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CONCEPT ENHANCER - CONSERVATION OF MECHANICAL ENERGY
Conservation of energy is an expression that is commonly used, especially as it relates to stewardship of
natural resources. The term conservation is concerned with conserving, protecting or keeping something
the same. Conservation of energy implies keeping something the same during the course of an event.
The event could be a student sliding down a water slide as illustrated in Figure 3.4 below. For example,
does the student have as much energy near the bottom of the water slide as at the top? Energy is
measured at the top, the event of water sliding takes place, and energy is measured near the bottom. The
two energies are compared to determine if the initial energy equals the final energy, or has energy been
lost or gained.
To consider conservation of mechanical energy, the
different forms of energy need to be recognized. When
you studied conservation of linear momentum in the last
Unit, there was only one kind of momentum, which made
the application of this principle comparatively easy.
However, in conservation of energy there are different
forms to consider. You have become acquainted with the
forms of mechanical energy in this unit. These forms
included gravitational potential energy, elastic potential
energy, and kinetic energy. Energy is transferred when
Figure 3.4
work is done on an object. Recall that work done on an
object is defined as the product of a force applied through a
displacement Fxd. In order for work done to be done, the directions of the applied force and the resulting
displacement must be parallel to one another. In case the force varies, then the average force is used in
the calculation. Friction, a force that acts on an object through a distance, typically does negative work on
an object since it opposes the motion of the object. Thus, the negative work done by friction on an object
needs to be considered in the conservation principle.
Summing Up Energies
Let’s go back to the student on the water slide as illustrated in Figure 3.4. Assuming she was sitting at
rest at the top of the water slide, then all her energy is in the form of gravitational potential energy energy due to her position. This gravitational potential energy may be calculated by the expression EG =
mgh, where m represents the mass (in kg), g represents the acceleration of gravity (in m/s2), and h is the
elevation (in m) relative to some reference point, which in this case is the bottom of the slide. The
resulting unit is Joules (J), which happens to be the same unit for work done. The units being the same is
not a coincidence. Recall that work being done on an object changes its energy.
Assume that the student has a mass of 50 kg (about 120 pounds) and is initially a height of 3.0 m above
the lowest position of the slide. Her gravitational potential energy can be calculated by using the formula
EGinitial = mgh
(50 kg)(9.8 m/s2)(3.0 m) = 1470 J
When the student has descended to a height of 1/3h, then she will have 2/3 the gravitational potential
energy she had at the top of the slide as indicated in the calculation below.
EGfinal = mgh
(50 kg)(9.8 m/s2)(2/3)(3.0 m) = 980 J
Where did the other 1/3 of the initial gravitational potential energy or 490 J go? Well, we know initially
the student was not moving. But we know at her new position, she is now moving. We also know from
experience that she is probably picking up speed as she goes down the slide. If she is moving and picking
up speed as she is going down, she is gaining kinetic energy.
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Kinetic energy, energy due to motion, has the formula EK = ½mv2, where m represents the mass (in kg) and v
represents the speed (in m/s). The resulting unit for kinetic energy, like gravitational potential energy or any other
form of energy, is Joules (J). You do not need to know the mass and the speed of the student in order to calculate
her kinetic energy at any position. You could calculate her kinetic energy at any position by using her calculated
gravitational potential energy at that position and by using the principle of conservation of mechanical energy. If
mechanical energy is conserved and none of the energy is lost to the surroundings (as a result of the work done by
friction or air resistance), the total mechanical energy of the student at any position on the slide must remain the
same. The total mechanical energy (ET ) of the student is the sum of her gravitational potential energy and kinetic
energy as indicated by
ET = EG + EK
Since she was not initially moving at the top of the water slide, her kinetic energy is zero and her total
energy at the top of the slide is equal to her initial gravitational potential energy.
ET = EG + EK = 1470 J + 0 J = 1470 J
If her total mechanical energy remains the same at any position on the water slide, the student’s total
energy at her new position at 2/3h remains 1470 J. Her kinetic energy at any position on the slide can be
found by subtracting her gravitational potential energy from her total mechanical energy at that point.
Thus, at 2/3h
EK = ET – EG = 1470 J – 980 J = 490 J
So if she descended to a height of 2/3h, her gravitational potential energy decreased by 1/3rd. In order for
her total mechanical energy to remain constant, her kinetic energy must increase by this same amount.
By now knowing her kinetic energy at 2/3h, you could determine her speed by manipulating the formula,
EK = ½ mv2. Her speed at this new height would be
v = (2EK/m)½ = [(2)(490 J)/(50 kg)] ½ = 4.4 m/s
What would the student’s EK be at the bottom of the slide? How fast would she be going at the lowest
position on the slide? Since she is at the lowest position of the slide, her height would be zero. As a
result, her gravitational potential energy at the bottom of the slide would also be zero.
Assuming that her total mechanical energy is conserved, her kinetic energy would be
EK = ET – EG = 1470 J – 0 J = 1470 J
And her speed at the bottom would be
v = (2 EK /m)½ = [(2)(1470 J)/(50 kg)] ½ = 7.7 m/s
which is the student’s maximum speed as she slides down the slide.
