant by the phrase state of motion? The state of motion of an object is defined by its

velocity - the speed with a direction. Thus, inertia could be redefined as follows:

Inertia: tendency of an object to resist changes in its velocity.

An object at rest has zero velocity - and (in the absence of an unbalanced force) will remain

with a zero velocity. Such an object will not change its state of motion (i.e., velocity) unless

acted upon by an unbalanced force. An object in motion with a velocity of 2 m/s, East will (in

the absence of an unbalanced force) remain in motion with a velocity of 2 m/s, East. Such an

object will not change its state of motion (i.e., velocity) unless acted upon by an unbalanced

force. Objects resist changes in their velocity.

As learned in an earlier unit, an object that is not changing its velocity is said to have anacceleration of 0 m/s/s. Thus, we could provide an alternative means of defining inertia:

Inertia: tendency of an object to resist accelerations.


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Since these two forces are of equal magnitude and in opposite directions, they balance each

other. The person is at equilibrium. There is no unbalanced force acting upon the personand thus the person maintains its state of motion. (Note: diagrams such as the one above

are known as free-body diagrams and will be discussed in detail in Lesson 2.)

Now consider a book sliding from left to right across a tabletop. Sometime in the prior

history of the book, it may have been given a shove and set in motion from a rest position.

Or perhaps it acquired its motion by sliding down an incline from an elevated position.

Whatever the case, our focus is not upon the history of the book but rather upon the

current situation of a book sliding to the right across a tabletop. The book is in motion and

at the moment there is no one pushing it to the right. (Remember: a force is not needed to

keep a moving object moving to the right.) The forces acting upon the book are shown

below.

The force of gravity pulling downward and the force of the table pushing upwards on the

book are of equal magnitude and opposite directions. These two forces balance each other.

Yet there is no force present to balance the force of friction. As the book moves to the right,

friction acts to the left to slow the book down. There is an unbalanced force; and as such,


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the book changes its state of motion. The book is not at equilibrium and subsequently

accelerates. Unbalanced forces cause accelerations. In this case, the unbalanced force is

directed opposite the book's motion and will cause it to slow down. (Note: diagrams such as

the one above are known as free-body diagrams and will be discussed in detail in Lesson 2.)

To determine if the forces acting upon an object are balanced or unbalanced, an analysis

must first be conducted to determine what forces are acting upon the object and in what

direction. If two individual forces are of equal magnitude and opposite direction, then the

forces are said to be balanced. An object is said to be acted upon by an unbalanced force

only when there is an individual force that is not being balanced by a force of equal

magnitude and in the opposite direction. Such analyses are discussed in Lesson 2 of this unit

and applied in Lesson 3.


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Lesson 2

The Meaning of Force

A force is a push or pull upon an object resulting from the object's interaction with another

object. Whenever there is an interaction between two objects, there is a force upon each ofthe objects. When the interaction ceases, the two objects no longer experience the force.

Forces only exist as a result of an interaction.

For simplicity sake, all forces (interactions) between objects can be placed into two broad

categories:

contact forces, and

forces resulting from action-at-a-distance

Contact forces are those types of forces that result when the two interacting objects are

perceived to be physically contacting each other. Examples of contact forces include

frictional forces, tensional forces, normal forces, air resistance forces, and applied forces.

These specific forces will be discussed in more detail later in Lesson 2 as well as in other

lessons.

Action-at-a-distance forces are those types of forces that result even when the two

interacting objects are not in physical contact with each other, yet are able to exert a push

or pull despite their physical separation. Examples of action-at-a-distance forces include

gravitational forces. For example, the sun and planets exert a gravitational pull on each

other despite their large spatial separation. Even when your feet leave the earth and you

are no longer in physical contact with the earth, there is a gravitational pull between you

and the Earth. Electric forces are action-at-a-distance forces. For example, the protons in

the nucleus of an atom and the electrons outside the nucleus experience an electrical pull

towards each other despite their small spatial separation. And magnetic forces are action-

at-a-distance forces. For example, two magnets can exert a magnetic pull on each other

even when separated by a distance of a few centimeters. These specific forces will be

discussed in more detail later in Lesson 2 as well as in other lessons.


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Examples of contact and action-at-distance forces are listed in the table below.

