3. +
+
What do you think?
• Imagine a mass moving back
and forth on a spring as
shown. At which positions (A,
B, or C) are each of the
following quantities the
greatest and the least?
• Force acting on the block
• Velocity of the block
• Acceleration of the block
• Kinetic energy
• Potential energy
• Mechanical energy
4. +
+
Hooke’s Law
F elastic
is the force restoring the spring to the
equilibrium position.
A minus sign is needed because force (F) and
displacement (x) are in opposite directions.
k is the spring constant in N/m.
k measures the strength of the spring.
5. +
Classroom Practice Problem
If
a mass of 0.55kg is attached to a
spring and stretches 2cm. What is the
spring constant?
m= 0.55kg x= 0.02 m
g= 10m/s2 k= ??
Felastic=kx and we know that Fg=mg
So…. mg=kx
6. +
Classroom Practice Problem
mg
k=
x
(0.55)( )
10
0.02
k= 275 N/m
7. +
+
Classroom Practice Problem
A slingshot consists of two rubber bands
that approximate a spring. The equivalent
spring constant for the two rubber bands
3
combined is 1.25 × 10 N/m. How much
force is exerted on a ball bearing in the
leather cup if the rubber bands are
stretched a distance of 2.50 cm?
Answer: 31.2 N
8. +
+
Simple Harmonic Motion
Simple harmonic motion results from
systems that obey Hooke’s law.
SHM is a back and forth motion that obeys
certain rules for velocity and acceleration
based on F = -kx.
9. +
+
The Simple Pendulum
The pendulum shown has a
restoring force F g,x.
A component of the force of
gravity
At small angles, F g,x is
proportional to the
displacement (θ), so the
pendulum obeys Hooke’s
law.
Simple harmonic motion
occurs.
10. +
+
Now what do you think?
• Imagine a mass moving back
and forth on a spring as
shown. At which positions (A,
B, or C) are each of the
following quantities the
greatest and the least?
• Force acting on the block
• Velocity of the block
• Acceleration of the block
• Kinetic energy
• Potential energy
• Mechanical energy
12. +
+
What do you think?
• The grandfather clock in the hallway
operates with a pendulum. It is a beautiful
clock, but it is running a little slow. You
need to make an adjustment.
• List anything you could change to correct the
problem.
• How would you change it?
• Which of the possible changes listed would
you use to correct the problem? Why?
13. +
+
Measuring Simple Harmonic Motion
Amplitude (A) is the maximum displacement
from equilibrium.
SI unit: meters (m) or radians (rad)
Period (T) is the time for one complete
cycle.
SI unit: seconds (s)
Frequency (f) is the number of cycles in a
unit of time.
SI unit: cycles per second (cycles/s) or s-1 or Hertz (Hz)
Relationship between period and frequency:
1 1
f = T=
T f
14. +
+
Period of a Simple Pendulum
Simple pendulums
small angles (<15° )
The period (T) depends only on the length (L) and
the value for ag.
Mass does not affect the period.
All masses accelerate at the same rate.
15. +
Classroom Practice Problem
You need to know the height of a tower,
but darkness obstructs the ceiling. You
note that a pendulum extending from
the ceiling almost touches the floor and
that it’s period is 12 sec. How tall is the
tower?
T= 12 sec ag= 10 m/s 2
L= ??
16. +
Classroom Practice Problem
Lookto the board for how to rearrange
the equation.
2
( 2 )( 0
1 1)
L= 2
4∏
L= 36.48m
17. +
+
Classroom Practice Problems
What is the period of a 3.98-m-long
pendulum? What is the period and
frequency of a 99.4-cm-long pendulum?
Answers: 4.00 s, 2.00 s, and 0.500 s-1 (0.500/
s or 0.500 Hz)
A desktop toy pendulum swings back
and forth once every 1.0 s. How long is
this pendulum?
