Download Adobe Photoshop PDF - Perimeter Institute

About the Resource
About the ANimatioNs
Welcome to Perimeter Institute’s Revolutions in Science,
The three 60-second animations serve to hook your
Alice & Bob in Wonderland animations.
far more fascinating than they may have realized. a classroom resource based on three serious-but-fun
students’ interest—to show them that the everyday world is
Challenge and inspire your students with the wonder and
Along with the two characters, Alice & Bob, students will
Wonderland” world where things are not always what they
profound shifts in our understanding of reality.
mystery of our universe—we really do live in an “Alice in
seem to be:
• Gravity is not a force pulling down; it is the ground
accelerating up in “curved spacetime.”
• Atoms cannot exist in a commonsense universe; they
require a strange “quantum” reality. • Energy has inertia; our energy to be alive comes from
literally eating the mass of the Sun!
Engage your students in the powerful, but surprisingly
accessible, creative and critical thinking processes that led
to three of the most profound revolutions in science. The resource focuses not only on basic scientific literacy—
what these enduring understanding are—but more importantly
how they were discovered. Using these discoveries as
discover that the simplest questions can lead to the most
Alice is a delightfully precocious little girl, brimming with
curiosity. Each episode opens with Alice wondering about
something that seems so obvious it sounds silly, such as,
“What keeps us stuck to the Earth?”
Bob is Alice’s older brother who feels it is his duty to
‘educate’ his sister. Without thinking, he blurts out the
commonsense answer to her ‘foolish’ questions.
Alice gives us reason to question the commonsense answer. Together, our characters use their imaginations and simple
reasoning to arrive at amazing insights into the universe. Your students are sure to enjoy their mind-warping
adventures with Alice & Bob in Wonderland!
exemplars of the power of inquiry, students can experience
for themselves how new scientific knowledge is created.
About the DVD: The accompanying menu-driven DVD contains the plug-and-play Alice & Bob in Wonderland
animations, as well as the following files, which can be accessed by closing the menu software and using your computer’s
file browser: this Teachers’ Guide in PDF format and the animations in various file formats. View the animations now!
The Student Worksheets and Assessments in editable DOC format can be found at www.perimeterinstitute.ca
Curriculum CoNNectioNs
2
What KeepS Us Stuck to the Earth?
3
Introduction
Teacher Demonstrations
Student Worksheets: SW1: Scientific Models: Gravity
SW2: Scientific Revolution: General Relativity
Student Assessments: SA1: Scientific Models: Gravity
SA2: Scientific Revolution: General Relativity
Answers
How CaN Atoms Exist?
17
Introduction
Student Worksheets:
SW1: Scientific Models: The Atom
SW2: Scientific Revolution: Quantum Mechanics
Student Assessments: SA1: Scientific Models: The Atom
SA2: Scientific Revolution: Quantum Mechanics
SA3: Applications of Quantum Mechanics
Answers
Where Does ENergy Come From?
Introduction
Student Worksheets:
Student Assessment:
Answers
CredIts
4
5-6
7- 8
9-11
12
13
14 -16
18 -19
20 -21
22-23
24
25
26
27- 28
29
SW1: Scientific Models: Time
SW2: Scientific Revolution: Special Relativity
SA: Scientific Revolution: Special Relativity
30 -31
32 - 34
35-36
37
38 - 40
41
Curriculum coNNectioNs
2
Topic
Connection to Resource
Module
Nature of Science
Science involves both creative and critical thinking, leading to new and sometimes
revolutionary ways of understanding nature. Educated guesswork and intuitive leaps
can lead the scientific imagination to very strange ideas, but as long as these ideas fit
the experimental evidence they must be taken seriously. The ultimate judge of a theory
is how well it matches the observations, not how well it matches our commonsense.
1, 2, 3
Process of Scientific
Modeling
We build scientific models to explain complex phenomena. Good models must be
logically self-consistent, explain the observations accurately, make testable predictions
of new observations, and give new insights into the phenomena.
1, 2, 3
Force and
Acceleration
Newton s second law of motion dictates that acceleration is the result of a net force. In
Newton s model, gravity is a force causing acceleration; in Einstein s model, gravity is
not a force so objects in freefall are not accelerating.
1
Weight
For Newton, weight is the force of gravity pulling down on you. For Einstein, there is no
force of gravity; weight is the magnitude of the force needed to accelerate you up along
with the accelerating ground.
1
Gravity
Students challenge the underlying assumption of Newton s mysterious “force of
gravity,” which has no known cause, and replace it with an alternative explanation for
gravity using Einstein s curved spacetime.
1
Frames of Reference
All observations and measurements are made relative to a frame of reference. If that
frame is moving with constant velocity, there is no experiment that can be done to show
that it is moving. If the frame is accelerating, the law of inertia seems to be violated so
we invent forces to reconcile our experiences.
Bohr-Rutherford
Model of the Atom
The Bohr-Rutherford model of the atom is an obsolete scientific model. The idea of
electrons orbiting around the nucleus is examined and shown to fail due to simple,
classical concepts that are within the students grasp.
2
Quantum Mechanical
Model of the Atom
The quantum mechanical model of the atom uses waves to describe the behaviour of
particles. Electrons can behave as if they are in many places at the same time, solving
the problems encountered by the classical (and Bohr-Rutherford) models.
2
Wave-Particle Duality
The electron is a point-like particle that behaves like a wave. This allows the electron to
act as if it is in many places, or traveling in many directions, at the same time.
2
Electromagnetic
Fields
The electron is charged so it is surrounded by an electric field. Accelerating electrons
have changing electric fields so they emit electromagnetic waves.
2
Relative Motion
Two observers watching the same event might have very different descriptions of the
event if they are moving relative to each other. There is no preferred frame of reference
in the universe so all motion is relative.
3
Time Dilation
The Newtonian concept of absolute time is wrong. Two observers moving relative to
each other will measure the other s time passing at a different rate—moving clocks
run slow.
.
3
Length Contraction
Two observers moving relative to each other will measure the other s space to be
contracted in the direction of motion—moving objects occupy less space.
3
Energy
Energy is not just “the ability to do work.” Closer inspection of energy leads to the
surprising result that all forms of energy have inertia—heating a cup of coffee increases
its resistance to acceleration.
3
Inertia
Inertia is not just “the ability to resist acceleration.” The inertia of even an object at rest
represents the presence of energy, as described by E=mc2.
3
1, 3
Introduction
4
Teacher Demonstrations
5-6
Student Worksheets:
SW1: Scientific Models: SW2: Scientific Revolution:
7- 8
Gravity
9-11
General Relativity
Student Assessments:
SA1: Scientific Models: 12
SA2: Scientific Revolution:
13
Gravity
General Relativity
Answers
14-16
This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: What Keeps Us Stuck to
the Earth? In this episode Alice and Bob ask questions about the nature of gravity and realize that there is a deep connection
between gravity and acceleration. Lesson 1 is an introductory level lesson (no prior knowledge of physics is required) that
guides students through a critical thinking activity to connect acceleration and gravity. Lesson 2 is a more advanced lesson
(prior knowledge of dynamics is an asset) that builds on concepts developed in Lesson 1 to show that the effects of gravity are
actually caused by curved spacetime.
LessoN 1:
LessoN 2:
SCIENTIFIC MODELS: GRAVITY
SCIENTIFIC REVOLUTION: GENERAL RELATIVITY
Use D1: Black Box to engage the students in the
Use D3: Toy and Bungee Cord to highlight the
Follow with D2: Sagging Rod to explore the force model
>> Show the Alice & Bob in Wonderland animation:
Distribute SW1: Scientific Models: Gravity after D2.
Distribute SW2: Scientific Revolution: General
creative process of building and evaluating models.
of gravity and introduce the acceleration model.
This worksheet walks the students through
an exercise in critical thinking about gravity
and acceleration.
>> Show the Alice & Bob in Wonderland animation:
What Keeps Us Stuck to the Earth?
SA1: Scientific Models: Gravity. This worksheet
includes additional questions to be done in class
differences between the two models.
What Keeps Us Stuck to the Earth?
Relativity. This worksheet guides the students into
a discovery of curved spacetime. In Part B, they
will use masking tape and beach balls to model
curved spacetime.
SA2: Scientific Revolution: General Relativity. This
worksheet includes additional questions to be done
in class or for homework.
or for homework.
3
WHAT KEEPS US STUCK TO THE EARTH?
Science is a process of building models to explain
wrong with Newton’s model—Mercury did not orbit the Sun
careful thought and experimentation. Good models explain
this discrepancy within the context of Newton’s model, but
observations and then refining those models through
existing observations and make testable predictions. This
Perimeter Institute classroom resource engages students
in this process by exploring models of a common real world
phenomenon—gravity. Students will exercise their critical
and creative thinking skills to demonstrate why Einstein’s
model of gravity is better than Newton’s.
Our everyday experiences of gravity suggest that the
Earth exerts an attractive force on nearby objects. Newton
successfully extended this force model of gravity to the Moon,
Sun and planets. Nevertheless, the force model of gravity
deeply troubled Newton because it did not explain the cause
of the force. Moreover, in the 1850’s, a more careful look at
existing observations suggested that something might be
quite as predicted. Scientists tried various ways to explain
all attempts failed. Newton’s model of gravity had reached
its limit.
Newton’s force model of gravity also troubled Albert Einstein.
In his “happiest thought,” Einstein realized that when you are
in freefall you do not feel your own weight, like an astronaut
floating weightlessly in deep space. However, when an
astronaut’s rocket accelerates, she feels as if there is a
force pulling her down toward the floor, like weight. In reality,
what the astronaut feels is the floor pushing up on her,
accelerating her up. Could gravity be like this? Could it be
that there is no force pulling us down, but instead the ground
is accelerating up? Yes! Einstein showed how curving
spacetime can make it possible for the ground to be forever
accelerating up without the Earth expanding faster and
faster! Students explore this idea through a simple, concrete
activity involving just tape and a ball.
ur theory is,
r how beautiful yo
“It doesn’t matte
e. If it doesn’t
how smart you ar
it doesn’t matter
ent, it’s wrong.”
agree with experim
MAN
– RICHARD FEYN
testable predictions that distinguish it from Newton’s force
model. Einstein’s model predicts that time passes more
slowly at the surface of a planet compared to farther away.
This effect has been precisely measured and is evident daily
in the Global Positioning System (GPS). Einstein’s model
other, at a
may act upon an
“…that one body
the mediation
vacuum, without
distance through
eir action and
by and through th
of anything else,
another, is to
eyed from one to
force may be conv
ve no man
surdity, that I belie
me so great an ab
a competent
sophical matters
who has in philo
o it.”
, can ever fall int
faculty of thinking
N
– ISAAC NEWTO
nt office in Bern,
chair at the pate
“I was sitting in a
rred to me:
en a thought occu
when all of a sudd
el his own
ely, he will not fe
If a person falls fre
thought made
rtled. This simple
sta
s
wa
I
t.
igh
we
lled me toward a
n on me. It impe
a deep impressio
tion.”
theory of gravita
t)
ough
TEIN (Happiest Th
– ALBERT EINS
Einstein’s curved spacetime model of gravity makes several
also correctly predicts the bending of light as it passes by a
massive object, such as a star. Such gravitational lensing
has become a powerful tool in astronomy. Einstein’s model
also provides a very accurate description of the orbits of all
the planets, including Mercury.
Einstein’s model of gravity has passed every experimental
test to date. These same tests have conclusively ruled out
Newton’s model. The old idea of gravity as a force may feel
right but it is wrong. The “force of gravity” is an inference, not
an observation. We observe the ground compressing under
our feet. We infer that gravity is a force pulling us down. In
reality, the ground is accelerating up in curved spacetime,
pushing up on us, forcing us to accelerate along with it.
Dropped objects don’t accelerate down: it is the ground that
accelerates up in curved spacetime. These statements may
strike us as odd, but they agree with experimental data.
Gravity is not a force. Our everyday experiences of gravity
are actually the effects of the ground accelerating up through
curved spacetime. Gravity is curved spacetime.
4
Teacher DemoNstratioNs
D1 - BLACK BOX: (see building instructions below)
1. Pull the top cords back and forth. Invite students to guess
how they are connected inside. Now pull one of the
bottom cords. Continue pulling different combinations
of cords while drawing students into the mystery.
2. Ask students to draw a picture of what they imagine is
inside the box. Encourage creative thinking!
3. Have students share their ideas on the board. Engage
the class in a discussion about the various models that are on the board. Verify that the models correctly explain
the observations. Highlight the following points:
• The same set of observations can generate
different models.
