The previous part of Lesson
2 discussed the relationship between work and energy
change. Whenever work is done upon an object by an external
force, there will be a change in the total
mechanical energy of the object. If only internal
forces are doing work (no work done by external forces),
there is no change in total mechanical energy; the total
mechanical energy is said to be conserved. Because
external forces are capable of changing the total mechanical
energy of an object, they are sometimes referred to as
nonconservative forces. Because internal forces do not
change the total mechanical energy of an object, they are
sometimes referred to as conservative forces. In this part
of Lesson 2, we will further explore the quantitative
relationship between work and energy.

The quantitative
relationship between work and mechanical energy is expressed
by the following equation:

TME_{i} +
W_{ext} = TME_{f}

The equation states that the initial amount of total
mechanical energy (TME_{i}) plus the work
done by external forces (W_{ext}) is equal to
the final amount of total mechanical energy
(TME_{f}). A few notes should be made about the
above equation. First, the mechanical energy can be either
potential energy (in which case it
could be due to springs or
gravity) or kinetic
energy. Given this fact, the above equation can be
rewritten as

KE_{i} +
PE_{i} + W_{ext} = KE_{f} +
PE_{f}

The second note which should be made about the above
equation is that the work done by external forces can be
a positive or a negative work
term. Whether the work term takes on a positive or a
negative value is dependent upon the angle between the force
and the motion. Recall from Lesson
1 that the work is dependent upon the angle between the
force and the displacement vectors. If the angle is 180
degrees as it occasionally is, then the work term will be
negative. If the angle is 0 degrees, then the work term will
be positive.

The
above equation is expresses the quantitative relationship
between work and energy. This equation will be the basis for
the rest of this unit. It will form the basis of the
conceptual aspect of our study of work and energy as well as
the guidingforce for our approach to solving
mathematical problems. A large slice of the world of
motion can be understood through the use of this
relationship between work and energy.

To begin our
investigation of the work-energy relationship, we will
investigate situations involving work being done by external
forces (nonconservative forces). Consider a weightlifter who
applies an upwards force (say 1000 N) to a barbell to
displace it upwards a given distance (say 0.25 meters) at a
constant speed. The initial energy plus the work done by the
external force equals the final energy. If the barbell
begins with 1500 Joules of energy (this is just a made up
value) and the weightlifter does 250 Joules of work
(F*d*cosine of angle = 1000
N*0.25 m*cosine 0 degrees = 250 J), then the barbell will
finish with 1750 Joules of mechanical energy. The final
amount of mechanical energy (1750 J) is equal to the initial
amount of mechanical energy (1500 J) plus the work done by
external forces (250 J).

Now consider a baseball catcher who
applies a rightward force (say 6000 N) to a leftward moving
baseball to bring it from a high speed to a rest position
over a given distance (say 0.10 meters). The initial energy
plus the work done by the external force equals the final
energy. If the ball begins with 605 Joules of energy (this
is just another made up value), and the catcher does -600
Joules of work (F*d*cosine of
angle = 6000 N*0.10 m*cosine 180 degrees = -600 J), then
the ball will finish with 5 Joules of mechanical energy. The
final energy (5 J) is equal to the initial energy (605 J)
plus the work done by external forces (-600 J).

Now consider a car which is skidding from
a high speed to a lower speed. The force of friction between
the tires and the road exerts a leftward force (say 8000 N)
on the rightward moving car over a given distance (say 30
m). The initial energy plus the work done by the external
force equals the final energy. If the car begins with 320
000 Joules of energy (this is just another made up value),
and the friction force does -240 000 Joules of work
(F*d*cosine of angle = 8000
N*30 m*cosine 180 degrees = -240 000 J), then the car will
finish with 80 000 Joules of mechanical energy. The final
energy (80 000 J) is equal to the initial energy (320 000 J)
plus the work done by external forces (-240 000 J).

As a final example,
consider a cart being pulled up an inclined plane at
constant speed by a student during a Physics lab. The
applied force on the cart (say 18 N) is directed parallel to
the incline to cause the cart to be displaced parallel to
the incline for a given displacement (say 0.7 m). The
initial energy plus the work done by the external force
equals the final energy. If the cart begins with 0 Joules of
energy (this is just another made up value), and the student
does 12.6 Joules of work (F*d*cosine
of angle = 18 N*0.7 m*cosine 0 degrees = 12.6 J), then
the cart will finish with 12.6 Joules of mechanical energy.
The final energy (12.6 J) is equal to the initial energy (0
J) plus the work done by external forces (12.6 J).

