The Universal Wave Equation Lab Report

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What does the speed of a mechanical wave depend on and how?
Theory Resource:
Data collection:
Use this simulation where you can change various characteristics

Complete the lab report using the template. Create a lab report and a script for a video presentation.

8.4
Determining Wave Speed
In this section, you will learn about the mathematical relationships involved with
wave speed, such as the universal wave equation. You will also learn what factors
influence the speed of a wave; in particular, a sound wave.
The Universal Wave Equation
Imagine you are standing on a dock on a lake so that you are able to observe the passing
waves. (You can assume that you have all the equipment necessary to allow you to
observe and measure properties of waves, such as distance and time.) First, by timing
the duration between crests passing your reference point, you can measure the period
of the wave. Then you can take a picture of the waveform and measure the wavelength
using the dock and other structures as distance references. From these measurements,
you can calculate wave speed using the kinematics definition for average speed.
l
v5
T
Using the fact that frequency is the reciprocal of period, a substitution can be made
for T in the wave speed equation:
Learning Tip
1
T
f5
Reciprocal
A reciprocal is a number that you
multiply by so that the result equals
1. For example, the reciprocal of 4 is
1
1
4 because 4 × 4 5 1.
v5
l
1
f
v 5 fl
universal wave equation v 5 f l
This important relationship is called the universal wave equation, and it is valid for all
waves and wave types.
The universal wave equation can also be derived as follows:
cycles
time
distance
wavelength 1l2 5
cycles
frequency 1f 2 5
cycles
distance
3
time
cycles
distance
5
time
frequency 1f 2 3 wavelength 1l2 5
Investigation
8.4.1
Investigating Two-Dimensional
Wave Motion (p. 403)
In this investigation, you will predict
the relationships between frequency,
speed, and wavelength.
Hence
5 wave speed 1v 2
v 5 fl
Tutorial 1 demonstrates how wave speed can be calculated using the universal wave
equation.
Tutorial 1 Using the Universal Wave Equation
Sample Problem 1: Calculating Wave Speed
A harp string supports a wave with a wavelength of 2.3 m and a frequency of 220.0 Hz.
Calculate its wave speed.
Given: l 5 2.3 m; f 5 220.0 Hz
Required: v
388   Chapter 8 • Vibrations and Waves
NEL
Analysis: In this example, both l and f are given. So, to solve this problem, substitute for
the variables and calculate the answer using the universal wave equation: v = f l
Solution:
v 5 fl
5 1220.0 Hz2 12.3 m2
v 5 506 m/s
Statement: The wave speed on the harp string is 506 m/s.
Sample Problem 2: Calculating Wavelength
A trumpet produces a sound wave that is observed travelling at 350 m/s with a frequency
of 1046.50 Hz. Calculate the wavelength of the sound wave.
Given: v 5 350 m/s; f 5 1046.50 Hz
Required: l
Analysis: Rearrange the universal wave equation to solve for wavelength: v 5 fl
Solution: v 5 fl
l5
5
v
f
350 m/s
1046.50 Hz
l 5 0.33 m
Statement: The wavelength of the sound wave coming from the trumpet is 0.33 m.
Practice
1. If a wave has a frequency of 230 Hz and a wavelength of 2.3 m, what is its speed? T/I
[ans: 530 m/s]
2. If a wave has a speed of 1500 m/s and a frequency of 11 Hz, what is its wavelength? T/I
[ans: 140 m]
3. If a wave has a speed of 405 m/s and a wavelength of 2.0 m, what is its frequency? T/I
[ans: 2.0 3 102 Hz]
(a)
Factors That Affect Wave Speed
The transfer of energy using waves is more efficient if the particle vibrations do not
absorb much energy. For example, a more rigid object such as a soccer ball tends to
bounce more effectively if it is fully inflated. If the atoms comprising an object are
linked by strong intermolecular forces, the wave energy is transmitted more efficiently and thus the wave speed is faster. If these forces are not as strong, then energy
transmission is less efficient and thus slower.
Temperature
In the case of gases, you might think that cooler gases are more effective at transmitting sound because they are denser. However, usually the converse is true because,
with an increase in temperature, the molecules move faster and transfer their kinetic
energy more efficiently (Figure 1).
NEL
Ontario Physics 11 U
0176504338
C08-F15a-OP11USB
FN
(b)
Figure 1 Comparing transmission of
sound through (a) a cool gas and (b) a
warm gas. The warm molecules jostle
neighbouring molecules more rapidly,
thus increasing the rate of sound energy
transfer.
8.4 Determining Wave Speed   389
Linear Density and Tension
linear density (m) the mass per unit
distance of a string; units are kilograms
per metre (kg/m)
The speed of a wave along a string, such as a violin or guitar string, is governed by the
properties of the string (Figure 2). A string’s linear density, or mass per unit distance,
determines how much force it will take to make the string vibrate. Linear density, m,
is calculated using the equation
m5
Figure 2 The diameters of the
guitar strings shown here are getting
progressively larger from left to right.
The linear density is therefore increasing
from left to right. The speed of sound is
progressively slower in these strings.
m
L
where m is the mass of the string, in kilograms, and L is its length, in metres.
Another variable affecting wave speed is tension. A loose string, for example, will
quickly absorb all of the energy. A taut (tight) string, however, will transmit energy
very effectively. Linear density and tension are the only variables that control the
speed that waves can travel along a string. The equation for the speed of a wave along
a string is
v5
FT
Åm
where FT is the tension in the string (in newtons) and m is the linear density (in