• All models that explain the observations are
equally valid.
• Models that fail to explain one or more observations
are wrong, or need revision.
4. Ask the students if the models on the board predict any
new observations that may help distinguish between
them. For example, shake the black box to see if it rattles.
Return to the models on the board and re-evaluate them,
emphasizing the role of testable predictions in
the process of developing robust scientific models.
Note: Never divulge what is inside the Black Box.
In science, we only ever have access to indirect
observations—we never “see reality” directly!
BUILDING YOUR BLACK BOX
Materials: (all dimensions are approximate)
• 2 pieces of 8 mm nylon rope, each 70 cm long
• 1 harness ring with a 4 cm diameter
• 35 cm long piece of drainage pipe (7.5 cm diameter)
• 2 drainage pipe end caps (7.5 cm diameter)
Tools:
• power drill with 3/8” drill bit
Procedure:
1. Drill the top holes directly across from one another, each 5 cm from the top. Repeat for
the bottom holes, each 5 cm from the bottom (see top Figure).
2. Thread one rope through the top holes and the harness ring (see middle Figure).
3. Tie a knot 15 cm from each end of the rope.
4. Thread the other rope through the bottom holes. Again, ensure that the rope passes
through the harness ring as indicated (see bottom Figure). Tie a knot 15 cm from each
end of the rope.
5. Secure the end caps.
Note: Variations on the design (without a ring for example) will enrich the discussion and
work equally well. You may also wish to encourage students to build their own versions of
the device with bathroom tissue tubes and string.
5
D2 - Sagging Rod: (a very flexible 2 m long rod with two
small masses on each end)
1. Hold the rod horizontally with your hand in the middle so
the rod sags. Ask students to explain why the rod is sagging–typically students will say “force of gravity!”
2. Place the rod on a table. Have two students apply
horizontal forces on the ends while you hold the middle
in place by applying an opposing horizontal force. The
class observes the same shape as in step #1. Reinforce
the concept that when opposing forces are applied to
the rod it will bend.
3. Emphasize the distinction between Observation (when
opposing forces are applied to the rod it bends) and Inference (the sagging rod is bent so there must be
opposing forces; a “force of gravity” opposes your hand).
4. Have students suggest ways to make the rod bend
without using opposing forces. Hold the rod vertically and accelerate it to the side. The ends of the rod will lag
behind the middle because of inertia. Emphasize that
your hand is applying a force but there is no opposing
“force of gravity.”
5. Distribute SW1: Scientific Models. Show the animation:
What Keeps Us Stuck to the Earth? after students have worked in small groups to complete the table and
discussion sections of the worksheet.
D3 - Toy, Bungee Cord and Board:
1. Show the animation: What Keeps Us Stuck to
the Earth?
2. Demonstrate Newton’s model of gravity by stretching
the bungee cord over the toy (see Figure). “According to
Newton gravity is a force, like an invisible bungee cord,
that pulls objects to the ground.” Pull the toy away from
the board and let it ‘snap’ back down. The bungee cord
exerts a force on the toy making it accelerate.
3. Demonstrate Einstein’s model of gravity by removing
the bungee cord, holding the toy in the air and
accelerating the board up to hit it. “According to Einstein,
gravity is not a force. The toy does not accelerate down;
rather, the ground accelerates up!” Place the toy on the
board and accelerate it up. Ask students to imagine that
they are in deep space (no gravity); what would it feel
like to stand on an accelerating board?
4. Distribute SW2: Scientific Revolution. Students work in
small groups to complete the worksheet.
D4 - Curved Spacetime Exemplar:
1. In SW2, the students will use masking tape and a beach
ball to model curved spacetime. Read through the
activity and make an exemplar on a large exercise ball,
if possible (see Figure).
2. The tape describing Alice’s path through spacetime
must lie flat. She is experiencing no “force of gravity”
and no acceleration so she must follow a straight path.
3. The tape describing Bob’s path must be crinkled. He is
experiencing the ground pushing up on him, accelerating
him up, and so he must follow a curved path.
4. Time dilation is demonstrated by comparing a length of
tape connecting the tops of the ladders with the length
of tape connecting the bottoms (Bob’s path).
6
Note: The time dilation demonstrated by this beach ball
analogy is actually reversed to the real time dilation–
analogies have limits.
SW1: Scientific Models: Gravity
Scientists use models to try to explain the observations they make. In this activity you are going to use two different models to
explain the same observations of an everyday phenomenon—gravity.
Force Model: You are standing in a room that is on the Earth; the Earth exerts a downward force on objects inside the room.
Explain the following phenomena using this downward force. Follow the sagging rod example.
Explain the
Sagging Rod
Explain Weight
Explain Freefall
(use words and arrows)
(use words and arrows)
- the Earth pulls
down on the rod and
your hand pushes up
- the rod bends because your hand is
only in the middle
- the rod does not accelerate because
the two opposing forces are balanced
Acceleration Model: You are standing in a room that is inside a rocket; the rocket is accelerating “upwards” in deep space.
Explain the following phenomena using this upward acceleration. Follow the sagging rod example.
Explain the
Sagging Rod
Explain Weight
Explain Freefall
(use words and arrows)
(use words and arrows)
Force Model
Acceleration Model
- the room is accelerating up; so are
you and the rod
- the rod accelerates up because there is
now only one force—your hand pushing up
- the rod bends because the ends
have mass, which resist acceleration
(inertia)
SUMMARIZE:
What is the “big idea” behind each
model? How does each explain
effects we call “gravity”?
7
Discussion:
1. Examine both of your explanations for freefall.
(a) What do you actually observe about an object in freefall?
(b) What can you infer about the nature of gravity from your observations of freefall?
2. A flexible rod bends when opposing forces act on it. The same rod bends when suspended horizontally from the middle.
Does this prove that gravity is a force? Explain.
3. A friend shows you a video on the Internet of a guy who can make objects “float” in the air. You know this is impossible—
how might you explain the video?
4. You wake up in a closed room with no windows, with no idea how you got there. Describe an experiment you could do to
determine if the room is on the Earth or inside a rocket accelerating in deep space.
>> Watch the animation: What Keeps Us Stuck to the Earth?
Thinking Deeper:
1. Both the force model and the acceleration model make claims that are hard to accept. What are they?
2. Both models of gravity explain everyday observations equally well. However, Newton’s force model fails to correctly
describe the orbit of Mercury, so it ultimately fails the test for a valid scientific model. Inspired by the acceleration model,
Einstein developed an alternative model of gravity. His curved spacetime model made several successful predictions that
have conclusively ruled out Newton’s model. Does this mean we should throw out Newton’s model? Does a model have
to be correct in order to be useful?
8
SW2: Scientific Revolution: General Relativity
Scientific models must make predictions that match our observations, or they must be revised or replaced. New scientific
models can be revolutionary. In this activity you are going to examine two models of gravity: Newton’s classical force model,
and Einstein’s revolutionary curved spacetime model.
Part A: Modeling Gravity
Complete this table after watching >> Alice & Bob in Wonderland: What Keeps Us Stuck to the Earth?
Force Model
Acceleration Model
Gravity: How
does it work?
What’s hard
to accept?
Alice steps off
In the boxes, sketch snapshots of Alice as she falls to the ground and Bob as he stands at the bottom of
tall ladder
Connect-the-dots of Alice’s position in SPACE
the top of a
Bob stands at
the bottom of
the ladder
the ladder, showing their progression in time. [Hint: Alice moves faster and faster as she falls.]
(height above the ground) as TIME goes on.
Is her path through spacetime straight or curved?
Connect-the-dots of Bob’s position in SPACE
(height above the ground) as TIME goes on.
Is Bob’s path through spacetime straight or curved?
According to
Newton...
Alice’s path through spacetime is ______________ because she is accelerating. She is accelerating
(straight/curved)
because gravity is a force pulling on her. Bob’s path through spacetime is ______________ because he is
(straight/curved)
not accelerating­—the force of gravity is balanced by the ground pushing up.
According to
Einstein...
There is no “force of gravity” pulling down on Alice so she _________ accelerating. Her path through
(is/is not)
spacetime should be ______________ . The ground pushes up on Bob and since there is no opposing
(straight/curved)
“force of gravity” to balance this force, he should accelerate up and follow a _______________ path
through spacetime.
(straight/curved)
Discussion:
1. Alice has a video camera in her hands as she falls. If she takes a video of herself as she falls, could she tell that she was
accelerating by viewing the video? (Ignore the background.)
2. Alice takes a video of Bob as she falls. Could she tell who was accelerating by viewing the video? (Ignore the background.)
3. Alice closes her eyes as she falls. What does she feel? Can she tell that she is accelerating?
4. Bob closes his eyes. What does he feel? Can he interpret this feeling as accelerating up?
9
Einstein knew that Newton’s model of gravity is wrong. For one thing, it fails to correctly predict the orbit of Mercury; for
another, it fails to obey the speed limit of the universe—the speed of light. In his search for a better model, the simple fact that
acceleration up mimics force down was too strong of a coincidence to ignore. Einstein needed to find a way to make sense of
the ground accelerating up without moving up. How can the ground be accelerating up when the Earth is not expanding? He
found the answer in the geometry of spacetime.
Part B: Bending Spacetime
In Part A, we used the fact that accelerating objects trace out curved paths in spacetime and non-accelerating objects trace
out straight paths. We also saw that Newton and Einstein would disagree on who is accelerating and who is not. In this part
of the activity you will use tape to transfer the spacetime diagram from Part A onto the surface of a large ball to reveal how
curving spacetime resolves the problem of who is accelerating.
1. Use a strip of tape to connect two points on your desk with a straight line. Use another strip of
tape to make a curved line. Compare the two pieces of tape. Which strip of tape lies flat on the
desk and which is crinkled?
2. Build your spacetime diagram on the surface of a large ball. Start with the space and time axes.
• The space axis is a strip of tape that runs vertically along a line of longitude.
• The time axis runs horizontally along a circle of latitude (about 15˚ above the equator).
3. Add three identical strips of tape to represent the ladder in three consecutive snapshots. The
ladders must follow lines of longitude on the surface, starting about 2 cm above the time axis and
ending about 10 cm from the top.
4. Alice’s path is a strip of tape that connects the top of the first ladder with the bottom of the last
ladder. Can you make it a straight line? Why would you want to?
5. Bob’s path runs parallel to the time axis along a circle of latitude. It will connect the bottoms of the
three ladders. Does the tape lie flat or is it crinkled? What does this indicate?
Curved
Spacetime:
When we transfer the spacetime diagram to the ball we find that the tape for Alice’s path can be
______________ , which means the line is ________________ so Alice is _____________________
(flat/crinkled)
(straight/curved)
(accelerating/not accelerating)
through curved spacetime. The tape describing Bob’s path is ______________________, which
(flat/crinkled)
means the line is _________________ so Bob is ____________________ through curved spacetime.
(straight/curved)
(accelerating/not accelerating)
Drawing the spacetime diagram on a curved surface reverses who is accelerating and who is not—
just what Einstein needed to make the acceleration model make sense. The ground can be forever
accelerating up without moving up! Gravity is not a force—it is curved spacetime.
6. The time elapsed for Bob at the bottom of the ladder is the length of his path (i.e. distance in the time direction). If Alice
stayed at the top of the ladder, would her elapsed time be the same? Einstein’s model predicts time dilation: time passes
at different rates depending on height about the ground, which has been verified by atomic clocks. Newton’s model makes
no such claim. Models cannot be proven right—but they can be proven wrong and time dilation proves that Newton’s
model of gravity is wrong!
Evaluating
Models:
Newton’s model fails to predict the orbit of Mercury accurately. Einstein’s model does and it also
accurately predicts time dilation and the bending of light. We must conclude that the best model of
gravity is __________________ ______________________ model.
(Newton’s/Einstein’s)
10
(force/curved spacetime)
By curving spacetime, Alice’s path changes from curved to straight—she experiences no “force of gravity” and no acceleration.
By curving spacetime, Bob’s path changes from straight to curved—he experiences the ground pushing up on him, continually
accelerating him up, but without him moving up. Einstein was able to show that gravity is not a mysterious, invisible force—it
is the curvature of spacetime. This curved spacetime model asserts that you feel heavy because the surface of the Earth is
forever accelerating up without actually moving up.
Part C: Accelerating Up without Moving Up
Consider the type of motion (accelerating or not) in each of the following scenarios:
In Deep Space
Rocket 1: Floating in deep
space, engines off
Rocket 2: Accelerating “up” in
deep space, engines on
Near the Ground
Rocket 3: In freefall near the
ground, engines off
Rocket 4: Hovering near the
ground, engines on
1. In Rocket 1, the astronaut knows she is not accelerating; the rod is straight and she is floating. In which other rocket does
she make these observations?