Contact Forces Action-at-a-Distance Forces

Frictional Force Gravitational Force

Tension Force Electrical Force

Normal Force Magnetic Force

Air Resistance Force

Applied Force

Spring Force

Force is a quantity that is measured using the standard metric unit known as the Newton. A

Newton is abbreviated by an "N." To say "10.0 N" means 10.0 Newton of force. One Newton

is the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s. Thus, the

following unit equivalency can be stated:

A force is a vector quantity. As learned in an earlier unit, a vector quantity is a quantity that

has both magnitude and direction. To fully describe the force acting upon an object, you

must describe both the magnitude (size or numerical value) and the direction. Thus, 10

Newton is not a full description of the force acting upon an object. In contrast, 10 Newton,

downward is a complete description of the force acting upon an object; both the magnitude

(10 Newton) and the direction (downward) are given.

Because a force is a vector that has a direction, it is common to represent

forces using diagrams in which a force is represented by an arrow. Such

vector diagrams were introduced in an earlier unit and are used

throughout the study of physics. The size of the arrow is reflective of the

magnitude of the force and the direction of the arrow reveals the direction

that the force is acting. (Such diagrams are known as free-body diagrams

and are discussed later in this lesson.) Furthermore, because forces are


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vectors, the effect of an individual force upon an object is often canceled by the effect of

another force. For example, the effect of a 20-Newton upward force acting upon a book is

canceled by the effect of a 20-Newton downward force acting upon the book. In such

instances, it is said that the two individual forces balance each other; there would be no

unbalanced force acting upon the book.

Other situations could be imagined in which two of the individual

vector forces cancel each other ("balance"), yet a third individual

force exists that is not balanced by another force. For example,

imagine a book sliding across the rough surface of a table from left

to right. The downward force of gravity and the upward force of the

table supporting the book act in opposite directions and thus

balance each other. However, the force of friction acts leftwards,

and there is no rightward force to balance it. In this case, an

unbalanced force acts upon the book to change its state of motion.

The exact details of drawing free-body diagrams are discussed later. For now, the emphasis

is upon the fact that a force is a vector quantity that has a direction. The importance of this

fact will become clear as we analyze the individual forces acting upon an object later in this

lesson.

Types of Forces

A force is a push or pull acting upon an object as a result of its interaction with another

object. There are a variety of types of forces. Previously in this lesson, a variety of force

types were placed into two broad category headings on the basis of whether the force

resulted from the contact or non-contact of the two interacting objects.

Contact Forces Action-at-a-Distance Forces

Frictional Force Gravitational Force


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Tension Force Electrical Force

Normal Force Magnetic Force


Applied Force

Spring Force

These types of individual forces will now be discussed in more detail. To read about each

force listed above, continue scrolling through this page. Or to read about an individual force,

click on its name from the list below.

Applied Force

Gravitational Force

Normal Force

Frictional Force


Tension Force

Spring Force

Type of Force

(and Symbol) Description of Force

Applied Force

Fapp An applied force is a force that is applied to an object by a person or another object. If

a person is pushing a desk across the room, then there is an applied force acting upon the

object. The applied force is the force exerted on the desk by the person.


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Gravity Force

(also known as Weight)

Fgrav The force of gravity is the force with which the earth, moon, or other massively large

object attracts another object towards itself. By definition, this is the weight of the object.

All objects upon earth experience a force of gravity that is directed "downward" towards the

center of the earth. The force of gravity on earth is always equal to the weight of the object

as found by the equation:

Fgrav = m * g

where g = 9.8 N/kg (on Earth)

and m = mass (in kg)

(Caution: do not confuse weight with mass.)

Normal Force

Fnorm 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 in order to support the weight of the book. On

occasions, a normal force is exerted horizontally between two objects that are in contact

with each other. For instance, if a person leans against a wall, the wall pushes horizontally

on the person.

Friction Force

Ffrict The friction force is the force exerted by a surface as an object moves across it or

makes an effort to move across it. There are at least two types of friction force - sliding and

static friction. Thought it is not always the case, the friction force often opposes the motion

of an object. For example, if a book slides across the surface of a desk, then the desk exerts

a friction force in the opposite direction of its motion. Friction results from the two surfaces

being pressed together closely, causing intermolecular attractive forces between molecules

of different surfaces. As such, friction depends upon the nature of the two surfaces and

upon the degree to which they are pressed together. The maximum amount of friction force

that a surface can exert upon an object can be calculated using the formula below:


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Ffrict = Fnorm

The friction force is discussed in more detail later on this page.


Fair The air resistance is a special type of frictional force that acts upon objects as they

travel through the air. The force of air resistance is often observed to oppose the motion of

an object. This force will frequently be neglected due to its negligible magnitude (and due to

the fact that it is mathematically difficult to predict its value). 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. Air resistance will be discussed in more detail in Lesson 3.