kilograms per metre). In Tutorial 2 we will demonstrate how this equation is used to
determine the properties of a string.
Tutorial 2 Determining String Properties
Sample Problem 1: Determining String Tension
On your class wave machine, you have a string of mass 350 g and length 2.3 m. You
would like to send a wave along this string at a speed of 50.0 m/s. What must the
tension of the string be?
Given: m 5 350 g or 0.350 kg; L 5 2.3 m; v 5 50.0 m/s
Required: FT
Analysis: First, calculate the linear density, m. Second, rearrange the equation for the
speed of a wave on a string to solve for the tension, F T: m 5
Solution:
m5
5
m
L
FT
m
;v5
L
Åm
0.350 kg
2.3 m
m 5 0.152 kg/m 1one extra digit carried2
v5
v2 5
FT
Åm
FT
m
FT 5 v 2m
5 150.0 m/s2 2 10.152 kg/m2
5 380.4 N
FT 5 380 N
Statement: The required tension of the string on the wave machine is 380 N.
390   Chapter 8 • Vibrations and Waves
NEL
Practice
1. If a 2.5 m long string on the same wave machine has a tension of 240 N, and the wave
speed is 300 m/s, what is the mass of the string? T/i [ans: 6.7 3 10–3 kg]
2. If a wave machine string has a linear density of 0.2 kg/m and a wave speed of 200 m/s,
what tension is required? T/i [ans: 8 3 103 N]
3. If a string on a wave machine has a linear density of 0.011 kg/m and a tension of 250 N,
what is the wave speed? T/i [ans: 1.5 3 102 m/s]
8.4 Summary
• The universal wave equation, v 5 fl, relates the speed of a wave to its
frequency and wavelength. The universal wave equation applies to all waves.
• More rigid intermolecular forces allow for a faster transfer of energy, and
therefore a higher wave speed in a medium.
• Waves travel faster in hotter gases than in cooler gases because of the increased
molecular motion caused by the higher temperature in a hotter gas.
• The speed of a wave on a string depends on the linear density of the string
How could you apply your understanding
of the speed of sound in different
materials to the Unit Task on page 486?
and the string’s tension: v 5
FT
Åm
8.4 Questions
1. A wave has a speed of 123 m/s and a frequency of 230 Hz.
What is its wavelength? T/i
earthquake zone
2. A guitar string has a tension of 37 N. The linear density is
0.03 g/m. What is the speed of sound along this string? T/i
3. The period of a sound wave from a piano is 1.20 3 10–3 s.
If the speed of the wave in the air is 3.40 3 102 m/s, what
is its wavelength? T/i
4. Earthquakes produce seismic waves, which travel through
Earth. Primary waves, or P-waves, are longitudinal. They
can travel through both solids and liquids. Secondary
waves, or S-waves, are transverse. They can travel through
solids only. P-waves travel at approximately 8.0 km/s,
and S-waves travel at approximately 4.5 km/s. Following
an earthquake, vibrations are recorded at seismological
stations around the world. K/U T/i A
(a) Calculate how long P-waves and S-waves take to travel
from an earthquake to a seismological station that is
(b) Why do you think that transverse waves are called
secondary waves?
(c) By referring to Figure 3, explain how observing P-waves
and S-waves helps geophysicists analyze the structure
of Earth’s interior.
5. Predict what happens to the wavelength of a wave on a
string when the frequency is doubled. Assume that the
tension in the string remains the same. Confirm your
prediction mathematically. K/U T/i
NEL
solid inner core
liquid outer core
mantle
P-waves
S-waves
Figure 3
6. Predict what happens to the speed of a wave on a string
when the frequency is doubled. Assume that the tension
in the string remains the same. Confirm your prediction
mathematically. K/U T/i
7. By what factor would you have to multiply the tension in
a taut spring in order to double the wave speed? Confirm
8. Develop the equation for wave speed on a string. Use
research if you wish. T/i C
8.4 Determining Wave Speed
391
SPH3U1 – Free Falling Motion Lab Creation Guide
Introduction: Given that the kinematics lab is your first lab you may be struggling with what to
actually do for the lab. Use the guide and the evaluation rubric to produce an excellent quality
lab report!
Section Title
MAX #
Slides
Guidance
Title
1
Give a title to your lab that indicates what relationship you
investigated. Include your name and the date.
Introduction
5
The first part of your introduction serves to introduce the
reader of the report to the concept being investigated. It
should include some real world applications and definitions of
key terms that will be used in the lab. You should include
some visual elements such as pictures or videos to support
your description of the background knowledge. You should
then describe what you saw in this lab.
So for this lab, you need to include a brief summary of free fall
motion that is supported with visuals and a description of what
we observed (had we done the lab in real life).
The second part is your intro is your table of quantities. This is
a description of all the variables in the environment and that
were measured that affect the lab. Remember there will only
be ONE independent variable, and ONE dependent variable
variable types.
(given in the lab description) and your stated hypothesis with a
rationale (why you believe what the results will be).
Procedure
2
● point format
● third person
● past tense
● refer to diagrams/videos for the important steps
● focus on safety
● apparatus setup
● controlling quantities
● measuring quantities
You will write your procedure as if you performed the
experiment IN REAL LIFE! There is no need to mention the
virtual setup such as “open lab data in Brightspace”.
Table of
Observations
1
This is where the work comes in!
Based on our testable question you need to create a data
table of time and vertical displacement.
To find time: We know that between each data point (red