2. In Rocket 2, the astronaut knows he is accelerating; the rod is bent and he feels the force of the floor pushing up on him.
In which other rocket does he make these observations?
3. The astronaut in Rocket 3 uncovers the window and looks out. She can see the ground and Rocket 4.
(a) What was her type of motion before looking out the window? (Accelerating or not accelerating)
(b) How would she describe her motion when she looks out the window?
(c) Combine your answers from (a) and (b) into a statement.
4. The astronaut in Rocket 4 uncovers the window and looks out. He can see the ground and Rocket 3.
(a) What was his type of motion before looking out the window? (Accelerating or not accelerating)
(b) How would he describe his motion when he looks out the window?
(c) Combine your answers from (a) and (b) into a statement.
We have discovered that astronauts in very different scenarios can experience the same type of motion. This insight is called
Einstein’s Equivalence Principle: Freefalling in a uniform gravitational field (Rocket 3) is physically identical to floating in deep
space (Rocket 1). Hovering in a uniform gravitational field (Rocket 4) is physically identical to constant acceleration in deep
space (Rocket 2). The mass of the Earth curves spacetime so that objects in freefall appear to accelerate down, but there is no
force causing this “acceleration”. It is the same kind of “acceleration” you feel when a car accelerates towards you. You are not
accelerating—the car is!
11
SA1: Scientific Models: Gravity
1. Observation: using your senses to gather information from your environment.
Inference: using logic to interpret the information gathered from your environment.
Identify the observations and inferences in the following narrative.
Bob wakes up and looks out the window. There are drops of water on the window. “It must have rained last night,” he
thinks. He goes downstairs and notices that the ladder is leaning against the house, so he goes outside to help his dad
with the roof repair work. “Hey Alice, what are you doing up there?” shouts Bob. Alice is so startled that she loses her
grip on the ladder. As she falls to the ground, she sees Bob getting closer and closer. “The force of gravity is making me
accelerate down at 9.8 m/s2,” yells Alice. Bob reaches out and catches her just before she hits the ground. “Good thing I
was accelerating up at 9.8 m/s2 so I could rescue you,” says Bob. Alice gives Bob a quizzical look and then she tells him
about how she was washing the windows when he made her fall.
OBSERVATIONS INFERENCES
2. True or False? Rewrite any false statements to make them true.
(a) There can only be one model that explains a set of observations.
(b) We prove a model is right when we observe the predictions it makes.
(c) Models that do not make new predictions are wrong.
(d) A model is valid if it can explain the observations.
(e) Any model that cannot explain the observations is useless and should be discarded.
(f) We design experiments to prove that a given model is correct.
12
SA2:
Scientific Revolution: General Relativity
1. Alice and Bob are arguing over whether gravity is a force or curved spacetime. Bob says, “You honestly believe the
ground is accelerating up? That’s weird!” Alice replies, “Mysterious invisible force? Who’s weird now, Bob?” Which side of
the argument do you hold? How would you convince someone to agree with you?
2. According to Newton, gravity is an invisible, attractive force that acts between massive objects. If his model of gravity is
wrong does that mean his equation for universal gravitation is also wrong?
3. According to Einstein, gravity is the curvature of spacetime. If Einstein’s model of gravity is better, why do we still use
Newton’s model? When do we have to use Einstein’s model?
4. How is the “force of gravity” similar to centrifugal force? Explain.
5. Newton and Einstein are looking at a book sitting on a table. How would each of them describe the forces acting on that
book and how would they justify their description?
13
14
- weight is the sensation of
pushing up on an object to force
it to accelerate up, along with the
accelerating room
- the object appears to accelerate
down but it is actually the room (and
you) accelerating up
- the floor continues to accelerate up
and meets the object
- an object in freefall has no forces
acting on it so it does not accelerate
- the room is accelerating up
- the room is accelerating up; so are
you and the object
- the object accelerates up because
there is now only one force—your
hand pushing up
Explain Freefall (use words and arrows)
Explain Weight (use words and arrows)
Acceleration Model
- the object accelerates because
there is no opposing force
- the Earth pulls down on the object
- the Earth pulls down on the object
and your hand pushes up
- the object does not accelerate
because the two opposing forces are
balanced
Explain Freefall (use words and arrows)
Explain Weight (use words and arrows)
Force Model
SW1: Answers
effects of this acceleration.
2. Newton’s model is still very useful. It gives a simple intuitive picture of
gravity that works for almost all situations. A model does not have to
be correct in order to be useful—there are many models that are useful
in limited contexts that ultimately fail. We don’t need to use Einstein’s
curved spacetime model to calculate the trajectory of a baseball; Newton’s
model is adequate for this task. Einstein’s model is necessary only to
understand what is really happening to the baseball. Using Newton’s model
is analogous to saying that the Sun revolves around the Earth—it is still a
convenient way of thinking, even if it is grossly incorrect.
1. The force model claims that there is a mysterious, invisible force that
reaches out through space to influence mass but cannot explain the
physical nature or cause of this force. The acceleration model claims that
the ground is forever accelerating up without moving up.
Thinking Deeper:
4. There is no simple experiment that you could do to distinguish between the
two scenarios.
3. There are several ways to do this: either a force is applied that you are
not able to see (magnets or wires), or the room is in freefall with all objects
falling at the same rate as the room. Astronauts train for weightlessness by
falling inside a plane that is diving.
2. The fact that a suspended rod will bend exactly like a rod that has opposing
forces acting on it does NOT prove that gravity is a force. We can also
make the rod bend this way without an opposing force by accelerating it.
The “force of gravity” could be a fictitious force we invent to explain the
observations made in an accelerating frame of reference.
1. (a) You observe that the distance between the ground and the object
decreases at an accelerating rate, regardless of the object’s mass.
(b) You can infer either that gravity is a force that causes the object to
accelerate or that gravity is our frame of reference accelerating.
Discussion:
like a mysterious invisible hand.
accelerating. Weight and freefall are
Our frame of reference is somehow
The Earth somehow exerts an
attractive force on nearby objects,
Acceleration Model
Force Model
Summarize:
15
Einstein...
According to
Newton...
According to
ladder
the bottom of the
Bob stands at
top of a tall ladder
Alice steps off the
accept?
What’s hard to
it work?
Gravity: How does
The ground is accelerating
Gravity is a mysterious
spacetime.
should accelerate up and follow a CURVED path through
is no opposing “force of gravity” to balance this force, he
STRAIGHT. The ground pushes up on Bob and since there
IS NOT accelerating. Her path through spacetime should be
There is no “force of gravity” pulling down on Alice so she
balanced by the ground pushing up.
because he is not accelerating—the force of gravity is
pulling on her. Bob’s path through spacetime is STRAIGHT
accelerating. She is accelerating because gravity is a force
up without moving up
Alice’s path through spacetime is CURVED because she is
invisible force
acceleration.
freefall are effects of this
that have mass.
accelerating. Weight and
emanates from objects
invisible force that
Acceleration Model
Our frame of reference is
Force Model
Gravity is a mysterious
Part A: Modeling Gravity
SW2: Answers
EINSTEIN’S CURVED SPACETIME model.
We must conclude that the best model of gravity is
ACCELERATING through curved spacetime.
which means the line is CURVED so Bob is
The tape describing Bob’s path is CRINKLED,
NOT ACCELERATING through curved spacetime.
which means the line is STRAIGHT so Alice is
we find that the tape for Alice’s path can be FLAT,
When we transfer the spacetime diagram to the ball
4. (a) The astronaut in Rocket 4 is accelerating.
(b) He sees that he is not moving relative to the ground.
(c) Objects NOT MOVING relative to the ground are ACCELERATING.
3. (a) The astronaut in Rocket 3 is NOT accelerating.
(b) She sees that she is in FREEFALL.
(c) Objects in FREEFALL are NOT accelerating.
2. The astronaut also feels a force and the rod is bent in Rocket 4.
1. The astronaut is also floating and the rod is straight in Rocket 3.
Part C: Accelerating Up without Moving Up
Evaluating Models:
Curved Spacetime:
1. The STRAIGHT TAPE is FLAT and the CURVED TAPE is CRINKLED.
Part B: Bending Spacetime
4. Bob will feel acceleration. He could just as well be inside a rocket
accelerating “up” in deep space.
3. Alice will not feel acceleration. She could just as well be floating
weightlessly in space, with a breeze blowing over her face. She cannot tell
that she is accelerating until she refers to something in a different frame of
reference (e.g. the ground) and even then she can only tell that something
is accelerating—not necessarily her.
2. Alice will be able to tell that one of them is accelerating but she can’t tell
which one.
1. If Alice ignores the background she cannot tell that she was accelerating.
Discussion:
16
- Bob is accelerating up
- the force of gravity pulls her down
- distance between Alice and
Bob decreases at 9.8 m/s2
- dad is fixing the roof
- “It must have rained”
INFERENCES
- notices the ladder
- drops of water on window
OBSERVATIONS Answers
1. Force is intuitively obvious. A falling object accelerates down, so there must be
a force pulling it down. The idea that the ground is accelerating up when the
Earth is not expanding just sounds absurd!
SA2:
FALSE: We design experiments to prove that a given model is wrong.
(f) We design experiments to prove that a given model is correct.
still be useful in a limited context.
discarded. FALSE: Models that do not explain all the observations can
(e) Any model that cannot explain the observations is useless and should be
(d) A model is valid if it can explain the observations. TRUE
explain the existing data to be valid.
FALSE: Good models make new predictions, but a model only needs to
(c) Models that do not make new predictions are wrong.
predictions it makes.
FALSE: We prove a model is wrong when we don’t observe the
(b) We prove a model is right when we observe the predictions it makes.
FALSE: There can be several models that explain a set of observations.
(a) There can only be one model that explains a set of observations.
2. True or False? Rewrite any false statements to make them true.
1. SA1: Answers
Einstein would agree that the book is “at rest”, but “at rest” relative to what?
The ground, which is accelerating up in curved spacetime. The only force acting
on the book is the table pushing up and since there is no “force of gravity”
opposing the force of the table, the book must accelerate up,
along with the accelerating ground. Ultimately, Einstein would justify his
description by appeal to experiments—time dilation has been observed
using atomic clocks.
5. Newton would say that the book is “at rest” and therefore not accelerating so
the forces acting on it are balanced. The compression of the table gives clear
evidence that the table is pushing up on the book so a downward force of
gravity is needed to balance the forces.
4. Centrifugal force is a fictitious force invoked when objects in a non-inertial
frame of reference experience inertia. For example, when a car turns a corner it
accelerates but the objects in the car want to keep going straight ahead so they
feel a “force” pushing them against the motion of the car. Similarly, the “force of
gravity” is a fictitious force created to explain inertial behaviour in a non-inertial
frame of reference. The Earth curves spacetime in such a way that the ground
is a non-inertial frame of reference. Falling objects seem to accelerate towards
the ground, but there is no force causing this “acceleration”; so we invent one—
the “force of gravity.” Both forces describe real effects caused by acceleration;
neither one describes an actual force.
3. We still use Newton’s model because it is intuitively simple and the math is
straightforward. We must use Einstein’s model when accuracy is very important
(e.g. space probes and GPS), where Newton’s model breaks down completely
(e.g. black holes and neutron stars), or when we are trying to get a clearer
picture for how the universe works.
2. Newton’s equation for universal gravitation makes reasonably accurate
predictions for the effects of weak gravity (e.g. the effects of the Sun on the
orbits of the planets), but gives grossly wrong predictions for the effects of very
strong gravity (e.g. near a black hole). The equation is also wrong in the sense
that it refers to a force, and gravity is not a force.
Acceleration is simpler. Objects “fall” because they have inertia. The frame of
reference is accelerating so it looks like objects fall but they don’t. They look
like they are accelerating in our frame but in spacetime they are actually not
accelerating. Experiments have confirmed the predicted curvature of spacetime,
which conclusively rules out the force model.
Introduction
18-19
Student Worksheets:
SW1: Scientific Models: 20- 21
SW2: Scientific Revolution: 22-23
The Atom
Quantum Mechanics
Student Assessments:
SA1: Scientific Models: 24
SA2: Scientific Revolution: 25
SA3: Applications of 26
The Atom
Quantum Mechanics
Quantum Mechanics
Answers
27- 28
This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: How Can Atoms Exist? In
this episode Alice and Bob ask questions about the structure of the atom and discover that the commonly accepted planetary
model of the atom (including the Bohr-Rutherford model) cannot possibly exist. Lesson 1 is an introductory level lesson (no
prior knowledge of physics is required) that explores why the planetary model fails. Lesson 2 is a more advanced lesson (prior
knowledge of waves is an asset) that extends the concepts developed in Lesson 1 to build the quantum mechanical model
of the atom—a model that explains how atoms can exist. An additional student activity sheet (SA3) is included that could be
combined with either lesson to address applications and implications of scientific discoveries.
