CHAPTER 6 NOTES - WORK AND ENERGY
Work in physics is defined as the product of a force and the
distance an object moves while the force is being applied to it.
The force and distance must be in the same direction. The
equation is:
W = FscosӨ
Where W is the work done, F is the magnitude of the applied
force, s is the distance moved during application of the force,
and Ө is the angle between the direction of the force and the
direction of the distance moved.
Example
Find the work done by a woman pulling a suitcase by exerting
a force of 94 N at an angle of 25 degrees with the horizontal for
a distance of 35 m.
Find the work done if the force is parallel to the horizontal.
The basic unit of work is the joule(J) and is equal to the
product of one newton and one meter. In the English system of
measurement, the unit of work is the foot pound, the product
of a force of one pound acting through a distance of one foot.
In order for work to be done, there must be a force acting and
a distance moved. If the force is zero, the work done is zero.
Also, if the distance moved is zero, the work done is zero. If the
angle between the direction of the force and the distance
moved is 90o, the work done is zero since the force and distance
must be in the same direction. If the force and distance are in
opposite directions, the work done by the force is negative
since the cos180o is negative one.
The work-energy theorem states that work done on an object
by a net force results in a change in the kinetic energy of the
object. This means that positive work done on an object will
result in an increase in the kinetic energy of the object and
negative work will result in a decrease in the object's kinetic
energy.
Kinetic energy is defined as energy of motion and is calculated
using the formula:
KE = ½MV2
Where KE is the kinetic energy, M is the mass of the moving
object, and V is the speed of the moving object. The kinetic
energy of an object can change due to changes in either the
mass or speed but we will usually be considering a change in
speed. This leads to the work-energy equation:
W = ΣFscosӨ = ½MVf2 - ½MVo2
Where W is the work done, ΣF is the net force, s is the distance
moved, Ө is the angle between the net force and the distance
moved, M is the mass of the object, Vf is the final speed of the
object and Vo is the initial speed of the object.
Example
A 0.045 kg golf ball is hit with a club that gives it an initial
velocity of 41.0 m/s. Find the work done on the golf ball.
If the ball and the club are in contact for a distance of 0.100 m,
find the average force exerted on the ball by the club.
Work Done By the Force of Gravity
When an object falls from a height ho to a height hf, the
equation used to calculate the work done by gravity is:
W = Mg(h0 - hf)
Where W is the work done, M is the mass of the falling object,
g is the acceleration due to gravity, hf is the final height, and ho
is the initial height. Mg is the force acting on the object and the
difference in height is the distance it moved. Since they act in
the same direction, the cosine of the angle between them is one
and does not affect the magnitude of the work. If an object
initially goes up and then comes down, we need only consider
the difference in initial height and final height. The total path
followed does not affect total work done if we ignore friction.
Gravitational potential energy is equal to the work done in
raising the mass that has this energy to the height associated
with this energy. Since the force that must be applied to move
the mass upward at a constant velocity is equal to its weight, its
GPE is equal to mgh where m is its mass, g is the acceleration
due to gravity, and h is the height to which it is raised.
Remember that the potential energy created here belongs to
the object and the earth as a system although we will often talk
about potential energy of the object alone, not the system.
Example
Find the gravitational potential energy of a 55 kg person at the
top of the Sears Tower, 443 m above the ground.
The gravitational force is a conservative force. Conservative is
defined as either of two ways:
(1) A force is conservative if the work done is independent of
the path taken by the object between its initial and final
positions.
(2) A force is conservative when it does no net work on an
object moving around a closed path so that it ends its motion at
the same place it started.
Some forces are not conservative. Friction is a nonconservative
force since the amount of work done by a frictional force
depends on the path taken by the object. Since in most normal
situations work is done by both conservative and
nonconservative forces, the equation for work done by a net
force is:
W = Wc + Wnc
Where W is the total work done, Wc is the work done by a
conservative force, and Wnc is the work done by a
nonconservative force.
Example
A 16 kg sled is pulled along a horizontal surface with a
horizontal force of 24 n. Starting from rest, the sled reaches a
speed of 2.0 m/s in a distance of 8.0 m. Find the coefficient of
kinetic friction between the sled and the surface.
With this equation, if gravity is the only conservative force
acting on the object, then we can write:
Wc = PE0 - PEf
And from the work-energy theorem:
Wnc + PE0 - PEf = KEf - KEo
which gives us:
Wnc = ΔKE + ΔPE
And tells us that any work done by an outside force on a
system results in a change in the potential energy of a system,
the kinetic energy of a system, or both.
