4.4 The Law of Conservation of Energy
and Efficiency
This section is by far the lengthiest and most important section in this unit. It features the law of conservation of energy,
examples of the law, an investigation to verify (or refute) the law, the concept and equation for the efficiency of an energy
transformation, an investigation that explores the efficiency of a ramp, and a case study that applies the law of conservation of
energy to sports activities.
Achievement Chart
Categories
Assessment Opportunities/Specific
Expectation Addressed
Assessment Tools
Knowledge/Understanding
Practice Questions
Understanding Concepts, q. 1–8
EW1.01, EW1.03, EW1.05
Section 4.4 Questions
Understanding Concepts, q. 1–5
EW1.01, EW1.03, EW1.05
Investigation 4.4.1
Analysis c–f, Evaluation g–j
EW2.01, EW2.03
Investigation 4.4.2
Analysis d–f, Evaluation g–i
EW2.01, EW2.03
Case Study
EW3.02
Section 4.4 Questions
Applying Inquiry Skills, q. 6, 7
EW1.03, EW2.02
Investigation 4.4.2
Synthesis j, k
EW2.01, EW2.03
Section 4.4 Questions
Making Connections, q. 8
EW1.03
Section 4.4 Questions
Reflecting, q. 9
EW1.03
Rubric 1: Knowledge/Understanding
Inquiry
Communication
Making Connections
Rubric 2: Inquiry Skills
Rubric 3: Communication
Rubric 4: Making Connections
Expectations Addressed
Overall Expectations—EWV.01, EWV.02, EWV.03
Overall Skills Expectations— SIS.01, SIS.02 SIS.03, SIS.04 SIS.05, SIS.07, SIS.08, SIS.09, SIS.10, SIS.11
Specific Expectations:
• EW1.01 define and describe the concepts and units related
• EW1.05 analyze, in quantitative terms, the relationships
to energy, work, and power (e.g., energy, work, power,
among percent efficiency, input energy, and useful output
gravitational potential energy, kinetic energy, thermal
energy for several energy transformations
energy and its transfer [heat], efficiency)
• EW2.01 design and carry out experiments related to
energy transformations, identifying and controlling major
• EW1.03 analyze, in qualitative and quantitative terms,
variables (e.g., design and carry out an experiment to
simple situations involving work, gravitational potential
identify the energy transformations of a swinging
energy, kinetic energy, and thermal energy and its transfer
pendulum, and to verify the law of conservation of energy;
(heat), using the law of conservation of energy
design and carry out an experiment to determine the power
produced by a student)
© 2002 Nelson Thomson Learning
Unit 2 Energy, Work, and Power 99
• EW2.02 analyze and interpret experimental data or
computer simulations involving work, gravitational
potential energy, kinetic energy, thermal energy and its
transfer (heat), and the efficiency of the energy
transformation (e.g., experimental data on the motion of a
swinging pendulum or a falling or sliding mass in terms of
the energy transformations that occur)
• EW2.03 communicate the procedures, data, and
conclusions of investigations involving work, mechanical
BACKGROUND INFORMATION
Although the law of conservation of energy applies to all
forms of energy, in this section we deal mainly with
mechanical energy. However, students should become aware
that the final form of nearly all energy transformations is
thermal energy. For example, friction results in the increased
agitation of molecules, which increases their thermal energy.
Various methods can be used to test the law of
conservation of energy. The method chosen for Investigation
4.4.1 involves the transformation of gravitational potential
energy of a pendulum bob into kinetic energy. Similar
transformations occur for a ball dropped vertically
downward from rest, or a cart on an inclined plane allowed
to accelerate from rest down the ramp.
The equation for efficiency (pages 141–42) is
straightforward to apply. To determine the efficiency of a
real example experimentally, a ramp was chosen (see
Investigation 4.4.2). Determining efficiency experimentally
is also addressed in a different context involving waste heat
(see Lab Exercise 5.2.1, pages 167–68).
Physics is applied in many ways to sports activities. The
application chosen for the Case Study (pages 144–45) relates
to the design and manufacture of equipment for various
sports. Pole vaulting is featured more than other sport
because it is obviously related to kinetic energy and
gravitational potential energy; it also involves elastic
potential energy of the pole. Information about pole vaulting
and many other sports activities is easy to find on the
Internet.