Answer: 0.25 m
18. +
+
Period of a Mass-Spring System
Greater spring constants = shorter periods
Stiffersprings provide greater force (Felastic = -kx) and
therefore greater accelerations.
Greater masses = longer periods
Large masses accelerate more slowly.
19. +
Classroom Practice Problem
The body of a 1275kg car is supported
on a frame by 4 springs. 2 people riding
have a total mass of 153kg when driven
over a pothole in the rode; the frame
vibrates with a period of 0.840sec. Find
the spring constant of a single spring.
m=(1275+153)/4= 357kg
T= 0.840sec k=??
20. +
Classroom Practice Problem
2
4∏ m
k= 2
T
2
4∏ (357)
2
(0.840 )
k= 19974.20N/m
21. +
+
Classroom Practice Problems
What is the free-fall acceleration at a location
where a 6.00-m-long pendulum swings exactly
100 cycles in 492 s?
Answer: 9.79 m/s2
A1.0 kg mass attached to one end of a spring
completes one oscillation every 2.0 s. Find the
spring constant.
Answer: 9.9 N/m
22. +
+
Now what do you think?
• The grandfather clock in the hallway operates with a
pendulum. It is a beautiful clock, but it is running a
little slow. You need to make an adjustment.
• List anything you could change to correct the
problem.
• How would you change it?
• Which of the possible changes listed would you use
to correct the problem? Why?
24. +
+
What do you think?
• Consider different types of waves, such as
water waves, sound waves, and light waves.
What could be done to increase the speed of
any one of these waves? Consider the
choices below.
• Change the size of the wave? If so, in what way?
• Change the frequency of the waves? If so, in what
way?
• Change the material through which the wave is
traveling? If so, in what way?
25. +
+
Wave Motion
A wave is a disturbance that propagates
through a medium.
What is the meaning of the three italicized
terms?
Apply each word to a wave created when a child
jumps into a swimming pool.
Mechanical waves require a medium.
Electromagnetic waves (light, X rays,
etc.) can travel through a vacuum.
26. +
+
Wave Types
The wave shown is a pulse wave.
Starts with a single disturbance
Repeated disturbances produce periodic
waves.
27. +
+
Wave Types
If a wave begins with a disturbance that is SHM, the
wave will be a sine wave.
If the wave in the diagram is moving to the right, in
which direction is the red dot moving in each case?
28. +
+
Transverse Waves
A wave in which the particles move perpendicular to
the direction the wave is traveling
Thedisplacement-position graph below shows the
wavelength (λ) and amplitude (A).
29. +
+
Longitudinal Wave
A wave in which the particles move parallel to the
direction the wave is traveling.
Sometime called a pressure wave
Try sketching a graph of density vs. position for the
spring shown below.
30. +
+
Wave Speed
Usethe definition of speed to determine the
speed of a wave in terms of frequency and
wavelength.
A wave travels a distance of one wavelength (λ) in
the time of one period (T), so,
Because frequency is inversely related to period:
v=fλ
31. +
+
Wave Speed
SI -1
unit: s × m = m/s
The speed is constant for any given medium.
If f increases, λ decreases proportionally.
Wavelength (λ) is determined by frequency and
speed.
Speed only changes if the medium changes.
Hot air compared to cold air
Deep water compared to shallow water
32. +
+
Waves Transfer Energy
Waves transfer energy from one point to
another while the medium remains in
place.
A diver loses his KE when striking the water but the wave
carries the energy to the sides of the pool.
Wave energy depends on the amplitude
of the wave.
Energy is proportional to the square of the amplitude.
If the amplitude is doubled, by what factor does the energy
increase?
Answer: by a factor of four
33. +
+
Now what do you think?
Consider different types of waves, such as
water waves, sound waves, and light waves.
What could be done to increase the speed of
any one of these waves? Consider the
choices below.
Change the size of the wave? If so, in what way?
Change the frequency of the waves? If so, in what
way?
Change the material through which the wave is
traveling? If so, in what way?
35. +
+
What do you think?