Tension Force

Ftens The tension force is the force that is transmitted through a string, rope, cable or wire

when it is pulled tight by forces acting from opposite ends. The tension force is directed

along the length of the wire and pulls equally on the objects on the opposite ends of the

wire.

Spring Force

Fspring The spring force is the force exerted by a compressed or stretched spring upon any

object that is attached to it. An object that compresses or stretches a spring is always acted

upon by a force that restores the object to its rest or equilibrium position. For most springs

(specifically, for those that are said to obey "Hooke's Law"), the magnitude of the force is

directly proportional to the amount of stretch or compression of the spring.


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Confusion of Mass and Weight

A few further comments should be added about the single force that is a source of much

confusion to many students of physics - the force of gravity. As mentioned above, the force

of gravity acting upon an object is sometimes referred to as the weight of the object. Many

students of physics confuse weight with mass. The mass of an object refers to the amount of

matter that is contained by the object; the weight of an object is the force of gravity acting

upon that object. Mass is related to how much stuff is there and weight is related to the pull

of the Earth (or any other planet) upon that stuff. The mass of an object (measured in kg)

will be the same no matter where in the universe that object is located. Mass is never

altered by location, the pull of gravity, speed or even the existence of other forces. For

example, a 2-kg object will have a mass of 2 kg whether it is located on Earth, the moon, or

Jupiter; its mass will be 2 kg whether it is moving or not (at least for purposes of our study);

and its mass will be 2 kg whether it is being pushed upon or not.

On the other hand, the weight of an object (measured in Newton) will vary according to

where in the universe the object is. Weight depends upon which planet is exerting the force

and the distance the object is from the planet. Weight, being equivalent to the force of

gravity, is dependent upon the value of g - the gravitational field strength. On earth's

surface g is 9.8 N/kg (often approximated as 10 N/kg). On the moon's surface, g is 1.7 N/kg.

Go to another planet, and there will be another g value. Furthermore, the g value is

inversely proportional to the distance from the center of the planet. So if we were tomeasure g at a distance of 400 km above the earth's surface, then we would find the g value

to be less than 9.8 N/kg. (The nature of the force of gravity will be discussed in more detail

in a later unit of The Physics Classroom.) Always be cautious of the distinction between mass

and weight. It is the source of much confusion for many students of physics.

Sliding versus Static Friction

As mentioned above, the friction force is the force exerted by a surface as an object moves

across it or makes an effort to move across it. For the purpose of our study of physics at The

Physics Classroom, there are two types of friction force - static friction and sliding friction.

Sliding friction results when an object slides across a surface. As an example, consider

pushing a box across a floor. The floor surface offers resistance to the movement of the box.

We often say that the floor exerts a friction force upon the box. This is an example of a

sliding friction force since it results from the sliding motion of the box. If a car slams on its

brakes and skids to a stop (without antilock brakes), there is a sliding friction force exertedupon the car tires by the roadway surface. This friction force is also a sliding friction force


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because the car is sliding across the road surface. Sliding friction forces can be calculated

from knowledge of the coefficient of friction and the normal force exerted upon the object

by the surface it is sliding across. The formula is:

Sliding Ffrict = Fnorm

The symbol represents the coefficient of sliding friction between the two surfaces. The

coefficient value is dependent primarily upon the nature of the surfaces that are in contact

with each other. For most surface combinations, the friction coefficients show little

dependence upon other variables such as area of contact, temperature, etc. Values of have

been experimentally determined for a variety of surface combinations and are often

tabulated in technical manuals and handbooks. The values of provide a measure of the

relative amount of adhesion or attraction of the two surfaces for each other. The more thatsurface molecules tend to adhere to each other, the greater the coefficient values and the

greater the friction force.

Friction forces can also exist when the two surfaces are not sliding across each other. Such

friction forces are referred to as static friction. Static friction results when the surfaces of

two objects are at rest relative to one another and a force exists on one of the objects to set

it into motion relative to the other object. Suppose you were to push with 5-Newton of

force on a large box to move it across the floor. The box might remain in place. A static

friction force exists between the surfaces of the floor and the box to prevent the box from

being set into motion. The static friction force balances the force that you exert on the box

such that the stationary box remains at rest. When exerting 5 Newton of applied force on

the box, the static friction force has a magnitude of 5 Newton. Suppose that you were to

push with 25 Newton of force on the large box and the box were to still remain in place.