LessoN 1:
LessoN 2:
SCIENTIFIC MODELS: THE ATOM
SCIENTIFIC REVOLUTION: QUANTUM MECHANICS
>> Show the Alice & Bob animation:
>> Show the Alice & Bob animation:
Distribute SW1: Scientific Models: The Atom.
Distribute SW2: Scientific Revolution: Quantum
How Can Atoms Exist?
This worksheet walks students through a critical
examination of atomic models using existing
knowledge and a computer simulation to reveal
the problems with classical models of the atom.
SA1: Scientific Models: The Atom. This worksheet
includes additional questions to be done in class or
for homework.
How Can Atoms Exist?
Mechanics. This worksheet engages the students in
the creative process of building a quantum model of
the atom. Students will use a computer simulation to
assist in visualizing the atom.
SA2: Scientific Revolution: Quantum Mechanics. This
worksheet includes additional questions to be done
in class or for homework.
17
How CaN Atoms Exist?
Science is a process of building models to explain
from? Another possibility would be to invent a new force
thought and experimentation. This Perimeter Institute
observed. In science, we exhaust all existing possibilities
observations and then refining those models through careful
classroom resource engages students in this process as
they explore models of the atom. Atoms are the building
blocks of matter. They are central to our existence; and yet,
there is no “commonsense” way to understand how they can
exist. The best commonsense atom we can imagine—the
one with electrons orbiting the nucleus, like planets orbiting
the Sun—would almost instantly self-destruct. Students
will exercise their critical and creative thinking skills as they
examine how various commonsense models fail and how the
very non-commonsensical quantum nature of our universe
makes atoms possible.
By the early 1900s, experiments had revealed that atoms
consist of particles much smaller than the atom itself: one
tiny, positively charged nucleus comprising almost all of
the atom’s mass, plus a number of even tinier, negatively
charged electrons. The challenge was to construct a working
model of the atom based on these particles and the forces
between them.
Electrons are attracted to the nucleus (since opposite
that acts inside the atom, but such a force has never been
before introducing a new type of matter or force.
If the atom cannot exist with static electrons, then the only
remaining possibility is a dynamic model where the electrons
are moving. In order for a moving electron to stay near the
nucleus, its trajectory must bend. The net attractive force
towards the nucleus—which defeated the static model—is
exactly the sort of force needed to bend an electron’s
trajectory into an orbit around the nucleus. But even in the
simplest case of a circular orbit, where the electron’s speed
is not changing, only its direction is continually changing—
the electron is accelerating. This is a problem. When a
charged object accelerates (changing its speed or direction),
it emits energy in the form of electromagnetic waves. For
instance, this is exactly how a cell phone works: electrons
in the antenna are accelerated, emitting radio waves. In the
atom, the accelerating electrons would emit electromagnetic
waves in the form of light. This light would carry energy away
from the atom, causing the electron to drop to lower energy
orbits, quickly spiraling into the nucleus.
charges attract) and repelled from each other (since like
So electrons can’t stand still (the static model fails); nor can
electrons don’t move is not stable; the attractive force always
result in all the atoms in your body collapsing in a blast of
charges repel). Any configuration of the atom in which the
wins and the electrons collapse into the nucleus. One way
to prevent this collapse would be to add “struts” that hold the
electrons in place, but we have never seen evidence of any
kind of support structure when we strip electrons off an atom.
And besides, what type of matter would the struts be made
they move (the dynamic model fails). Both models would
light energy on par with an atomic bomb. There is no way
to escape the catastrophic collapse of any commonsense
atom. This raises the question: if the electrons in an atom
can’t stand still, and can’t move, what could they possibly
be doing?
As a first step towards a working model, imagine spreading
an orbiting electron into a rotating ring. A perfectly smooth
knowledge
sm, all scientific
“If, in some catacly
e sentence
oyed, and only on
were to be destr
of creatures,
next generation
passed on to the
ost
uld contain the m
what statement wo
fewest words?
information in the
.. that all things
omic hypothesis.
at
e
th
is
it
ve
lie
I be
s...”
are made of atom
MAN
– RICHARD FEYN
rotating ring is moving but you cannot see any motion—it
appears to be static; this is what physicists call a stationary
state. A charged rotating ring is stationary so it does not emit
electromagnetic waves and would be a simple solution to
the energy loss problem. However, such a spreading out of
a particle is fraught with severe problems of its own. Each
part of the ring would be repelled from all the other parts
(since like charges repel), and there would be very strong
electrostatic forces tending to make the ring fly apart. We
would have to invent a new kind of matter or force to hold it
together. Also, whenever we “look” at an electron we always
18
“see” a point-like particle, with the full mass
Copernican revolution, with equally vast and far-reaching
of a spread out electron in the shape of a ring, or any
have allowed us to not only understand how atoms can
and full charge of one electron. We never see evidence
other shape.
Nature’s solution to the unstable atom problem is very
strange. An atomic electron does something very much in
the spirit of spreading itself out into a rotating ring (avoiding
consequences that go well beyond the atom. Quantum ideas
exist, and how they work; they also underlie a huge array
of technologies from cell phones and computers, to laser
surgery and the Internet, representing millions of jobs and
trillions of dollars of the world’s economy.
the energy loss problem), without literally spreading
out its matter (avoiding the other severe problems
mentioned above).
How can an electron spread out, and not spread out? In an
atom, an orbiting electron can be thought of as a particle, like
a very tiny baseball, but unlike a baseball, one that doesn’t
ly say that nobody
“I think I can safe
nics.”
quantum mecha
understands
MAN
– RICHARD FEYN
move along a definite trajectory. It exists in a profoundly
weird state in which, at any instant of time, it is not definitely
at any location in its orbit. Instead, it is only potentially at
each location in its orbit (all at the same time), with an equal
potential of being found at any particular location if we were
to “look” at the atom (e.g. shine light on it). The very act
of light hitting an electron somehow forces the electron to
“take a stand”—to assume a definite location. (How this
happens is still a mystery today.) This potential, or indefinite,
location is described by a fuzzy donut-shaped wave that
circulates around the nucleus. It’s not a physical wave, like
a sound wave or a water wave; nor is it the electron’s matter
physically spread out; instead, it’s a mathematical wave that
describes the probability of finding the electron (as a whole
point-particle) here or there if we were to “look.”
In short, the electron is a particle that behaves like a wave.
This weird blending of “particle” and “wave” properties into a
single entity is called quantum mechanics. At the foundations
of everything we currently know about matter and forces
is the discovery of the quantum nature of our universe.
The quantum nature of the atom is non-commonsensical.
An orbiting electron behaves like a wave, effectively allowing
it to be in many places and moving in different directions
at the same time! If you wiggle both ends of a Slinky
simultaneously, you will create two waves moving along
the Slinky in both directions at the same time, resulting in
a standing wave. In exactly the same way, we can have
two quantum waves circulating in opposite directions
around the nucleus. The resulting quantum standing wave
describes a single electron behaving as if it is orbiting both
clockwise and counter-clockwise at the same time! The
mathematics of these waves is well understood. What is not
well understood, and still the subject of much debate, is what
this mathematics implies about the ultimate nature of reality.
The quantum model results in a stable atom and has been
experimentally verified to unprecedented precision—it’s
decidedly strange, but it works.
This breakthrough was a 20th century equivalent to the
19
SW1: Scientific Models: The Atom
By the early 1900s, experiments had revealed that atoms consist of
particles much smaller than the atom itself: one tiny, positively charged
nucleus comprising almost all of the atom’s mass, plus a number of even
tinier, negatively charged electrons, such that the total electric charge is
zero. In this activity you will build and evaluate possible configurations of
these particles to try to produce a stable model of the atom.
Part A: Static Model
The Law of Static Electricity states that OPPOSITE charges ATTRACT and LIKE charges REPEL.
1. Hydrogen is the simplest atom. It has one negatively charged electron and a positively charged nucleus. What would
happen if you put the electron near the nucleus and “let go”?
2. How can Hydrogen exist as a stable atom if its electron and nucleus are attracted to each other? Can you think of a fix for
this problem?
Part B: The Planetary Model
If electrons in the atom cannot be standing still, then they must be
moving. Maybe the atom looks like a tiny solar system, with electrons
orbiting around the nucleus, like planets around the Sun. As you
consider this model, recall that objects that are moving will continue
moving on a straight path unless pushed or pulled to the side.
1. What has to happen to a moving electron to change its direction
of motion?
2. How might the positively charged nucleus of an atom bend the path of a moving electron?
3. A circular path, or orbit, is the simplest trajectory that an electron could follow. What would happen to the electron’s orbit if
we gradually removed energy from the atom?
20
Part C: The Failure of the Planetary Model
Any charged object is surrounded by an electric field. It is this field of the nucleus that exerts an attractive force on an electron
inside the atom. The electron, too, is surrounded by an electric field. Let’s use the PhET simulation (http://phet.colorado.edu/
en/simulation/radio-waves) to investigate what happens to that field when the electron accelerates (wiggles around).
1. Begin with the following settings: Manual, Full Field, Electric Field, Static Field. What happens to the electric field when
you wiggle the electron in the transmitting antenna?
2. Change the settings to: Manual, Full Field, Electric Field, Radiated Field. What happens when you wiggle the electron in
the transmitting antenna?
3. Change the settings to: Oscillate, Full Field, Electric Field, Radiated Field. Watch the electron in the receiving antenna.
Where does it get the energy to move?
4. An electron orbiting around the nucleus is accelerating just like the electron you wiggled in the antenna. (Imagine looking
at the atom from the side. As the electron orbits, it will appear to move up and down.) What would be emitted by the
electron as it orbits around the nucleus?
5. Whenever a charged object accelerates (changes its speed or direction of motion), it emits electromagnetic (EM) waves.
It takes energy to create these waves, and the waves carry this energy away. Why would this be a problem for the
Planetary Model of the atom?
Summary:
1. Electrons can’t stand still because:
“How wonderful that we have met a paradox.
Now we have some hope of making progress.”
– NIELS BOHR
2. Electrons can’t move because:
There is no way to escape the catastrophic failure of any commonsense atom. This raises the question: if the electrons in
an atom can’t stand still, and can’t move, what could they possibly be doing? The answer lies in Quantum Mechanics—a
completely new set of laws that describe how nature behaves at a deeper level.
21
SW2: Scientific Revolution: Quantum Mechanics
Any commonsense model of the atom is destined to fail. In static
models, the atom collapses due to the electrostatic force of
attraction the nucleus exerts on the electrons. Dynamic models,
like the planetary model, also fail because the atom loses energy
as the accelerating electrons emit EM waves, again collapsing
the atom. We need a model in which the electron is somehow
dynamic (orbiting) but at the same time static (not emitting EM
waves)—something physicists call a stationary model. For example,
a perfectly smooth spinning top is dynamic (rotating), but appears
to be static—you can’t tell that it’s spinning because nothing is
changing; it always looks the same.
Part A: The Rotating Ring
The electron cannot orbit around the nucleus as a point-like particle. What if we spread the mass and charge of the electron
out into a rotating ring?
1. A rotating ring of charge behaves like a current-carrying wire. Would the rotating ring emit EM waves? Why or why not?
2. Consider the electrostatic forces acting inside the ring. Would such a structure be stable? Why or why not? Would we be
able to observe it?
Part B: Standing Waves
The rotating ring idea is on the right track, but we have never
observed such rings. We always “see” electrons as point-like
particles. In preparation for Part C we will need to review some
facts about waves: (1) A wave can be in many places at the same
time, and (2) Two waves can exist simultaneously in the same
place.
1. Stretch a coiled spring (e.g. a Slinky) between two people, on
a smooth, horizontal surface (hard floor or table). Wiggle one
end of the spring at a constant rate. Where is the wave? What is the direction of the wave?
2. Wiggle both ends of the spring at the same rate. This creates two waves travelling in opposite directions along the spring,
existing simultaneously in the same place. Adjust the rate until you get a stable pattern. Notice that the combined wave
is not travelling in either direction. It is a wave—it oscillates side to side—but it is not travelling. This is called a standing
wave. What happens to the standing wave as you gradually increase the frequency of vibration? Can you create standing
waves at higher frequencies of vibration?