diamond) 9 frames of video have passed. Every 240 frames of
video is 1.0 seconds of real time. So how long does 9 frames
of video take? Once you’ve figured this out you will be able to
easily determine the total time elapsed for each data point.
To find displacement: We know that each dotted box in the
background is 10.0cm so we can figure out the total downward
displacement for each data point.
You should now have a properly formatted table of
observations that includes units and direction!
Graphs
2
You can plot 3 graphs for this lab but only the last one is
necessary.
Position v Time for the small ball.
● Position v Time for the large ball.
● Position v Time for both balls.
Make sure to give the graphs a title, include axis labels with
units (and direction if applicable) and include a caption with
the equation of the lines produced.
Diagrams and
Videos
1
Include any diagrams and/or videos the you would like to refer
to in your work. Make sure to format them correctly with a
figure reference.
For example you might have the following caption below the
picture of one of the falling balls:
Fig 1 – Vertical displacement of the falling ball was
collected by video capturing 240 frames per second
marked at every 9 frames.
Analysis
2
Show calculations for any values that you can use to compare
in your error section of the lab.
In this lab you can calculate the acceleration of the ball and
compare it to the value of acceleration due to gravity near the
Earth’s surface.
Conclusions
1
diagrams to validate or invalidate it. Indicate future studies that
could be completed to further gather data to further investigate
your testable question or improve the experimental design.
Errors
2
Create a table of errors identifying errors in the lab and
here).
Determine the percentage error for your experimental versus
error calculations).
References
1
Include APA formatted references for any information used in
Lab Report Format for SPH3U
slides. Remember to save your slides in PDF format before you submit them
to the dropbox.
Title page
● Title (specific and is related to the testable question and actual
experiment)
● Date
● Author
● Team (if applicable)
Introduction
Prior knowledge – background information required as knowledge & cited as per APA
requirements
Images to support background information & cited as per APA requirements
Testable Question – this has been provided to you for each lab. Copy directly from the lab
Properly formatted table of quantities affecting your study
Hypothesis (properly formatted with all parts – If…..then…..because….)
Purpose (written in past tense)
Sample of quick introduction (change me)
As you can see in figure 1, the forces affecting the motion are weight and air resistance.
Based on quick observation was noticed that coffee filter moves at constant speed
almost immediately.
Forces include gravity, friction, and applied force. Force causes changes in the
speed or direction of motion. There are four main types of contact forces.(World
Atlas, 2022) The first one, the normal force is when nothing is happening. A book may
be sitting on a table and gravity is pulling it to the ground, but there is the table
underneath it, preventing that from happening. Applied force, is the force that is
applied to move that same book from one position to another. Tension is when two
pulling forces, directly opposite of each other stretch an object (World Atlas, 2022).
Imagine you and your sibling pulling on the book, fighting over it, stretching its covers
and pages until it pulls apart. The final type of contact force is called spring force, and
this is the force that is created by a compressed, stretched spring on any object that is
directly attached to it (World Atlas, 2022).
All motion depends on an object changing position over a period of time. There are
numerous quantities that can affect the motion of an object in this lab.
Ball in
projectile
motion
Horizontal
ramp
Fig. 1 Data was collected using video tracker
and a video with 240 fps.
Meter stick
for
calibrating
video
Table of quantities
Table 1. Quantities affecting or being affected by the vertical falling motion of the coffee filter through air
Quantity
Variable
Unit
Type
Position vertical
y
m
dependent
Position horizontal
x
m
controlled
Time
t
s
independent
shape
D
No units
controlled
mass
m
kg
Controlled
Air density
ρ
Kg/m3
Controlled
Testable Question & Hypothesis
Testable question: How does position depend on time for a coffee filter with constant mass and
shape falling through air near earth’s surface?
Hypothesis: The coffee filter will reach terminal velocity almost immediately since it has a very small mass
and an aerodynamic shape that will be increase the drag force at the very instant it moves. Therefore, the
vertical position of the coffee filter falling through air will increase linearly proportional with time. The position
time graph will yield a linear relationship.
Procedure
Point or paragraph format
Third person
Past tense
Refer to diagrams for crucial steps and setup
Apparatus setup
How quantities are controlled
How is independent quantity changed and measured
How is dependent quantity measured
No other steps are added to the procedure
Table of observation
Table 1 Position of a ball in projectile motion collected using video tracker at 120 fps
Time t (s)
Range x (m)
Vertical position y (m)
0.83
-3.24
1.80
0.87
-3.35
1.71
0.90
-3.42
1.62
0.93
-3.53
1.52
0.97
-3.68
1.41
1.00
-3.80
1.28
1.03
-3.93
1.14
1.07
-4.03
0.98
1.10
-4.15
0.82
1.13
-4.21
0.65
1.17
-4.33
0.47
1.20
-4.45
0.27
Data shows a nonlinear relationship
variable, unit
Table is
indexed
Proper number of
significant digits
Caption
Title
Graphs
Descriptive title
Error bars
Labelled axis
Fig 2. The experimental data produced the following
equation matching the pattern: y = -5.2t2 + 6.3t + 0.1
Caption
Analysis