Static friction now has a magnitude of 25 Newton. Then suppose that you were to increase

the force to 26 Newton and the box finally budged from its resting position and was set into

motion across the floor. The box-floor surfaces were able to provide up to 25 Newton of

static friction force to match your applied force. Yet the two surfaces were not able to

provide 26 Newton of static friction force. The amount of static friction resulting from the

adhesion of any two surfaces has an upper limit. In this case, the static friction force spans

the range from 0 Newton (if there is no force upon the box) to 25 Newton (if you push on

the box with 25 Newton of force). This relationship is often expressed as follows:

Ffrict-static frict-static Fnorm


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The symbol frict-static represents the coefficient of static friction between the two

surfaces. Like the coefficient of sliding friction, this coefficient is dependent upon the types

of surfaces that are attempting to move across each other. In general, values of static

friction coefficients are greater than the values of sliding friction coefficients for the same

two surfaces. Thus, it typically takes more force to budge an object into motion than it doesto maintain the motion once it has been started.

The meaning of each of these forces listed in the table above will have to be thoroughly

understood to be successful during this unit. Ultimately, you must be able to read a verbal

description of a physical situation and know enough about these forces to recognize their

presence (or absence) and to construct a free-body diagram that illustrates their relative

magnitude and direction.

Drawing Free-Body Diagrams

Free-body diagrams are diagrams used to show the relative magnitude and direction of all

forces acting upon an object in a given situation. A free-body diagram is a special example of

the vector diagrams that were discussed in an earlier unit.

These diagrams will be used throughout our study of physics.

The size of the arrow in a free-body diagram reflects the

magnitude of the force. The direction of the arrow shows the

direction that the force is acting. Each force arrow in the

diagram is labeled to indicate the exact type of force. It is

generally customary in a free-body diagram to represent the

object by a box and to draw the force arrow from the center

of the box outward in the direction that the force is acting.

An example of a free-body diagram is shown at the right.

The free-body diagram above depicts four forces acting upon the object. Objects do not

necessarily always have four forces acting upon them. There will be cases in which thenumber of forces depicted by a free-body diagram will be one, two, or three. There is no

hard and fast rule about the number of forces that must be drawn in a free-body diagram.

The only rule for drawing free-body diagrams is to depict all the forces that exist for that

object in the given situation. Thus, to construct free-body diagrams, it is extremely

important to know the various types of forces. If given a description of a physical situation,

begin by using your understanding of the force types to identify which forces are present.

Then determine the direction in which each force is acting. Finally, draw a box and add

arrows for each existing force in the appropriate direction; label each force arrow according


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to its type. If necessary, refer to the list of forces and their description in order to

understand the various force types and their appropriate symbols

Determining the Net Force

If you have been reading through Lessons 1 and 2, then Newton's first law of motion ought

to be thoroughly understood.

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.

In the statement of Newton's first law, the unbalanced force refers to that force that does

not become completely balanced (or canceled) by the other individual forces. If either all

the vertical forces (up and down) do not cancel each other and/or all horizontal forces do

not cancel each other, then an unbalanced force exists. The existence of an unbalanced

force for a given situation can be quickly realized by looking at the free-body diagram for

that situation. Free-body diagrams for three situations are shown below. Note that the

actual magnitudes of the individual forces are indicated on the diagram.

In each of the above situations, there is an unbalanced force. It is commonly said that in

each situation there is a net force acting upon the object. The net force is the vector sum of

all the forces that act upon an object. That is to say, the net force is the sum of all the

forces, taking into account the fact that a force is a vector and two forces of equal

magnitude and opposite direction will cancel each other out. At this point, the rules for

summing vectors (such as force vectors) will be kept relatively simple. Observe the following

examples of summing two forces:


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Observe in the diagram above that a downward vector will provide a partial or full

cancellation of an upward vector. And a leftward vector will provide a partial or full

cancellation of a rightward vector. The addition of force vectors can be done in the same

manner in order to determine the net force (i.e., the vector sum of all the individual forces).

Consider the three situations below in which the net force is determined by summing the

individual force vectors that are acting upon the objects.


1. Free-body diagrams for four situations are shown below. For each situation,determine the net force acting upon the object. Click the buttons to view the answer


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2. Free-body diagrams for four situations are shown below. The net force is known for each

situation. However, the magnitudes of a few of the individual forces are not known. Analyze

each situation individually and determine the magnitude of the unknown forces. Then clickthe button to view the answers.

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