22
Part C: The Quantum Model
In the quantum model of the atom, the electron is a point-like particle
whose behaviour is described by a wave. If the wave is moving,
the electron is moving. Wherever the wave exists, the electron can
potentially exist. The weird thing is that the electron does not exist at
any definite location until its location is measured. Left undisturbed,
the electron behaves as if it is spread out like a wave, and
stationary states similar to the rotating ring become possible. Use
this simulation (http://www.falstad.com/qmatom/) to visualize these
waves. Note that these waves are mathematical—the electron’s
mass and charge are not physically spread out.
1. Start the simulation. In the top-right drop down menu select
“Complex Combos (n=1-4)”. Click on “Clear” then move your
mouse over the little circles in the bottom-left panel, noting the yellow text that appears just above the panel. Click on
the “n=2, l=1, m= –1” circle, which is the top circle in the second column. Finally, rotate the view by clicking on the z-axis
in the top right corner of the main panel and dragging it down until the z disappears at the origin and the y-axis points
straight up. This is a “top down” view of a single electron “orbiting” the nucleus of a Hydrogen atom. (The nucleus is at the
centre, but not shown.) What do you see?
2. The colours represent the “phase” of a donut-shaped wave circulating around the nucleus, showing that the wave
“crests” and “troughs” are moving. Observe that the moving electron is behaving as if it is in two places at once—actually
everywhere at once, wherever the wave is non-zero! Select the “View” drop down menu from the top menu bar and
deselect “Phase as Color.” You will now see a probability pattern: the probability, at any instant of time, of finding the
electron at various locations around the nucleus. In what way is the electron static? In what way is it dynamic? Do you
think the electron is emitting EM waves? Draw comparisons with the rotating ring in Part A.
3. Reselect “Phase as Color,” click on “Clear,” and then choose the “n=2, l=1, m=+1” circle. Note the direction of rotation of
this wave. Now click on the “n=2, l=1, m=–1” circle. You have just combined two waves circulating in opposite directions
around the nucleus to produce a standing wave. This standing wave describes an electron behaving as if it is moving
both clockwise and counterclockwise at the same time! Is the electron “moving”? Click on the x-y-z coordinate system
and rotate it to view this standing wave from different angles. Deselect “Phase as Color” to reveal the corresponding
probability pattern. In what way is the electron static? In what way is it dynamic? Do you think the electron is emitting EM
waves? Why or why not?
By describing the behaviour of a particle using a wave, anything a wave can do a particle can do. A wave can be in many
places at once, or be moving in different directions at once—so can a particle! This leads to very non-commonsensical
behaviour of electrons inside atoms, and yet these are the lengths scientists have gone to in order to construct a working
model of the atom—one that allows us to understand how atoms can exist in our universe.
23
SA1:
Scientific Models: The Atom
1. Why does the Hydrogen atom collapse if the electron isn’t moving?
2. Lithium has 3 electrons and a nucleus with a +3 charge. Show that there is no way to put electrons near the nucleus in a
stable, static arrangement.
3. Explain how having the electrons move improves the model.
4. The PhET simulation shows a radio station transmitting EM waves. The energy it takes to create these waves is carried
off by the waves. Describe some other examples of EM waves and identify the sources of energy.
5. What is the major problem with the planetary model of the atom? Why do we need new “quantum” rules?
Thinking Deeper:
1. If the planetary model doesn’t work, why is it included in almost every introductory Chemistry textbook?
2. What problem does the Law of Electrostatics have for the nucleus of the atom? Suggest a possible solution.
3. All matter is made out of atoms, but there is no way to build a commonsense model of the atom. Summarize the problems
and identify properties that a new model must have.
24
SA2: Scientific Revolution: Quantum Mechanics
1. Explain how the rotating ring solves the dilemma of orbiting electrons emitting EM waves.
2. Why does the rotating ring idea fail?
3. A traveling wave is a wave pattern that moves. How does describing the “orbiting” electron by a traveling wave circulating
around the nucleus solve the problem of the electron emitting EM waves?
4. A standing wave is composed of two oppositely-directed traveling waves. How is the behaviour of the “orbiting” electron
described by a standing wave similar to its behaviour in the static model of the atom? How is it different?
Thinking Deeper:
1. Both standing (and traveling) waves can only exist around the nucleus when an integer number of wavelengths fit around
the “orbit”. How can this property be used to explain discrete energy levels in an atom?
2. Start the simulation from SW2 (http://www.falstad.com/qmatom/) and use similar settings (and deselect “Phase as Color”).
The state produced by selecting “n=2, l=1, m=–1” and “n=2, l=1, m=+1” at the same time is an example of an excited state
of the atom. The ground state of the atom (the state of minimum energy) is given by the “n=1, l=0, m=0” circle (top circle
in the first column). Both of these are stationary states—they do not emit EM waves. Click on the “n=1, l=0, m=0” circle to
put the electron in both states at once—an excited state and the ground state. Increase the Simulation Speed using the
slider in the right panel. Do you think the electron is emitting EM waves? Draw comparisons with the antenna simulation in
Part C of SW1. What is the quantum atom in the process of doing?
3. Quantum mechanics is often referred to as weird or strange. What is so strange about it?
25
SA3: Applications of Quantum Mechanics
Quantum Mechanics is one of the most successful scientific models ever created. Not only has it passed every experimental
test to date, but it has become the basis for a huge number of applications, resulting in trillions of dollars of economic activity
every year.
1. The quantum mechanical model of the atom says that light is emitted when electrons go from a higher energy state to a
lower energy state. A light emitting diode (LED) is a device that uses this property to produce light very efficiently. LEDs do
not get hot, do not burn out and do not contain any harmful materials.
(a) Where do you find LEDs being used?
(b) For a typical home, about 15% of its electricity bill is for lighting. How much money would you be willing to invest in
new lighting technology in order to reduce your energy consumption?
(c) Research the LED bulb technology that is currently available for residential use. How much would it cost to convert
your house over to LED bulbs? How many years would it take for this investment to pay off?
(d) What are the factors that you would consider when choosing which technology is the best for you?
2. The quantum mechanical model of the atom says that electrons can only occupy certain energy levels and that the
atom will absorb or emit light as the electron changes energy levels. In 1917 Albert Einstein used the laws of quantum
mechanics to predict that excited atoms could be stimulated with light to emit their extra energy as more of the same kind
of light, thereby amplifying the light. Forty-three years later the first functioning laser was made. (a) Lasers produce very intense, coherent, monochromatic light. List all the applications of lasers that you know of and
describe how the properties of laser light are well suited for that application.
(b) The scientist who coined the term laser (Light Amplification by Stimulated Emission of Radiation) spent 27 years
fighting with the patent office. What would he gain by winning the patent for this technology?
3. The quantum mechanical model of the atom says that electrons are particles that behave like waves. Waves can reflect
and produce standing waves. This wave behaviour of electrons is essential for the functioning of transistors, which are the
basis for all electronics.
(a) Consider your bedroom. List all the devices that contain electronic components.
(b) Research the electronics industry. How much money was generated last year by the production of transistors alone?
How much money was generated by the production of devices that use transistors?
(c) Look up Moore’s Law on the Internet. What does Moore’s Law say and why is it important to the electronics industry?
4. The quantum mechanical model of the atom says that electrons can behave as if they are in more than one place or
state of motion at the same time. This strange behaviour of electrons is being explored to design a new type of computer
called a quantum computer. Quantum computers will be able to do certain complicated tasks extremely quickly and will
allow scientists to make very sophisticated models of quantum systems. As scientists gain the ability to model quantum
systems, they will be able to design new and more powerful quantum technologies. Think back over the last century and
reflect on how discoveries in basic science and their applications have worked together to produce the world we live
in. Where do you think these new applications of quantum mechanics will take us in this next century? Use historical
examples to support your insights.
26
27
3. The electron in the receiving antenna gets its energy from the wave emitted
by the transmitter.
2. The electric field wiggles, creating a wave pattern in the field that moves
away from the electron.
1. The electric field changes as the electric field pattern around the electron
moves with the electron.
Note for teachers: An electron at rest is surrounded by a static, radial electric field
pattern. This field stores energy in the space around the electron (electrostatic energy).
When the electron moves with a constant speed and direction, this field pattern (and
energy) moves with the electron, like flies buzzing around a moving garbage truck.
But when the electron accelerates (changes its speed or direction of motion), some
of this energy is “shaken off” (like flies shaken off an accelerating garbage truck) in
the form of electromagnetic waves. According to Maxwell’s equations, a changing
electric field creates a magnetic field, and vice-versa, setting up a chain reaction
that is an electromagnetic wave. (The simulation shows only the electric part of the
electromagnetic wave.) In question #1 below, students see just the static part of the
field pattern that moves along with the electron. In question #2, students see the
radiated part of the electric field—the “flies that are shaken off”. (Note that the energy
in the space around the electron is immediately replaced with energy from the “hand”
that is wiggling the electron, i.e., it takes more effort to wiggle a charged particle than
a neutral particle of equal mass!) When the acceleration is a simple up-and-down
oscillation, the EM waves form a simple pattern that radiates outwards from the
electron, and carry with them the energy required to make other electrons move. This
is what students see in question #3.
Part C: The Failure of the Planetary Model
3. The electron would spiral into the nucleus as the energy is gradually removed
from the atom. Note for teachers: The electron would actually speed up
as it spirals in, increasing its kinetic energy; but the electrostatic potential
energy of the electron-nucleus system would decrease by a greater amount,
resulting in a net decrease in the atom’s total energy.
2. The nucleus will pull sideways on the moving electron, bending its path.
1. We must exert a sideways force on the moving electron to change its
direction of motion.
Part B: The Planetary Model
2. The Hydrogen atom cannot be stable if the electron is static. The only other
option is to make it dynamic—we need to make the electron move.
1. The electron would be attracted to the nucleus and accelerate towards it,
emitting a flash of light.
Part A: Static Model
SW1: Answers
Answers
3. The electron is behaving as if it is moving in two opposite directions at once.
In this sense it is not moving (there is no angular momentum), and is like the
classical static model of the atom. The difference from the classical static
model is that the electron does not get pulled into the nucleus because it is
actually moving! As in #2 above, the electron is static in that the probability
pattern does not change at all. It is dynamic in that the phase is circulating,
albeit in two opposite directions at once! The electron is behaving like two
classical, static “blobs” of charge—it will not emit EM waves.
2. The electron is static in that the probability pattern (the “amplitude” of the
wave) does not change at all. The electron is dynamic in that the “crests” and
“troughs” (the “phase” of the wave) is circulating. The probability pattern tells
us that the “potential location” of the electron is spread out into a perfectly
smooth ring. The circulating phase tells us that this ring is rotating. So the
electron is behaving exactly like a classical, stationary charged rotating ring—
it will not emit EM waves.
1. You see a colourful fuzzy ring that slowly rotates in a clockwise direction. The
rotation shows the motion of the wave “crests” and “troughs” as the wave
circulates around the nucleus.
Part C: The Quantum Model
2. As you gradually increase the frequency of vibration, the standing wave
will disappear and the Slinky will appear ‘chaotic,’ with random vibrations.
Eventually you will reach a frequency which produces another stable standing
wave pattern; this pattern will have one more node.
1. The wave is everywhere in the Slinky at the same time. While each part
of the Slinky moves side-to-side only, the wave pattern travels in the
perpendicular direction, from one end of the Slinky to the other.
Part B: Standing Waves
2. The charge inside the ring would repel itself, and the ring would tend to fly
apart. Classically, at least, we would be able to observe such a structure by
using a microscope with light of sufficiently short wavelength.
1. A rotating ring will not emit EM waves because the electric (and magnetic)
fields surrounding the ring are not changing. It is changing electric or
magnetic fields that produce EM waves.
Part A: The Rotating Ring
SW2:
5. The EM wave would remove energy from the atom causing the electron to
spiral into the nucleus; the atom would collapse in a flash of light.
4. The electron would emit an EM wave as it orbits around the nucleus because
the accelerating electron would create a changing electric field in the
reference frame of the atom.
28
1. If an orbiting point-like electron is spread out into a rotating ring, nothing would
be “waving” back and forth, or side to side, and so it would not emit EM waves.
SA2: Answers
3. The static model collapses due to electrostatic forces. The planetary model
collapses due to EM waves draining energy from the atom. We need a model in
which electrons somehow “orbit” without emitting EM waves.
2. The protons in the nucleus electrostatically repel one another very strongly.
The idea of the strong nuclear force, which holds the nucleus together, could
be introduced to students. Here nature does use a new force to solve a
stability problem!