ADDRESSING ALTERNATIVE
CONCEPTIONS
Students can be excused for thinking that the law of
conservation of energy is something they must learn in
science class but that doesn’t apply to real-life situations.
Few energy transformations conserve the original form of
energy. For example, if you drop a ball to the floor, its total
mechanical energy is never conserved; that is evident
because the ball never bounces back up to its original drop
height. However, students should be aware that the original
gravitational potential energy changes into kinetic energy
and other forms of energy, such as sound and thermal
energy.
100 Chapter 4 Energy, Work, Heat, and Power
energy, power, thermal energy and its transfer (heat), and the
law of conservation of energy, using appropriate means (e.g.,
oral and written descriptions, numerical and/or graphical
analyses, tables, diagrams)
• EW3.02 analyze and explain improvements in sports
performance, using principles and concepts related to
work, kinetic and potential energy, and the law of
conservation of energy (e.g., explain the importance of the
initial kinetic energy of a pole vaulter or high jumper)
If any students are aware of the law of conservation of
mass-energy, which is summarized by Einstein’s famous
equation E = mc2, and they ask about it, commend them on
their learning and let them know that the law of conservation
of mass-energy involves nuclear reactions, which we do not
consider here.
By definition, a heater should be considered 100%
efficient because all the input energy becomes thermal
energy. However, the true efficiency depends on how the
heater is used. For example, an electric room heater placed
on a ledge near the top of a wall will be less efficient in
heating a room than the same heater placed near the floor.
In considering the sports of high jumping and pole
vaulting, it is interesting that the jumper’s entire body does
not have to clear the bar at the same time. Only the centre of
mass of the body has to clear the bar. When this principle
was first discovered, the records for the heights increased
dramatically because jumpers could glide over the bar in an
inverted V-shape, allowing greater heights to be achieved.
Related Background Resources
F.H. Froes, “Is the Use of Advanced Materials in Sports
Equipment Unethical?” JOM 49, 2 (1997),
15–19. (Also available online at
www.tms.org/pubs/journals/JOM/9702/
Froes-9702.html.)
Karim Nice, “How Pole Vaulting Works.” (Available
online at www.howstuffworks.com/pole-vault.htm.)
Peter J. Brancazio, Sport Science: Physical Laws and
Optimum Performance (New York: Simon & Schuster,
1984).
The following product can be purchased from Boreal
Laboratories (www.boreal.com or 1-800-387-9393):
• The Physics of Toys, #65841-00, about $250
This highly recommended set includes 23 different
toys, some of which apply to the law of conservation of
energy.
Nelson Web site:
www.science.nelson.com
for specific Web links
© 2002 Nelson Thomson Learning
PLANNING
Suggested Time
Narrative/Practice Questions—30 to 40 minutes
Investigation 4.4.1—30 minutes
Investigation 4.4.2—30 minutes
Case Study—30 minutes
Section Questions—30 minutes
Core Instructional Resources
• Solutions Manual
• Colour Transparencies
• Lab and Study Blackline Masters
• Reference to the Appendixes: Appendix A1 and
Appendix A4
TEACHING SUGGESTIONS
Figure 1(a)
“Canned earthquake” device
• This section will take two full periods to complete. A good
way to split the topics is to focus on the law of
conservation of energy and the corresponding experiment,
Investigation 4.4.1, for the first class, and then to focus on
the rest of the section for the second class.
• Begin the class discussion by showing several simple
examples of energy transformation. For example, show a
pendulum bob swinging on the end of a string, with the
amplitude of the swing decreasing gradually. Also let
several balls with different bounce properties fall from the
same height and compare how high they bounce back.
Relate the observed actions to the law of conservation of
energy.
• You can design and build your own “canned earthquake”
device (see Figure 1(a)). The device can be used for a
“black box” demonstration in which the students try to
infer what causes the action observed. To create the
“earthquake,” swirl the can around repeatedly while
holding it vertically, and then immediately put the can
down on a hard surface. The can will wobble violently at
first and then less so. To create more interest and as a
greater challenge, wind the device up before class and
invert it immediately (see Figure 1(b)). The elastic
potential energy will remain “potential” until an
appropriate moment in class. Note that the elastic band
tends to wear out after several violent quakes and may
have to be changed each year.