The distribution of the student’s mechanical energy can be represented in a bar graph. Figure 3.5
illustrates the student’s ET, EG, and EK for the various positions discussed above.
Energy Distributions
Figure 3.5
ET
EG
h = 3.0 m
EK
ET
EG
h = 2.0 m
EK
ET
EG
h=0m
EK
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The energy distributions illustrated in Figure 3.5 are for ideal situation in which no energy is lost to the
surroundings. Notice that at all three positions, the distribution of the two forms of mechanical energy
varies but the total mechanical energy remains the same. As the student slides down, she loses
gravitational potential energy but gains the same amount of kinetic energy. As a result, the total
mechanical energy is constant.
Identify the System
The total mechanical energy of a system, in this case water slide and a person, cannot change of itself.
There is nothing the person can do to gain or lose her total amount of energy. It is possible for a force
outside the system to change the amount of energy. Another person standing along side the water slide,
could stick out his arm, exert a force, and hinder the motion of the person sliding down the slide.
Therefore, an outside agent has done work on the water slider, taking energy from her and slowing her
down. Looking just at the water slider, her energy as illustrated in Figure 3.5 appears to be conserved.
But when we include the influence of the outside agent doing work on her against her motion and
removing some of her energy, then it may appear that the total work and energy are not conserved. For
example, if friction does work on an object sliding on a horizontal surface, the result is a loss of kinetic
energy since eventually the object will come to stop. Where did the kinetic energy go? It did not change
to gravitational potential energy since the object’s relative height did not change. As a result of the work
done by the surface and the surrounding air on the object, eventually all the kinetic energy of the block
was transferred to other forms of energy which may include thermal and sound energies.
In a more realistic situation, the student will lose some her initial mechanical energy in the form of
thermal and sound energies as she slides down the water slide as a result of friction doing work on her.
Since friction opposes her motion, the net result is that she will move down the slide and will not be able
to obtain the maximum speed of 7.7 m/s calculated above. Recall that when work is done, energy is
changed. Work, a force acting through a distance, can also be done in the same direction of the person
sliding, and therefore energy is gained and she speeds up. Or the work can be done against the person,
slowing her down. But the law of conservation of energy requires that nothing in the system itself can
change the energy of that system. The change has to be done by an external agent.
Extending the Concept
Another factor to keep in mind is that energy exists in many forms. By mechanical energy we are limiting
our discussion now to kinetic energy, gravitational potential energy, and elastic potential energy. But
consider for a moment the operation of a light bulb. Electrical energy enters the lamp and it is changed to
light energy. But does all the energy change to light energy? No, some of it changes to heat energy,
which is usually not what we desire from the lamp. But in the bookkeeping of energies, the electrical
energy will always be equal to the sum of the light energy and the heat energy. So we need to consider
the entire system, and be aware of what forms of energy are present.
In the application activity, HOW HOT ARE YOUR HOT WHEELS, you will have the opportunity to
apply the Principle of the Conservation of Mechanical Energy to a toy car going around a loop-the-loop.
You will need to determine the gravitational potential energy required at the beginning of the event to
have the car complete the loop. In addition, you will have to consider friction, which is part of the real
world.
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Period _________ Name _________________________
CONCEPTUAL PRACTICE
1. Consider the skier in Figure 3.6 with a mass of 60.0 kg. The elevations at A, B and D are: hA =
120.0 m, hB = 80.0 m and hD = 40.0 m. The skier is at rest at position A and has a gravitational
potential energy (EG) of 72,000 J. Use 10.0 m/s2 as the approximate value of g.
F
A
hA
B
D
hD
hB
C
E
Figure 3.6
a. If the skier looses negligible energy to friction from A to B, what is the kinetic energy of the skier
at B?
b. If the skier has expended 20,000.0 J to heat by friction by the time he reaches point D, what is his
kinetic energy at point D?
c. On another run the skier expends only 12,000 J into heat by friction by the time he reaches C, the
bottom of the run. How fast would the skier be traveling at C?
d. Suppose the skier starts from point A, goes across D and then up the hill toward F. By the time
the skier has stopped he has transferred 28,000 J of energy to the snow in the form of heat and
plowing snow. What elevation do you predict he will reach?
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2. How does doubling the height that a barbell is lifted affect its gravitational potential energy?
3. How does doubling the velocity of a car affect its kinetic energy? How does tripling the velocity of a
car affect its kinetic energy?
4. How does doubling the velocity of a car affect its stopping distance?
5. Describe the energy changes that take place in a pendulum when it is swinging from side to side,
specifically the energies at the ends of the swing and the middle of the swing. How does the total
energy change at each point along the path? How does work enter the energy picture?
6. Describe the energy changes on a mass hanging from a spring when the mass oscillates up and down.
List the types of mechanical energy present, and show where each is a maximum and where each is
zero.
7. When friction causes a moving object to slow down, is work being done on the object? Explain your
reasoning.
.
8. Consider the hunter, his bow and arrow to all be part of the same system. Describe the energy
transfers and the conservation of energy within the system from the time the hunter draws the bow to
the time the arrow is in flight.
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