1. The planetary model (including the Bohr-Rutherford model) gives a simple, intuitive
picture for the atom. It is a good starting point for understanding simple chemical
reactions. Models can be useful in a limited context, even if they are wrong.
Thinking Deeper
5. An orbiting electron is continually accelerating (toward the nucleus) due to its
continually changing direction of motion. This causes the atom to radiate energy
in the form of EM waves, and the electron to spiral into the nucleus. Newton’s
laws, together with the laws of electromagnetism, predict the collapse of the
atom, so atoms cannot exist in a classical universe. Quantum ideas are needed
to explain how atoms can exist.
4. Light is emitted when an electron in an atom “drops” to a lower energy level; the
energy comes from the excited atom. Cell phones emit EM waves just like a radio
station; the energy comes from a battery. X-rays are emitted when electrons are
slowed down by a collision; the energy comes from the moving electron.
3. The previous problem points out that there is a net force pulling electrons
towards the nucleus. Instead of allowing this force to simply pull static electrons
into the nucleus, we use this force to bend the path of the moving electrons,
causing them to orbit the nucleus.
2. In any multi-electron atom, each electron will be repelled
from the other electrons, and will try to move as far away
from the others as possible, in a symmetric way (see Figure
for Lithium). But in all cases, each electron experiences
a stronger force of attraction towards the nucleus than
the net force of repulsion from the other electrons. These
unbalanced forces cause the atom to collapse.
1. The nucleus attracts the electron, pulling it into the nucleus. (The nucleus is
much heavier, and so hardly moves.)
SA1: Answers
3. The weirdness of quantum mechanics may be stated as the wave-particle
duality: the idea that all quantum particles (e.g. electrons and photons) exhibit
both wave and particle properties. An electron can exhibit the wave properties
of being in two locations at once, or moving in two directions at once, which
is natural for waves, but not for classical particles. Once we accept this
weird wavelike behaviour of particles (and vice-versa), other quintessentially
quantum aspects of nature follow naturally. For example, quantization of
atomic energy levels is not weird—it is a simple consequence of electrons
behaving like waves.
2. The simulation shows a probability pattern that is changing—one that is
sloshing back and forth, exactly like an electron moving up and down in
an antenna wire. An electron in this non-stationary state is emitting (or
absorbing) EM waves, i.e., a photon. In the case of emission, the atom is
in the process of “dropping” from the excited state to the ground state; and
the reverse in the case of absorption. Note to teachers: Electrons do not
mysteriously “jump” between atomic energy levels! There is a very sensible
physical process involved.
1. Each energy level corresponds to a different “harmonic,” like the harmonics
on a violin string. For the first energy level, one wavelength fits around the
“orbit”. For the second energy level, two wavelengths fit around the “orbit”,
and so on. The electron is never found between these discrete energy levels
because you don’t get stable standing (or traveling) waves there.
Thinking Deeper
4. It is the same in that a standing wave does not move, like an electron in the
classical static model. It is different in that the electron does not get pulled
straight into the nucleus. This is because the electron is actually “moving”
(albeit in two directions at the same time!), and so the net force towards the
nucleus just bends the path of the electron into two simultaneous, counterrotating “orbits”!
3. The point-like electron behaves as if it is in many places at once (wherever
the wave is non-zero), and so it is effectively spread out exactly like a rotating
ring. The wave is moving (the electron has angular momentum) but the
corresponding probability pattern is not changing—a state called a stationary
state. Electrons in such stationary states do not emit EM waves.
2. Different parts of a spread out electron would repel each other, tending to make
the ring fly apart. We would have to invent a new type of matter or force to hold
the ring together. Also, whenever we “look” at an electron, we always see a
point-like particle. If an electron took the form of a ring, it would have to turn into
a point-like particle the instant we “look” at it. This would be absurd.
It would create static electric and magnetic fields, but EM waves are produced
only when these fields change.
Introduction
30-31
Student Worksheets:
SW1: Scientific Models: 32-34
SW2: Scientific Revolution:
35-36
Time
Special Relativity
Student Assessment:
SA: Scientific Revolution:
37
Special Relativity
Answers
38- 40
This module contains two single-period lessons based on the Alice & Bob in Wonderland animation: Where Does Energy
Come From? In this episode Alice and Bob ask questions about energy and discover a deep connection between mass and
energy. Lesson 1 (introductory level; some prior knowledge of physics is an asset) introduces the concept of relativity, and
shows how it led to Einstein’s model of relative time and a universal speed limit. Lesson 2 is a more advanced lesson (prior
knowledge is expected) that starts with relative time to show how energy has inertia.
LessoN 1:
LessoN 2:
SCIENTIFIC MODELS: TIME
SCIENTIFIC REVOLUTION: SPECIAL RELATIVITY
>> Show the Alice & Bob animation:
>> Show the Alice & Bob animation:
Distribute SW1: Scientific Models: Time. This
Distribute SW2: Scientific Revolution: Special
Where Does Energy Come From?
worksheet presents the students with several
thought experiments that will help them develop
Einstein’s Special Theory of Relativity.
Where Does Energy Come From?
Relativity. This worksheet presents the students
with several thought experiments that will help guide
them to develop E=mc2.
SA: Scientific Revolution: Special Relativity is a
worksheet that includes additional questions to
be done in class or assigned for homework.
29
Where Does ENergy Come From?
Science is a process of building models to explain
But is relativity universal? Is it really true that you can’t detect
careful thought and experimentation. Good models explain
inside the room? Ever since Galileo first suggested the
observations and then refining those models through
existing observations, make testable predictions, and give
deeper insights into the phenomena. This Perimeter Institute
classroom resource engages students in this process by
exploring models of two common real world phenomena—
time and energy, with an emphasis on the role of thought
experiments in science. Students will exercise critical and
creative thinking to discover how Albert Einstein’s intuitive
belief that an observer in a closed room cannot tell whether
the room is moving leads to a radical new understanding of
time and energy, known as the Special Theory of Relativity.
We have all experienced relativity. When you are inside a
closed room, such as an airplane with the window shades
the inertial motion of a closed room by any experiment done
concept of relativity in 1632, it has been accepted as true for
mechanical experiments (such as drinking coffee, or juggling
balls). But in the early 1800s, strong experimental evidence
emerged to show that light behaves like a wave, and this
presented a problem for relativity. The problem was that,
like sound waves, light waves presumably could not travel
in empty space. So space must be filled with a light-wave
medium they called the “ether”: an immobile substance
whose vibrations constitute light, but through which matter
could move freely. The wave nature of light created the
possibility of detecting motion relative to the medium. Ether
might represent a state of absolute rest.
drawn, you can’t tell that the room is in motion. You can’t feel
Students will explore the
speed with no rotation). Everything that happens inside the
for relativity using a thought
inertial motion, (i.e. motion in a straight line at a constant
room (e.g. drinking coffee or juggling balls) happens the
same way it does when the room is at rest, no matter how
fast the room is moving. From this we learn that “at rest” and
“in motion” are relative concepts—they make sense only
when compared to objects outside the room, such as the
ground moving beneath an airplane. There is no such thing
as absolute rest.
wed that the very nature of
Did you know? Einstein sho
tive
from detecting motion rela
time and space prevents us
it
and
ht
mig
It
ts.
exis
it
if
w
to the ether. We cannot kno
might not.
hes us, however, that
“More careful reflection teac
y does not compel us to
the special theory of relativit
STEIN
deny ether.” – ALBERT EIN
only
“Speed of Light Principle”
Did you know? Einstein’s
ion
mot
the
of
t
t is independen
asserts that the speed of ligh
e-in
wav
the
is obvious for
of the source of light, which
of
not suggest that the speed
s
doe
It
t.
ligh
ether model of
t.
ligh
of
r
erve
motion of the obs
light is independent of the
and
r,
eve
ement is true, how
This absurd-sounding stat
his
an obvious consequence of
is
it
how
Einstein showed
ce.
new model of time and spa
is,
the very concept of time, that
“My solution was really for
STEIN
defined...” – ALBERT EIN
that time is not absolutely
challenge that waves pose
experiment: Alice is inside
a spaceship floating in
deep space, at rest in the
hypothetical
ether. She sends
a pulse of light
(a single wave
front) upwards
from the floor,
which moves at
speed c relative
to the ether.
She measures
the time it takes to reflect off the ceiling and return to the
floor. Alice then repeats exactly the same experiment in her
“closed room”, except now it is drifting through the ether
(causing an “ether wind” to blow through the spaceship). She
still sees the pulse of light travel vertically up and down, but
Bob—floating at rest in the ether, sees the light pulse move
diagonally up and down. It is an established fact that waves
of any kind move at a fixed speed relative to the medium.
Once created, a wave propagates on its own, independent
of any motion of the source. Even though Alice’s light source
is in motion, the wave front it creates will move with speed
c relative to the ether. And since it travels a greater distance
in the second experiment, Bob will measure a greater
return time.
30
The crucial question is, “What elapsed time will Alice
and clock. Bob will see the ball covering less than one metre
But, since this time is different from Alice’s “at rest” time, we
The additional speed of 1 m/s that Alice has given to the ball
measure?” The obvious answer is, “The same as Bob!”
would have to admit that it is possible to use an experiment
with light inside a closed room to detect the inertial motion of
that room. Universal relativity would not hold. For universal
relativity to hold, Alice must measure the same time whether
she is at rest or moving, but then Isaac Newton’s model
of absolute time would be wrong. (Absolute time says
when one second elapses for you, one second elapses for
everyone in the universe, regardless of their location or state
of motion; in this case, Alice must measure the same time as
Bob.) Students see that if we adopt a wave-in-ether model of
light (as most physicists did until after 1905), then absolute
time and universal relativity are incompatible. One is wrong,
and must be jettisoned from our thinking.
For many years scientists, including Einstein, struggled with
the tension between relativity—which seemed so simple and
natural that it ought to be universal—and the nature of light.
The definitive breakthrough came in 1905 when Einstein
realized that the problem had nothing to do with light,
but rather the nature of space and time. In particular, he
realized that absolute time was just an assumption that had
never really been tested beyond everyday experience. He
immediately jettisoned Newton’s model of absolute time and
worked out the logical consequences of universal relativity.
Students will work through the following rational and intuitive
progression:
(length contraction) in more than one second (time dilation).
will be less than 1 m/s for Bob. As Alice’s rocket approaches
the speed of light, this effect becomes more pronounced,
so that it is never possible for Bob to see the ball reach (or
exceed) the speed of light. By universal relativity, no matter
how fast Alice is moving relative to Bob, she can consider
herself to be “at rest”, and can throw the ball forward as
close to the speed of light as she wants, relative to herself.
Unified Model of Energy: E=mc2. Using only time dilation,
students engage in a thought experiment to conclude that
a fast moving ball is more difficult to deflect sideways than
one at rest. Its inertia relative to someone at rest increases
along with its kinetic energy. By placing two balls connected
to the ends of a spring inside a box, and letting the balls
oscillate rapidly, students see that the balls’ increased inertia
relative to the box increases the mass of the box—kinetic
energy has inertia. As the system energy oscillates between
kinetic energy and spring potential energy, students realize
that potential energy has inertia too. Extending this to the
molecules in a hot object, students learn that thermal energy
has inertia. Since a block can be heated with light, students
also discover that electromagnetic energy has inertia. All
forms of energy possess inertia (resistance to changes in
motion). Students also explore the converse: how Einstein
correctly guessed that the inertia (i.e. mass) of even an
object at rest is equivalent to an enormous amount of energy.
Time Dilation. According to the second part of our thought
This resource introduces students to Einstein’s Special
Bob’s. Note that, according to universal relativity, from Alice’s
facts, but rather as a logical argument based on the simple
experiment, Alice’s moving clock runs slowly relative to
perspective she is “at rest”, and it is Bob’s clock that is moving,
and running slowly (not faster!). Time dilation is reciprocal.
Theory of Relativity not as a sequence of counterintuitive
and natural Principle of Universal Relativity, supported by
ample experimental evidence.
Length Contraction. During the time measured on his
clock, Bob sees Alice cover a certain horizontal distance as
measured in his frame of reference. But for Alice, it takes
less time. For both to be moving at the same relative speed,
Alice must measure this distance to be less in her frame of
reference. She must see Bob’s space—and everything in it,
contracted. And again, by universal relativity, this effect must
be reciprocal: Bob must see Alice’s space—and everything
in it, equally contracted.