© 2002 Nelson Thomson Learning
Figure 1(b)
Energy is stored until an appropriate time.
• Another “black box” device, the favourite potential energy
device of many physics teachers, is the “come-back” can
(see Figure 2). Starting with the can in its natural state
(unenergized), roll it horizontally on a flat surface and
observe it come to a stop and then return. The energy
transforms from kinetic to elastic potential, then back to
kinetic. An alternate way to use the device is to wind it up
without letting the students know what you are doing, and
then allow the can to roll uphill on an low-incline plane.
Unit 2 Energy, Work, and Power 101
Assessment:
• Students can be assessed on their inquiry and
communication skills.
Student Preparation
• To learn more about controlled experiments and reporting
on labs, students can refer to Appendix A1, pages 548–49,
and Appendix A4, pages 560–64.
DURING
• Students should try to reduce the sources of error as much
as possible during this investigation. For example, it is
important that the height measurements (Procedure steps 2
and 5) are the perpendicular distances from the reference
level to the bottom of the bob. It is also important to
reduce parallax error as much as possible.
AFTER
• (c) The average speed at the bob’s lowest position is
Figure 2
The “come-back” can
INVESTIGATION 4.4.1
Testing the Law of Conservation of Energy
• The basic idea of this investigation is simple: compare the
maximum kinetic energy of a swinging pendulum bob
with the maximum gravitational potential energy of the
bob at its highest position, and explain any observed
discrepancy.
•
•
BEFORE
Teacher Preparation
Time: 30 minutes
Materials and Equipment:
Each group of three or four students will need:
a 50-g (or smaller) mass
strong string or wire
a tall stand (more than 1 m) and clamp
a millisecond timer (or computer)
a light source and photocell
Safety and Disposal:
• It is important to set up the apparatus so the swinging
pendulum does not come close to the light source or the
photocell.
• The stands should be clamped securely to the lab bench.
• Care should be taken whenever using electrical equipment.
102 Chapter 4 Energy, Work, Heat, and Power
•
•
•
•
simply the bob’s diameter divided by the time interval
taken by the bob to break the light beam to the photocell.
The kinetic energy of the bob can be found using the

mv 2 
kinetic energy equation  E k =
.

2 

(d) The gravitational potential energy is calculated at the
highest position of the bob (step 5) relative to the lowest
position (step 2) using the gravitational potential energy
equation (Eg = mgh).
(e) In most cases, the maximum gravitational potential
energy exceeds the maximum kinetic energy by a slight
amount. However, due to experimental error, it sometimes
happens that the two values are equal or, even worse, that
the maximum kinetic energy exceeds the maximum
gravitational potential energy.
(f) There is a small amount of friction at the top of the
pendulum string, and the mechanical energy gradually
becomes thermal energy as the friction slows the
pendulum down. A very small amount of energy goes to
overcoming air resistance.
(g) The reply depends on the original question.
(h) The answer depends on the students’ hypothesis and
prediction.
(i) The main sources of error in this investigation are
• random parallax error in measuring the diameter of the
pendulum bob and in measuring the lowest and highest
heights of the bob above the reference level
• random error in measuring the heights of the bob above
the reference level at an angle to the perpendicular
• systematic error caused by the fact that the diameter of
the pendulum bob may not be the true distance that
breaks the light beam aimed at the photocell
• systematic error if the bob is either too high or too low
when it passes by the light beam
© 2002 Nelson Thomson Learning
• (j) To reduce sources of error,
• Use a plumb bob or other device to make sure the
vertical distances measured to the pendulum bob are
perpendicular to the reference surface.
• Use a very fine light beam centred exactly on the middle
of the bob.
• Use a cylindrical bob.
DURING
• Students should gather data and record them in an
appropriate table. (They can be given the Blackline
Master.) Table 1 is a sample table of data with
experimental values.
AFTER
• (c) A typical set of data is shown in Table 1.
Extensions and Modifications:
• Students could test the law of conservation of energy
Table 1 Data for Ramp Investigation
using some other method, such as dropping a ball
vertically from rest or analyzing the motion of a cart
released from rest from the top of a ramp.