Paragraph format

Explanation of results

Sample calculation

All calculations in table form

Percent error calculations (if appropriate)
Conclusions

Restate hypothesis

Conclude whether you were able to verify or not your hypothesis

Base your conclusions on the tables, diagrams, videos, graphs, analysis

Classify errors (in a properly formatted table)

Include further studies based on the same setup
Error calculations
Error analysis
Table 3 – Possible errors that can occur in this experiment
Description
Type of error
Solution/Reduction of Effect
Tracking the same point on the ball
Random
This error is due to the irregular shape of the ball and its size.
Possible solutions:
Repeat measurements to increase number of trials
Track a point at the lowest edge to be consistent
Zoom in to be able to get the same point over each measurement
The image of the 3.0 m marker is not
sharp. It is used to calibrate the
whole scene
Systematic
Zoom in to account for the gap between edges and the posts.
Sample of a conclusion
In projectile motion, the hypothesis stated that the motion of the object in the horizontal direction is constant velocity motion
and in the y direction it is acceleration motion because in the horizontal direction there is no force acting on the
object, whereas in the y direction there is a gravitational force acting on it. Therefore, in the x direction we don’t
have acceleration and in the y direction we have acceleration.
According to the data (table 1) and the graph (figure 2) that was obtained, we have a linear relationship (x=4.556t+0.492) between the x
position and time, which indicated that we have a constant velocity motion in the x direction. Whereas, for the
second graph, we got a second order relationship (y=-5.0537t^2 + 5.7317t + 2.5043) which indicated that the motion
in the vertical direction is acceleration motion. The results are consistent with our hypothesis.
The acceleration we got from this experiment is a=10.1 m/s^2 meaning it is greater than what we expected. The errors
made are equivalent to 3%. The results that we obtained verified our hypothesis.
A study of the positions (horizontal and vertical) of a basketball could be performed similar to a canon. In both
apparatuses, the horizontal position has a constant velocity and the vertical component is affected by gravity.
References
● All references listed in alphabetical order by author’s last name (if
available)
● All references include date of copyright and/or date of retrieval
● All references properly formatted as per APA requirements
Please note that any submissions without references will be assigned a zero
Last step
Download the completed report as PDF and submit it in the appropriate
assignment folder