• Imagine two water waves traveling toward each other
in a swimming pool. Describe the behavior of the two
waves when they meet and afterward by considering
the following questions.
• Do they reflect off each other and reverse
direction?
• Do they travel through each other and continue?
• At the point where they meet, does it appear
that only one wave is present, or can both
waves be seen?
• How would your answers change for a crest
meeting a trough?
36. +
+
Wave Interference
Superposition is the combination of two
overlapping waves.
Waves can occupy the same space at the
same time.
The observed wave is the combination of the
two waves.
Waves pass through each other after forming
the composite wave.
37. +
+
Constructive Interference
Superposition of waves that produces a resultant
wave greater than the components
Both waves have displacements in the same direction.
38. +
+
Destructive Interference
Superposition of waves that produces a resultant
wave smaller than the components
The component waves have displacements in opposite
directions.
39. +
+
Reflection: Free End
The diagram shows a wave
reflecting from an end that is
free to move up and down.
The reflected pulse is
upright.
Itis produced in the same way as
the original pulse.
40. +
+
Reflection: Fixed End
This pulse is reflected
from a fixed boundary.
The pulse is inverted upon
reflection.
The fixed end pulls downward
on the rope.
41. +
+
Standing Waves
Standing waves are produced when two
identical waves travel in opposite
directions and interfere.
Interference alternates between constructive
and destructive.
Nodes are points where interference is
always destructive.
Antinodes are points between the nodes
with maximum displacement.
42. +
+
Standing Waves
A string with both ends fixed
produces standing waves.
Only certain frequencies are possible.
The one-loop wave (b) has a
wavelength of 2L.
The two-loop wave (c) has a
wavelength of L.
What is the wavelength of
the three-loop wave (d)?
2/3L
43. +
+
What do you think?
• Imagine two water waves traveling toward each other
in a swimming pool. Describe the behavior of the two
waves when they meet and afterward by considering
the following questions.
• Do they reflect off each other and reverse
direction?
• Do they travel through each other and continue?
• At the point where they meet, does it appear
that only one wave is present or can both waves
be seen?
• How would your answers change if it was a
crest and a trough?
Hinweis der Redaktion
When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students may recall the information on energy from previous work. They have studied all of the material necessary to answer these questions correctly. However, it is sometimes difficult to sort out the meaning of all these terms and apply prior knowledge to a new example. For example, even though they know F = ma, it is likely some will say the force is greatest at the ends (A and C) and the acceleration is zero at these same points. This may result from confusion regarding acceleration and velocity. Listen and seek clarification of their ideas.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful. Use F = -kx and make sure students are careful about the units (cm and m).
The next slide works through a detailed example of SHM for a mass-spring system.
The next slide reinforces the dependence on small angles by having students calculate displacements for small and large angles.
Students should now be able to answer these quickly and even explain their answers based on Hooke’s law, Newton’s 2nd law, and the conservation of energy. Try to connect the answers to these three rules so students understand WHY the force, acceleration, velocity, KE, PE, and ME act as they do.
When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students are likely to believe that the mass, the angle of the swing (or displacement), and the length will all affect the period. If you hear these responses, be sure you ask them how each factor alters the period. For instance, do larger masses increase the amount of time or decrease it?
It is important to stress the difference between frequency and period. One is seconds/cycle and the other is cycles per second, so each is the inverse of the other. In order to keep units straight when using frequency, the easiest unit is s-1. When giving students frequencies, instead of saying the frequency is 20 Hz or 20 cycles per second, it is useful to say the frequency is 20 per second. This will get them accustomed to the terminology. When entering the value into equations, they should enter it as 20/second. This facilitates the problem solving aspect. It should eventually become automatic for them to equate Hz with “per second” and vice versa.