Example
Rocket Man starts from rest on the ground and is propelled
straight up. At a height of 16 m he has a speed of 5.0 m/s.
His mass including his propulsion unit remains approximately
constant at 136 kg. Find the work done on him and his
propulsion unit.
If Wnc is zero, then we have a truly conservative system and
any change in potential energy leads to an equal and opposite
change in kinetic energy.
The principle of conservation of mechanical energy arises from
the work energy theorem. If an outside force does work on a
system then the total energy of the system must change. The
equation is :
Wnc = Ef - Eo
which indicates that work done by a nonconservative force is
equal to the change in total energy of the system. This change
can occur either in the potential or kinetic energy of the
system. It can also occur in both.
If there is no work done on the system by a nonconservative
force, then the total energy of the system does not change and,
in fact, is equal to a constant value. In this case energy can be
changed from kinetic to potential or potential to kinetic in a
fashion that keeps the total of the two constant.
E = KE + PE
is the equation for this relationship. An increase in KE will
result in an equal decrease in PE and an increase in PE will
result in an equal decrease in KE.
Example
A gymnast is swinging on a high bar. The distance between his
waist(center of mass) and the bar is 1.1 m. If his speed at the
top of his swing is zero and his gain in speed is due entirely to
his change in gravitational potential energy, find his speed at
the bottom of his swing.
Another concept associated with work and the transfer of
energy is power. Power is defined as the rate at which work is
done. The equation for the calculation of power from its
definition is:
P = ΔW/Δt
Where P is the average power, ΔW is the work done, and Δt is
the time required to do the work. The SI unit of power is the
watt which equals one joule per second. 1000 watts is called a
kilowatt and other metric prefixes are used to express very
large or small numbers of watts(mega, giga, milli, etc.).
Power is also used to express the rate of energy consumption. A
light bulb rated at 60 watts uses 60 joules of electrical energy
every second.
Example
Find the average power required to accelerate a car from rest
to 20.0 m/s in 5.6 s along a level stretch of road if the weight of
the car is 9000 n.
The rate of energy production of an automobile is generally
given in an English system power unit, the horsepower which is
approximately 746 watts. We expect more horsepower to mean
greater change in velocity but if we are accelerating a more
massive load, a smaller acceleration will result. The equation
relating force, power and velocity comes from the basic
definition of power.
P = ΔW/Δt
P = FaveS/Δt
P = FaveV
Remember that these values are average or constant values.
Other forms of energy are conserved as well as mechanical
energy. It turns out that the total amount of energy that exists
before an event is equal to the total amount of energy that
exists after the event. However, some of the energy may have
changed from one form to another.
Consider the case of a candy bar that is eaten by a hiker. The
candy bar contains chemical potential energy which is taken in
by the hiker. The hiker's body converts that chemical potential
energy into heat and kinetic energy. The heat is lost to the
atmosphere and the kinetic energy can be changed into
gravitational potential energy as the hiker climbs. If we add
the heat energy and gravitational potential energy together we
get the total energy originally in the candy bar.
This is an example of the Law of Conservation of Energy
which states:
ENERGY CAN BE NEITHER CREATED NOR
DESTROYED, BUT CAN ONLY BE CONVERTED FROM
ONE FORM TO ANOTHER.
So the total energy does not change.
The one notable exception to this occurs during nuclear
reactions when matter and energy are converted according to
Einstein's equation E = mc2.
When work is done by a variable force, the equations become a
little more complicated. If the force varies in a linear fashion
with distance, an average force can be used to find the work
done.
Consider a spring being stretched where the magnitude of the
force = kx, with k as the spring constant in newtons per meter
and x is the distance stretched. When x = 0, F = 0 and when
x=xmax, Fmax = kxmax. The equation for the work done becomes:
W = Favexmax
Since Fave = (0 + Fmax)/2,
W = ½(Fmaxxmax)
W = ½(kxmax2)
This is the equation for the energy stored in a stretched spring
if we do not stretch it beyond its elastic limit.
If the equation governing the relationship between the force
and the distance over which it is applied becomes more
complex, it becomes necessary to use more advanced forms of
mathematical analysis(calculus, etc.).
The amount of work can be approximated using a graph of
force vs distance. The area bounded by the X axis and the
graph of the force function represents the work done. If you
count the blocks and partial blocks in that area you get a good
approximation for the work done.
W = 9.00 X .0278 = .25J for one block.
There are 242 blocks under the curve.
Total work = 242 X .25 = 60.5J
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Page 189, Problems 1, 3, 7, 9, 13, 15, 17, 20, 23, 25, 29, 31, 32,
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