Variables
mass of cart, m (kg)
weight of cart, Fg (N)
length of ramp, ∆d (m)
height of ramp, h (m)
angle of ramp (°)
force parallel to ramp, F (N)
work input, Ein (J)
useful energy output, Eout (J)
efficiency (%)
INVESTIGATION 4.4.2
The Efficiency of a Ramp
• In this controlled experiment, students determine how
various factors affect the efficiency of a simple machine,
in this case an inclined plane or ramp.
• (d) Answers are as follows:
BEFORE
Teacher Preparation
Time: 30 minutes
Materials and Equipment:
Each group of three to five students will need
a dynamics cart
a ramp (a straight board or piece of plywood)
a beam balance
a spring scale or force meter calibrated in newtons
several bricks or books to support the ramp
a metre stick
extra masses (500 g, 1000 g)
Safety and Disposal:
• Make sure the ramp support is stable; clamp it if
necessary.
• Any extra masses mounted on the cart should be secured
safely.
Assessment:
• Students can be assessed on their inquiry and
communication skills.
Student Preparation
• Students should have read and understood Sample
Problem 2 on page 142 and completed Practice question 5
on page 143 before starting this investigation.
• To learn more about controlled experiments and reporting
on labs, students can refer to Appendix A1, pages 548–49,
and Appendix A4, pages 560–64.
© 2002 Nelson Thomson Learning
Trial 1
0.86
8.4
2.44
0.75
18
2.7
6.6
6.3
95
•
•
•
•
(i) Changing the mass of the cart has little or no effect
on the efficiency.
(ii) Changing the slope of the ramp has little or no
effect on the efficiency.
(iii) Increasing the friction between the cart and the
ramp causes a decrease in efficiency.
(e) The most important factor to control is the friction
between the cart and the ramp. To reduce this friction, we
could use smooth surfaces and, obviously, we could use
rolling friction rather than sliding friction.
(f) We can work backwards from the calculated data to
determine the force of friction. If the frictional resistance
were zero, the efficiency would be 100%. Thus, using the
data provided in (c), Ein = Eout = 6.3 J. The force parallel to
the ramp with no friction added can be calculated to be
2.6 N (W = F∆d), which is 0.1 N less than the measured
value. Therefore, the frictional resistance has a magnitude
of 0.1 N.
(g) The answer depends on the hypothesis and prediction.
Usually students predict efficiency values lower than the
measured values.
(h) Typical sources of error are
• parallax errors in measuring distances and forces
(random errors)
• difficulty in keeping the metre stick vertical when
measuring the height of the ramp (random error)
• difficulty in keeping the force on the cart parallel to the
ramp (random error)
• difficulty in keeping the cart moving at a constant speed
Unit 2 Energy, Work, and Power 103
• (i) The answer depends on the equipment used and the
success of the investigation. For example, students who
used a spring scale are likely to suggest using a force
meter.
• (j) The main advantage is that the force required to move
the object up the ramp is less than the force needed to lift
it vertically.
• (k) The disadvantage is that the distance moved up the
ramp is correspondingly greater than the distance moved
vertically.
Extensions and Modifications:
• Some students may be interested in determining the
efficiency of other machines, such as a pulley, a pulley
system, a lever, or a wheel-and-axle. Refer to Applying
Inquiry Skills question 6, page 147.
Setup:
Students should read the case study and discuss it with other
members of their group or the class. They should then
answer the Practice questions on page 146.
Assessment:
• Students can be assessed on their research and
communication skills.
Student Preparation
• Students should have a basic understanding of the topics
presented earlier in this chapter.
DURING
• Students can find out more about the topics presented on
the Internet (refer to the Related Background Resources).
AFTER
• After students have answered Practice Questions 6–11 on
page 146, you can discuss their answers in class.
CASE STUDY
Physics and Sports Activities
• This case study explores some of the ways in which the
study of physics is applied to sports activities, with a
particular focus on pole vaulting.
Extensions and Modifications:
• Students interested in a different sport could focus on that
sport rather than on pole vaulting. For example, the
Internet has a lot of information on tennis, golf, and
baseball.
BEFORE
Teacher Preparation
Time: 30 minutes
104 Chapter 4 Energy, Work, Heat, and Power
© 2002 Nelson Thomson Learning