Point out to students that the two pictured pendulums have the same angular displacement, so the restoring force is the same for each of the pendulums (Fr = mg sinθ) . Therefore, the accelerations will be the same for both, but since the longer pendulum has a greater distance to travel, it takes more time. With regard to mass, the restoring force is greater if the mass is greater, but since a = F/m, the accelerations will be the same. This is the same situation as falling bodies. Greater masses have more force acting on them. However, since acceleration is the ratio of force to mass, the resulting acceleration is always the same. The fact that the period of a pendulum on Earth only depends on the length is NOT intuitive. Many students believe that the mass and displacement will have a significant effect on the period. Do the following demonstration to help students understand this concept. Set up two pendulums as shown in the diagram, and let each swing from about a 10° angle. Students should see the difference in period for the longer pendulum. Then adjust them so they are the same length and change the mass on one so it is two or three times more massive. This can be done easily by using washers on a paper clip and adding a couple. Students should now see the same period for each pendulum. Now adjust the displacement by pulling one back about 5° and the other about 15°. These values do not need to be carefully measured. Students should see the period is the same for each, regardless of the displacement. Finally, try pulling one back so the displacement is 60° or so and they should see that the periods are no longer the same. That is because a pendulum only satisfies the SHM equations if the displacement is small. It no longer obeys Hooke’s law if the displacement is large. See if students can explain WHY a smaller displacement still requires the same amount of time for one complete cycle.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful. The 2nd part of the first problem asks for frequency as well as period (f = 1/T). Notice on these problems that the period keeps cutting in half (4 --> 2 --> 1) while the lengths are reduced by one-fourth (3.98 ---> 0.994--->0.25). When the unknown is under the radical sign, an easy approach is to simply square both sides of the equation : T2 = 4π2 (L/g).
Point out to students that these are square root relationships. Quadrupling the mass provides a period that is twice as large.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Students should now say that the only two factors that affect the period are the length of the pendulum and the acceleration of gravity. The only one they can easily control is the length, so the solution to the slow clock is to shorten the pendulum so that the period is slightly less.
When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. One common student idea is that changing the size (amplitude) of the wave will affect the speed. If this is mentioned, ask them why it will change the speed and in what way (i.e. Do bigger waves move faster or slower?). Others may expect frequency to affect the speed. Ask them to clarify why they believe that to be true. Facilitate a discussion between those that think one of these factors will affect the speed and those that do not think so.
A disturbance is a change from equilibrium position. This occurs when the child pushes down on the water. Propagates means that the wave continues to travel outward from the disturbance. This occurs as the wave moves toward the other side of the pool. The medium is the material through which the wave travels. In this case, it is the water. Be sure students understand that “through a vacuum” means that no medium is required.
Students may struggle and think, for example, in (a) that the red point is moving downward because it looks like it is going “downhill.” Help them visualize where the rope will be in the next instant. Answers: moving upward (b) stopped but will move downward (c) moving downward (d) stopped but will move upward
Ask students to draw the wave on the left, and to identify in their diagrams the directions of wave motion (a horizontal line) and particle motion (a vertical line). Point out to students that, even though the graph on the right looks like the wave, it is really the measured values for displacement as a function of position. This is important because a subsequent slide will show a longitudinal wave and a graph of density-position, and that graph does not look at all like the wave. Instead, it looks like a transverse wave. It is important that students understand what is being graphed, and that graphs are mathematical models rather than pictoral representations.
After the white rectangle appears on the slide, do not click again until after students have had a chance to draw the graph. At that point, click again to reveal the graph for students. To help students draw the graph, you might show the axis for density and position (it is covered in the diagram to hide the answer). Point out to students that even though the graph looks “transverse,” is really just a mathematical model of the density of the longitudinal wave. It is a good idea to show students both types of waves on a slinky after discussing the definitions of the waves. The spring is most dense (or compressed) at x1 and x3, while it is least dense at x2 and x4. In the next chapter, students will find that sound waves are longitudinal. Later they will study the properties of light waves, which are transverse.
Have students work through the derivation before showing them the steps on this slide. The next slide explores this equation in more detail.