Universal Speed Limit. Now suppose Alice throws a ball
forward inside her moving spaceship so that it is covering a
dilation and length
What makes time
y
Did you know?
ect called relativit
ocal is a subtle eff
t
contraction recipr
no
is
b
Bo
us for
hat’s simultaneo
,
of simultaneity. W
rsa! Time dilation
ve
e
vic
d
an
,
Alice
rk
simultaneous for
wo
y
eit
an
ult
sim
n, and relativity of
lflength contractio
y in a logically se
vit
ati
rel
l
rsa
ive
un
ce
for
en
to
er
togeth
er.
consistent mann
and mechanics
of electrodynamics
“The phenomena
g to the idea of
rties correspondin
possess no prope
IN
ALBERT EINSTE
absolute rest.” –
distance of one metre every second according to her ruler
31
SW1:
Scientific Models: Time
In this activity you will conduct several thought experiments and use logic to discover something fascinating about the nature
of time and space. You will work in groups of three using chart paper or a large whiteboard. Each group will have a Reader
(who keeps the group on task), a Recorder (who writes the answers on the chart paper), and a Reviewer (who asks questions
to check for understanding).
Part A: Relativity—“Can you tell you are moving?”
Commonsense dictates and experience confirms that there is no mechanical experiment that can be done in a closed room to
tell whether or not the room is moving at a constant speed in a straight line.
1. Imagine Alice sitting in a parked car, tossing an apple straight up. Later, she repeats
this experiment while the car is moving at a constant speed.
(a) Sketch the path of the apple, as seen by Alice, both when the car is parked and
when it is moving.
(b) Can Alice use this experiment to tell that she is moving?
(c) Sketch the path of the apple, as seen by Bob standing on the sidewalk, when the
car is moving.
2. Imagine Alice on a raft made from two pontoons joined by a couple of poles. She
anchors the raft in a still pool of water and sends a wave from one pontoon across to the
other. It reflects back and she measures the total time taken. Later, the raft is propelled
through the water at a constant speed and she repeats her experiment (see Figure). An
important fact to consider is that the speed of a wave produced by a moving source will
be the same as the speed of a wave produced by a stationary source.
(a) Sketch the path of the wave Alice observes when the boat is anchored.
(b) Sketch the path of the wave, as seen by Bob floating at rest in the water, when the boat is moving through the water.
(c) Can Alice use the times measured in this experiment to tell that she is moving?
3. Imagine Alice inside a rocket deep in space with a clock that measures the time taken
for a pulse of light to go up to a mirror on the ceiling and back again. Alice has learned
that light behaves like a wave, and assumes it travels through some medium in space
that flows freely through the moving rocket. Suppose she starts at rest in that medium
and measures the time taken for the pulse to go up and back again. Later, she repeats
the experiment when she is moving at a constant speed. Bob is just floating in space, at
rest in the medium. He also times the light pulse as Alice cruises by.
(a) Sketch the path of the light pulse Alice observes when she is at rest.
(b) Sketch the path of the light pulse Bob observes when Alice is moving.
(c) Can Alice use this experiment to tell that she is moving?
Part B: Newton vs Einstein
Newton believed in absolute time: the rate at
which time passes is the same for everyone
regardless of their motion. Einstein believed in
universal relativity: there is no way to tell, from
inside a closed room, that the room is moving.
The natural assumption that light waves travel
through some medium in space means that the speed of light along both paths
in the diagram is the same. Since the path observed by Bob is longer he must
record a longer time than Alice did when she was at rest. This puts absolute
time and universal relativity into direct conflict—only one of them can be right.
32
1. Absolute time dictates that in the moving experiment Alice should measure the same time as Bob does. What are the
implications for universal relativity if this is true?
2. Universal relativity dictates that in the moving experiment Alice should measure the same time as she did when she was
at rest. What are the implications for absolute time if this is true?
Absolute time and universal relativity are incompatible. One of them must be wrong. The only way to resolve the
conflict is through experiment.
3. The Large Hadron Collider (LHC) is the largest scientific experiment in history. It is a 27 km long particle accelerator that
smashes protons together with unprecedented energy. Neutral kaons are unstable particles produced during the collisions
that decay with a half-life of 8.9x10-11s. Use this half-life as the decay time for the kaons.
(a) How far would you expect the kaons to travel before decaying, if they are travelling at 0.995c?
(b) Kaons are detected 27 cm from the centre of the collision. How does this data refute Newton?
(c) How would Einstein interpret these results?
4. Consider again the kaons produced at the LHC. From the perspective of the kaons, they are at rest and they survive for
8.9x10-11s. The kaons “see” the detector rushing by them at 0.995c. What length of the detector rushes by during this
time? How can you reconcile this with the 27 cm mentioned above?
A logical consequence of universal relativity is time dilation—“moving clocks run slow.” The flip-side of time dilation
is length contraction—“moving objects occupy less space.”
Part C: Time Dilation and Length Contraction
Experimental evidence supports Einstein’s predictions of time dilation and length contraction. With this as motivation, let’s take
a closer look at the diagram above for the rocket experiment.
1. According to universal relativity, Alice cannot tell that she is moving. The time taken by the light pulse must be the same
for her, whether she is moving or not. Write the algebraic expression for the time taken, tAlice , when Alice is not moving.
2. Bob sees the light pulse travel up and down in the vertical direction, with a vertical speed of
). Write the algebraic expression for the time Bob measures, tBob.
(which simplifies to
3. Compare tAlice and tBob. What is the “time dilation” factor that relates Alice’s time to Bob’s time?
4. In the moving case, Bob sees the vertical speed of the light pulse to be
vertical speed of light to be less than c? Explain, using time dilation.
. Does this mean that Alice sees the
5. When Alice is moving relative to Bob, Alice and Bob disagree on how much time elapses for the light pulse to return. How
will this disagreement affect their understanding of how far Alice has travelled in her rocket?
Time and length are both changed by the same amount, the Lorentz factor, and it shows up in so many relativity
calculations that it gets its own symbol, γ. Note that γ is always ≥ 1.
33
Part D: Speed Limit
One of the misunderstandings about relativity is that Einstein began with this statement that “nothing goes faster than the
speed of light” and derived everything from that premise. While this statement is true, it is not fundamental—it is a logical
consequence of time dilation and length contraction.
1. Alice is inside her rocket, moving relative to Bob, when she starts running forward at 1 m/s. She covers 1 m in 1 s
according to her ruler and clock.
(a) How far is Alice’s 1 m as measured by Bob? (Express your answer in terms of γ.)
(b) How long is Alice’s 1 s as measured by Bob? (Express your answer in terms of γ.)
(c) How fast is Alice running inside the rocket as measured by Bob? (Express your answer in terms of γ.)
2. As Alice begins to run, her speed inside the rocket changes from zero to 1 m/s. How will this speed change appear to
Bob? How will this speed change be affected by the speed of the rocket as it gets closer to c?
Time dilation and length contraction “enforce” a universal speed limit, and allow it to make sense.
3. Suppose Alice needs to apply a force F, for one second, to get herself running. Now imagine, instead, that Bob “reaches”
into her moving rocket and pushes her with the same force, F. (He stays at rest, but his hand moves very fast!) How long
does he need to apply the force, according to his clock, to have the same effect on Alice?
It’s harder for Bob to accelerate Alice, as if her mass somehow increases. Actually, nothing happens to Alice’s mass.
It is time dilation and length contraction that make Alice’s effective inertia greater, relative to Bob.
4. Plot the following historical data for particle accelerators with Energy of the proton (in GeV) on the x-axis and Speed of the
proton (as a % of c) on the y-axis. What happens to the speed as more and more energy is given to the particle? Where is
the energy going, if it’s not going into increasing the proton’s speed?
Proton accelerator
Energy (GeV)
Speed (%c)
CERN Linac 2
0.050
31.4
TRIUMF
0.48
75
CERN PS Booster
1.4
91.6
BNL Cosmotron
3.3
97.5
CERN PS
25
99.93
Suppose we give a particle some energy to accelerate it from rest up to speed v. By universal relativity we can catch
up with the particle and see it as “at rest” again. We can then repeat this process—again and again, forever. While its
speed is limited (by the nature of time and space), the amount of energy we can give a particle is unlimited. This is
just one of the fascinating ideas contained in Einstein’s most famous equation, E=Mc2.
Bringing It All Together
1. Review the work that you have done as a group and discuss any points that need clarification.
Summarize the concepts in your notebook. Be sure to address the following points:
• What is universal relativity and absolute time, and how do they conflict with each other?
• What is time dilation and length contraction, and how are they related?
• Why is there a universal speed limit?
• List some of the experimental evidence for Special Relativity.
• What happens to an object’s effective inertia as its speed increases?
34
SW2:
Scientific Revolution: Special Relativity
Following the logical consequences of universal relativity, we have so far discovered time dilation, length contraction and a
universal speed limit. In this activity you will conduct several more thought experiments and use logic to discover something
fascinating about energy. You will work in groups of three using chart paper or a large whiteboard. Each group will have a
Reader (who keeps the group on task), a Recorder (who writes the answers on the chart paper), and a Reviewer (who asks
questions to check for understanding).
Part A: A Head-on Collision
Imagine that two Super Balls collide head-on.
1. Both balls have speed v going in and speed v going out of the collision. How do their
masses compare?
2. Both balls have speed v going in, but one gains speed as a result of the collision. How
do their masses compare?
3. One ball has speed v going in and out of the collision, and the other has a greater
speed, V, going in and out of the collision. How do their masses compare?
Part B: A Glancing Collision
Two identical Super Balls undergo a very fast glancing collision that is perfectly
symmetrical. Alice is riding on the upper ball. She sees the dashed line whizzing
by her with a very large horizontal speed as she drifts toward and then away from
it with a very small vertical speed, V. Bob, riding on the lower ball, sees the same
thing for himself.
1. Imagine that you are now moving to the left with enough speed that Bob
has no horizontal speed relative to you (see lower Figure). Would your
new perspective change the vertical speeds that Alice and Bob observe for
themselves?
2. Alice is now moving even faster relative to you, and her time is dilated
compared to you by a factor of γ. If it takes Alice one second to move the
vertical distance to the dashed line according to her clock, will it take more or
less time for her to cover the same vertical distance according to your clock?
How does time dilation change Alice’s vertical speed, from your perspective?
3. Alice’s vertical speed does not change as a result of the collision, neither
does Bob’s. If we ignore Alice’s horizontal motion, this collision is the same as
#3 in Part A. How do their masses compare?
By time dilation, a moving object has greater “effective inertia” for sideways deflection: M = γ m
Part C: All Forms of Energy have Inertia
Alice comes across a closed box “at rest” in deep space.
1. (a) Nothing enters or leaves the box. Can the mass of the box suddenly
change from M to M’? Why or why not?
(b) Bob is drifting by and sees the box moving at a constant speed v. How
can Bob use conservation of energy and/or momentum to explain that such
a change in mass is impossible?
35
2. Alice opens the box and finds two balls of mass m connected by a spring. The balls are
oscillating back and forth very quickly. When they are moving fastest, their time dilation
factor is γ.
(a) Using the concept of “effective inertia” for sideways deflection, how does the motion
of the balls affect the mass of the box? Does kinetic energy have inertia?
(b) As the balls move outwards, their kinetic energy changes into elastic potential
energy stored in the spring. The total mass of the box cannot change. How does
this show that potential energy has inertia?
Kinetic Energy
3. A brick is made of atoms connected by spring-like inter-atomic bonds. As the brick is heated,
would you expect its mass to increase? Explain.
4. A brick can be heated with the energy in light. The “before” picture shows a box containing a
brick of mass M and two pulses of light in midflight heading towards the brick. In the “after”
picture, the block has absorbed the light energy and is warmer. Does the mass of the box
change when the light is absorbed? What does this say about the inertia of light?
All forms of energy have one property in common: inertia. This is a powerful unifying
principle in unravelling the mystery of what “energy” is.
Part D: E=mc2
We have just discovered that various forms of energy (kinetic, potential, thermal, and electromagnetic) inside a box contribute
to the mass (or inertia) of the box. So any change in the energy, ΔE, inside a box must produce a corresponding change in its
mass, Δm. The exact relationship is very simple: ΔE = Δmc2. This mass-energy equivalence applies to all physical processes,
including chemical and nuclear. All forms of energy have inertia; but do all forms of inertia have energy? Does even mass at
rest have energy?
1. The “before” picture shows a box containing two particles at rest, each of mass m; one is
matter and the other is antimatter. In the “after” picture, the matter and antimatter have been
transformed entirely into light.
(a) The box is sitting on a weigh scale. Does its weight change? Explain.
(b) We uncover a window on the box and let out all the light. What is the change in mass
of the box? How is this related to the amount of energy that left the box?