In each of these examples, an external
force does work upon an object over a given distance to
change the total mechanical energy of the object. If the
external force (or nonconservative force) does positive
work, then the object gains mechanical energy. The
amount of energy gained is equal to the work done on the
object. If the external force (or nonconservative force)
does negative work, then the
object loses mechanical energy. The amount of mechanical
energy lost is equal to the work done on the object. In
general, the total mechanical energy of the object in the
initial state (prior to the work being done) plus the work
done equals the total mechanical energy in the final
state.

The
work-energy relationship presented here can be combined with
the expressions for potential and kinetic energy to solve
complex problems. Like all complex problems, they can be
made simple if first analyzed from a conceptual
viewpoint and broken down into parts. In other words, avoid
treating work-energy problems as mere mathematical problems.
Rather, engage your mind and utilize your understanding of
physics concepts to approach the problem. Ask "What forms of
energy are present initially and finally?" and "Based on the
equations, how much of each form of energy is present
initially and finally?" and "Is work being done by external
forces?" Use this approach on the following three practice
problems. After solving, click the button to view the
answers.

Practice Problem
#1

A
1000-kg car traveling with a speed of 25 m/s skids to a
stop. The car experiences an 8000 N force of friction.
Determine the stopping distance of the car.

Practice Problem
#2

At
the end of the Shock Wave roller coaster ride, the 6000-kg
train of cars (includes passengers) is slowed from a speed
of 20 m/s to a speed of 5 m/s over a distance of 20 meters.
Determine the braking force required to slow the train of
cars by this amount.

Practice Problem
#3

A
shopping cart full of groceries is sitting at the top of a
2.0-m hill. The cart begins to roll until it hits a stump at
the bottom of the hill. Upon impact, a 0.25-kg can of
peaches flies horizontally out of the shopping cart and hits
a parked car with an average force of 500 N. How deep a dent
is made in the car (i.e., over what distance does the 500 N
force act upon the can of peaches before bringing it to a
stop)?

All three of the above
problems have one thing in common: there is a force which
does work over a distance in order to remove mechanical
energy from an object. The force acts opposite the object's
motion (angle between force and displacement is 180 degrees)
and thus does negative work.
Negative work results in a loss of the object's total amount
of mechanical energy. In each situation, the work is
related to the kinetic energy change. And since the distance
(d) over which the force does work is related to the work
and since the velocity squared (v^2) of the object is
related to the kinetic energy, there must also be a direct
relation between the stopping distance and the velocity
squared. Observe the derivation below.

TME_{i} +
W_{ext} = TME_{f}

KE_{i} +
W_{ext} = 0 J

0.5•m•v_{i}^{2}
+ F•d•cos(Theta) = 0 J

0.5•m•v_{i}^{2}
= F•d

v_{i}^{2}
d

The above equation depicts stopping
distance as being dependent upon the square of the
velocity. This means that a twofold increase in velocity
would result in a fourfold (two squared) increase in
stopping distance. A threefold increase in velocity would
result in a nine-fold (three squared) increase in stopping
distance. And a fourfold increase in velocity would result
in a sixteen-fold (four squared) increase in stopping
distance. This is one more example in which an equation
becomes more than a mere algebraic recipe for solving
problems. Equations can also be powerful guides to thinking
about how two quantities are related to each other. In the
case of a horizontal force bringing an object to a stop over
a some horizontal distance, the stopping distance of the
object is related to the square of the velocity of the
object.

The above principle - that stopping
distance is proportional to velocity squared - is often the
focus of a popular physics lab. A Hot wheels car is rolled
down an inclined plane to the floor below. Once reaching the
floor, it strikes a computer diskette box and skids to a
stop as a result of the friction between the car/box system
and the floor. A photo gate time is used to determine the
speed of the car prior to striking the box. Several trials
are performed and a data set is collected and plotted. As
the speed of the car is increased, the stopping distance is
increased. If the data are plotted, then a clear power
relationship is seen. If power regression is performed on
the data set, the results tend to show that d =
k•v^{2} where k is a constant of
proportionality.

The examples mentioned on this page involve the
application of the work-energy relationship to situations
involving external or nonconservative forces doing work. An
entirely different outcome results in situations in which
there is no work done by external forces. The next
part of Lesson 2 involves an analysis of these
situations.