Point out that speed is not in any way dependent on the amplitude. Also, even though the equation makes it appear that speed depends on frequency, it does not. Increasing the frequency does NOT change the speed. Instead, it decreases the wavelength so that the speed remains constant. Students should note that at an outdoor concert, they hear all the instruments at the same time even though they are producing different frequencies and amplitudes. The web site listed below has an excellent simulation of transverse waves. http://phet-web.colorado.edu/web-pages/index.html Choose “Go To Simulations” then choose “Waves and Sound” then “Wave on a string.” Set the simulation for “No End” to avoid reflections and “oscillate” to produce a periodic wave. Turn off damping. Adjust the tension to the center and frequency to 15 so the waves move more slowly and it is easier for students to observe: (1) The particles move up and down in SHM while the wave moves to the right. (2) Increasing the amplitude does not change the wave speed or the wavelength. (Try different amplitudes and observe the speed.) (3) Increasing the frequency does not change the speed but does decrease the wavelength. (Try increasing the frequency and observe the speed and wavelength.)
The only change that will affect the speed is a change in the medium, such as adjusting the tension in a string or spring, changing the temperature of the air, changing the depth of the water, or changing the material through which light waves are traveling. Changing the size (amplitude) or frequency will not affect the wave speed. A change in frequency will cause a change in wavelength to keep the speed constant for a given medium.
When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. After listening to students’ ideas regarding the collision between two waves, move to the next slide. If possible, use the simulation recommended in the notes before beginning the slide. This simulation will allow students to “see” the answers to the questions above, and discover the principle of superposition and observe how it applies to constructive and destructive interference.
IF POSSIBLE, USE THE SIMULATION BELOW BEFORE SHOWING THIS SLIDE. This simulation of superposition, constructive interference, and destructive interference will show students how waves behave when colliding. http://www.phy.ntnu.edu.tw/ntnujava/ Choose “Wave” and then choose “superposition of pulses”. This simulation allows you to see the two individual pulses as they pass through each other, as well as the sum of the two waves. The sum of the waves is the only wave an observer would actually see. You can also invert either wave by simply clicking the mouse underneath the wave. This allows you to demonstrate the addition of two troughs (constructive) or the addition of a crest and a trough (destructive). Stop the simulation shortly before the waves meet and ask the students what will happen to the medium while they overlap and afterward. You can also pause the simulation at any time to discuss the wave behavior.
Walk students through the 4 images (a through d) and show them that the total is just the sum of the two waves. You can use a meter stick and measure the two components and the composite and show they are equivalent.
Walk students through the 4 images (a through d) and show them that the total is still just the sum of the two waves (one is negative). You can use a meter stick and measure the two components and the composite and show they are equivalent.
Before going through this slide, there is an excellent web site that allows you to show reflection from a “fixed” end and from a “free” end. http://phet-web.colorado.edu/web-pages/index.html Choose “Go to Simulations” and then choose “Sound and Waves” and then choose “Wave on a string.” Turn off damping. Choose either fixed end or loose end. Grab the wrench and send a pulse, and observe it as it reflects. Students can be asked what to expect when you switch. See if they can predict how the behavior will change.
If possible, use the web site below prior to this slide. http://www.phy.ntnu.edu.tw/ntnujava/ Choose “Wave” and then choose “Superposition Principle of Wave.” Change the frequency of each wave to 5 (this will be easier to see). Observe the two waves as they overlap. Point out to students that the standing wave pattern is the only thing an observer would see because it is the resultant of the two component waves. The red wave appears to be standing still (not moving right or left). Pause the simulation and show them the nodes (points that never move because the two components always cancel) and the antinodes (points that have the maximum displacement in each direction).
In order to see the wavelengths, trace a single path on the loop so students are not trying to analyze all five lines. Explain that these diagrams are like strobe photos showing five different positions for the wave as it moves.
Students should now understand that the waves pass through each other and form a resultant wave during the process.