Experiments with elementary particles confirm Einstein’s intuition that even mass at rest has
energy: E=mc2. The general form of Einstein’s mass-energy equivalence relation is: E=Mc2,
where M= γm is the relativistic mass and m the rest mass. This mass-energy equivalence can be rewritten as a general relation
between energy and momentum: E2 = m2c4 + p2c2, where p = γmv is the relativistic momentum of the system. When the
system has no rest mass (e.g. a photon) the general relation reduces to E=pc, a result that agrees with both a wave model of
light and a particle model of light. When the system is at rest (p=0), the general relation reduces to E=mc2.
It is a remarkable fact of nature that matter can transform into light, and vice versa, but notice that in such processes
both the total mass and the total energy stay the same. Mass is not “converted” into energy, or vice versa. The
energy in light has inertia, and the inertia in matter has energy—as described by E=mc2.
Putting It All Together:
1. Review the work that you have done as a group and discuss any points that need clarification. Summarize the concepts in
your notebook. Be sure to address the following points:
• How does time dilation result in moving objects having extra “effective inertia”?
• How is an increase in the “effective inertia” different from an increase in the mass of the object itself?
• How would you explain to your friend that kinetic and potential energy have inertia?
• Does a cup of coffee weigh more when it is hot? Explain.
• Why is it incorrect to say that mass is converted into energy, and vice-versa?
• As the Sun emits light energy, what must happen to its mass? Where has the mass gone?
36
SA:
Scientific Revolution: Special Relativity
1. The scientists at CERN accelerate protons to 99.9999991% c as the protons travel around a 27 km long ring.
How does this extreme speed create challenges for them?
2. One of the most famous equations in science is E=mc2. There are several ways to
understand where this equation comes from. Here is one published by Albert Einstein:
(a) Start with a block of mass M at rest. Let it absorb two pulses of light, each with
energy E/2, so the block heats up (see Figure).
(b) Consider this same experiment from the point of view of an observer drifting by
with speed v. To this observer, the block has momentum Mv, and the pulses of light
approach the block at an angle.
(c) Draw a diagram to show how this angle is related to the relative speed of the block
and the speed of the light pulses. Remember that light moves at speed c.
(d) Express the sine of the angle as a ratio of these two speeds.
(e) The momentum of light is given by Maxwell’s equations as: p=
.
What is the momentum of each pulse of light approaching the block? What is the vertical component of this momentum?
(f) What is the change in momentum of the block when it absorbs the two pulses of light?
(g) The block does not speed up when it absorbs the light. Why not?
(h) If momentum changes but speed does not, then the mass of the block must increase from M to M’.
Using conservation of momentum, find an expression for E in terms of the change in mass.
3. In the animation Where Does Energy Come From? Alice and Bob discover that we are literally eating the Sun!
The Sun provides the energy that is the basis for virtually every food chain. We measure the energy output of the Sun by
its luminosity and find that the Sun emits 3.8x1026 W.
(a) Use E=mc2 to determine how much mass the Sun is losing every second.
(b) The Sun has a total mass of 2x1030 kg. How long will it take to use up the mass of the Sun?
(c) The Sun is expected to survive for another 5 billion years. Will the mass loss be significant by then?
4. The general relation between energy and momentum in Special Relativity is E2 = m2c4 + p2c2 , where m is the rest mass
for the system and p is the relativistic momentum, p = γmv.
(a) Start with the general energy-momentum relation and use common factors to derive E = γmc2.
(b) Start with E = γmc2 and use the binomial expansion of to show that E ≈ mc2 + ½mv2.
(c) What does this last equation contribute to our understanding of energy?
37
38
Answers
(b)
2. (a) (b)
3. (a)
2. If Alice measures the same time as when she was at rest, then she would
not know that she is moving. But since this time is different from Bob’s, time
is not absolute. Two observers moving relative to each other will disagree on
how much time has elapsed.
1. If Alice measures the same time as Bob, which is different from when she
was at rest, then she would know that she is moving. If she can tell that
she is moving then relativity is not universal. Relativity would not apply to
experiments done with light in a closed room.
Part B:
(c) This problem is in principle the same as #2, and students might expect
that the return time will be different—Alice can use this experiment to tell that
she is moving. In fact she cannot, and students challenge the reason for their
expectation (absolute time) in Part B.
3. (a) (c) Alice can tell that she is moving relative to the water because the time
taken for the wave to return is longer when she is moving than when she is at
rest. The time is longer because the path in longer, but the speed of the wave
relative to the still water is the same in both cases.
(c)
(b) Alice cannot tell that she is moving. Both paths will be identical.
1. (a) Part A:
SW1:
The detector rushes by at 0.995c for 8.9x10-11 s. During this time the kaons
will “see” 2.7 cm of the detector pass by. But actually they have travelled
through 27 cm of the detector. From the kaons’ perspective, the detector
must be contracted in the direction of motion by a factor of 10, so the 27 cm
of detector occupies a space of only 2.7 cm in the kaons’ frame of reference.
4. Bob sees the light moving up and down more slowly than speed c, and so it
might seem that Alice would see a vertical speed of less than c for the light
pulse in her frame. But Alice’s time is also passing more slowly relative to Bob,
and so she sees the normal speed for light—c. It is the nature of time and
space, and not the nature of light, that makes all observers measure the same
speed for light (or anything else moving at the universal speed limit). Alice
cannot use the return time or the speed of the light pulse to detect her motion!
3.
2.
1. If d is the length of the path (up and down), then
Part C:
4.
(c) Einstein would interpret these results by saying that when 8.9x10-11 s
elapses in the detector frame, less time elapses for the kaons. While the kaons
don’t feel it, time passes more slowly for them relative to the detector. The
kaons still have time to travel further (in fact, 10 times as far) before they decay.
(b) The kaons are travelling 10 times further than they “should”—27 cm
versus 2.7 cm. According to Newton, when 8.9x10-11 s elapses in the detector
frame, the same amount of time should have elapsed for the kaons. They
“should” decay after travelling only 2.7 cm.
39
b) (c)
4. Giving more energy to a very fast
moving particle has little effect
on its speed. If we catch up with
the particle, giving it energy will
dramatically increase its speed in this
new reference frame, but this speed
increase is small as seen in the frame
of the accelerator. (See #2.) As in
#3, time dilation causes the effective
inertia of the particle to increase as its
speed approaches c.
0
25
50
75
100
0
16.7
Energy (GeV)
8.3
25.0
3. One second for Alice is γ times one second for Bob. He will need to apply
the force for a longer time, according to his clock, to have the same effect
on Alice. Alternatively, if he applies a force F for some time according to his
clock, Alice will claim that he applied that force for less time. Time dilation
makes his effort to accelerate Alice less effective than he thinks!
2. Bob will measure Alice’s change in speed to be less than 1 m/s—her speed
change will be reduced by a factor of γ2. As the speed of Alice’s rocket
approaches c, γ becomes larger and larger so her changes in speed will
appear to Bob to be less and less. That “nothing goes faster than the speed
of light” is a consequence of the very nature of time and space.
1. (a)
Part D:
will not have gone as far as Bob thinks.
Alice’s perspective, Bob’s space—and everything in it is contracted so she
Bob moving at the same speed at which that Bob sees Alice moving). From
tBob), she must see this distance as less: LAlice < LBob (in order that Alice see
will see Alice travel a distance LBob. Since Alice’s travel time is less (tAlice <
5. Alice and Bob will disagree on how far she travels. During the time tBob, Bob
Speed (%c)
Answers
3. As the brick is heated the atoms will vibrate, much like the balls in #2, so the
inertia of the system will increase and the hot brick will have a greater mass.
2. (a) The moving balls have more effective inertia for sideways deflection,
which means the box will present a greater resistance to upward
acceleration. (It can be shown that this effect is the same for all directions of
acceleration.) The kinetic energy of the balls increases the inertia (or mass)
of the box.
(b) The mass (or inertia) of the box cannot change so the potential energy
stored in the spring must have an equal amount of inertia as the kinetic
energy that was previously stored in the moving balls.
1. (a) The mass of the box cannot change without gaining or losing something
to the surroundings.
(b) The mass of the box cannot change because that would violate both
energy and momentum conservation laws. For example, the kinetic energy of
the moving box is ½ Mv2. The kinetic energy can’t change because there is
no outside force acting on the box. So if v doesn’t change, M can’t change.
Part C:
3. If Alice’s vertical speed in and out is lower than Bob’s, then she must have
more mass (or inertia). Note to teachers: Ignoring the horizontal motion of the
ball may be challenging for students. This difficulty is alleviated in Part C by
effectively putting the ball in a box and letting it move side-to-side very rapidly.
2. It will take Alice longer from your perspective to cover the same vertical
distance, so you see her moving more slowly than she does.
1. A change in your perspective will not affect the speeds observed by Alice
and Bob.
Part B:
3. If two balls of differing speeds collide and maintain their original speeds then
their masses must be different by the same ratio. The slower ball must have
more mass than the faster moving ball.
2. One ball gains speed as a result of the collision so it must have less mass
than the other.
1. The speeds going in and out are equal so the masses must also be equal.
Part A:
SW2:
40
Answers
2.
The speed of the block does not change in its rest frame. The speed of the
block in the moving frame is due to the motion of the observer, and so obviously
it will not change.
1. The protons are travelling so fast that there is a huge increase in their effective
inertia for sideways deflection (γ=7500). They need a magnetic field of over
8 Tesla to bend their trajectories into a circle—this requires superconducting
magnets.
SA:
(b) When all the light has escaped, nothing remains, and the change in the
mass of the box is Δm = 2m, where 2m is the mass of the original particle/
anti-particle pair. Using ΔE = Δmc2, the change in the energy of the box is
ΔE = (2m)c2. But since energy is conserved, we must have ΔE = E,
where E is the energy in the escaped light. So E = (2m)c2 in this case. In
general, an object with mass m at rest has energy E=mc2. Even mass at
rest has energy!
1. (a) The mass of the box will not change. The inertia of the light must be the
same as the mass of the particle/anti-particle pair. Mass is conserved.
Note to teachers: While students might not make the connection, the fact
that the weight of the box also does not change follows from the nature
of gravity: the weigh scale is accelerating up in curved spacetime, and is
measuring the resistance of the box to acceleration!
Part D:
4. The box contains the brick and the light (the light is not being added from
outside the box) so the total mass of the box cannot change when the brick
absorbs the light. According to #3 the warmer brick has more mass, which
means that light must have inertia! A box with light bouncing around inside
will resist acceleration more than the same box when it’s empty.
Even after 5 billion years the amount of mass lost by the Sun is negligible.
(c) Einstein showed that even when an object is at rest and has no
kinetic energy (½mv2=0), it still has energy—a new kind of energy
called “rest energy”.
(b) Binomial Expression:
4. (a)
(c)
(b)
3. (a)
This thought experiment shows that adding an amount of energy E to a system
will increase the mass of the system by E/c2. As a general formula we write
ΔE = Δmc2. A change in the energy of a system is always accompanied by a
corresponding change in its mass. Energy has inertia.
CREDITS
Authors Executive Producer
Scientific Advisor
Dr. Richard Epp Manager of Educational Outreach Perimeter Institute for Theoretical Physics Greg Dick
Director of Educational Outreach
Perimeter Institute for Theoretical Physics
Dr. Niayesh Afshordi
Associate Faculty
Perimeter Institute for Theoretical Physics
Dave Fish Senior Physics Teacher
Sir John A. Macdonald Secondary School Educational Advisors
Joan Crawford Glenview Park Secondary School Cambridge, Ontario
Rob Crawford
Turner Fenton Secondary School
Brampton, Ontario
Philip Freeman Richmond Secondary School Richmond, British Columbia
Olga Michalopoulos
Georgetown District High School
Georgetown, Ontario
Dennis Mercier Turner Fenton Secondary School Brampton, Ontario
Dr. Damian Pope
Manager of Educational Outreach Perimeter Institute for Theoretical Physics
Melissa Reist Elizabeth Ziegler Public School
Waterloo, Ontario
David Vrolyk
Sir John A. Macdonald Secondary School
Waterloo, Ontario
Brandon Lloyd PIP Animation Services Director & Design Justin Aresta
PIP Animation Services
Design
Cynthia DeWit Graphic Design
Sara Leblanc
Illustrations
Kevin Donkers Preston High School Cambridge, Ontario
Elisa Gatz
Sterling High School
Sterling, Illinois
Animation Production
Frank Taylor Title Entertainment Executive Producer Document Production
Susan Fish Editor Corporate Sponsor
Steve Kelly
DVD Authoring