Physics Lab Report 3 Final
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School
University of Alabama *
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Course
520
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
6
Uploaded by BarristerFang1573
Title of Experiment:
Standing Waves in a String
Name:
James Clear
Date Performed:
February 6th, 2023
Date Handed in:
February 20th, 2023
Name of Partner(s):
Alexia Luciano
Introduction (Purpose/Hypothesis/Theory):
The objective of this experiment was to test a theory on different mechanical waves,
specifically those occurring in two types of strings: a thick white elastic string labeled as string 1,
and a thin neon string noted as string 2. The study focused on analyzing the behavior of standing
waves on these strings. Mechanical transverse waves start as vibrations along a string, with
particles of the medium vibrating perpendicular to the direction of motion. The experimental
setup involved utilizing a Wave Driver to oscillate the strings while applying a specific hanging
mass to provide tension. Waves were identified by the presence of antinodes or nodes, with
nodes observed at the ends of the strings or at specific points along their length. Various
equations and formulas were utilized to describe different aspects of the collected data. For
instance, the speed of a standing wave was calculated using the equation v = fλ, where frequency
is multiplied by wavelength. The equation λ = 2 L÷n represents the standing wave formation,
with wavelength equaling twice the length of the string divided by the number of segments (n).
Additionally, the velocity of the standing waves was determined using the equation √FT/μ, where
the square root of string tension is divided by linear mass. By changing the frequency of the
standing wave and measuring the resulting changes in wavelength and speed, the experiment
facilitated accurate data recording. Prior to conducting the experiment, it was hypothesized that
increasing the frequency would lead to a decrease in wavelength and velocity for both string 1
and string 2. Furthermore, for the second part of the experiment involving changing the mass, the
hypothesis under investigation suggested that a higher mass would result in a greater frequency
value for string 2.
Data
STRING 1 (thick white elastic string)
Mass on end of string: 0.3 kg
|
Tension in string: 2.94 N
|
L = 2.07 m
n
f
(hz)
Uncertainty of
f?
(hz)
λ (m)
v
(m/s)
1
9.3
0.6
4.14
38.51
2
16.3
0.2
2.07
33.74
3
24.3
0.3
1.38
33.53
4
32.3
0.2
1.04
33.43
5
39.3
0.3
0.83
33.62
6
48.3
0.4
0.69
33.33
7
55.3
0.6
0.59
32.62
8
63.3
0.4
0.52
32.91
9
71.3
0.3
0.46
32.79
STRING 2 (thin neon string or green-and-white braided string)
Mass on end of string: 0.3 kg
Tension in string: 2.93 N
L = 2.07 m
n
f
(hz)
Uncertainty in
f?
(hz)
λ (m)
v
(m/s)
1
28.9
+/- 1.9
4.14
119.23
2
59.6
+/- 1.2
2.07
123.37
3
87.1
+/- 0.9
1.38
119.92
4
113.9
+/- 0.7
1.04
118.45
5
141.7
+/- 0.8
0.83
118.44
6
171.1
+/- 0.1
0.69
118.06
Average
v
: 119.58 m/s | σ
v
= 1.97 m/s
Part 2
Length of vibrating string:
2.07 m
Hanging mass (kg)
Frequency.
f
(hz)
f
2
(Hz
2
)
0.100
66.1
4369.21
0.150
82.2
6756.84
0.200
93.1
8667.61
0.250
103.2
10629.61
0.300
113.4
need this
0.350
122.4
14981.76
0.400
131.4
17265.96
0.450
139.1
19348.81
Sample Calculations
Part 1:
1.
Using the wave speed equation, calculate the speed of the wave in your string for
n
= 1.
-
v = f λ
-
v = 9.3 Hz × 4.14 m
-
v = 38.51 m/s
2.
Calculate the average speed of all values of n.
-
v
avg
= 1/n (v
1
+ v
2
+ v
3
+ v
4
….v
9
)
-
v = (38.51 + 33.74 + 33.53 + 33.43 + 33.62 + 33.33 + 33.62 + 32.91 + 32.79)/9
-
v
avg
= 33.94 m/s
3.
Calculate the standard deviation in the speed.
-
σ
n-1
=
√
(1/n-1)∑(x
i
- x)
-
σ
n-1
=
√
25.94/8
-
σ
n-1
=
1.80 m/s
4.
Calculate
u
for each string.
-
u
= mass between knots / length between knots tensions
-
u
= 0.0128/2.68
-
String 1
u
= 4.7 x 10
-3
kg/m
-
u
= 0.000889/2.93
-
String 2
u
= 3.04 x 10
-4
kg/m
5.
Calculate your percent differences.
-
String 1: Average wave speed: 33.94 m/s | Average theoretical wave speed: 25.01 m/s
-
String 2: Average wave speed: 119.58 m/s | Average theoretical wave speed: 4.91 m/s
-
Percent Difference Equation: (E
1
- E
2
)/ (E
1
+ E
2
)/2 x 100
-
String 1 Percent Difference = 30%
|
String 2 Percent Difference = 184%
6.
Calculate an experimental value for the linear mass density,
u
, of the string. What is the
percent difference? (Number 10 in Manual)
-
Slope of graph = 4g /
u
L
2
= 42,518
-
Experimental
u
= (4g / slope) / 1.91
2
|
Experimental
u
= 2.61 x 10
-4
kg/m
-
String 2
u
= 3.04 x 10
-4
kg/m
-
Calculated Percent Difference (using percent difference equation): 26.7%
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Figure 1 (Graph):
This graph shows a graph of the frequency squared,
f
2
, vs. the hanging mass
(kg). The data on the graph illustrates a linear line and a slope of the line of 42,518.
Questions/Conclusions:
2.
Is the speed fairly consistent for all your values of n? If so, what type of relationship does
that suggest exists between frequency and wavelength of a wave? Does the speed of a
standing wave in a string depend upon the frequency of vibration of the string or the
wavelength of the wave? Explain.
-
Yes, the speed demonstrates a fairly consistent value across all of the nine different
values of n. As the frequency rises on the device, the wavelength would decrease in direct
proportion to that. Likewise with velocity, which relies on frequency and wavelength,
remains steady due to the inverse relationship, with more vibration of the string the
wavelength decreases, frequency went higher, but speed remained relatively the same.
4.
Calculate the theoretical value for the speed of the wave in each string using equation 2.
-
v =
√
F
t
/
u
|
F
t
=
mass x 9.8 (g)
-
v =
√
2.94/ 4.7 x 10
-3
kg/m = 25.01 m/s
-
String 1 v = 25.01 m/s
-
v =
√
0.00893/ 3.7 x 10
-4
kg/m = 4.91 m/s
-
String 2 v = 4.91 m/s
5.
How do your average speeds calculated from the experiment compare to the theoretical
values found in question 4? Calculate your percent differences. Do the theoretical values
fall within one standard deviation of the average found from your measurements? Are your
results precise?
-
For
String 1
the values of average wave speed and theoretical wave speed fall very
closely to one and other but yield a percent difference of 30%, leading to the results being
somewhat precise. But they are not within one standard deviation. For the
String 2
, the
values are much further away with a very large percent difference of 184%, meaning they
may have not been precise. And are not within one standard deviation.
6.
Which of the two springs has a higher
u
? Given this, which spring would you expect the
waves to travel faster in? Do the speeds you calculated from question 1 corroborate this?
Explain.
-
Of the two strings the thicker white elastic string yielded a higher linear mass density, μ,
which indicates a greater resistance to the change in velocity along the length. But in turn
this suggests a slower velocity due to a higher μ. This means the thin string 2 would show
a higher average wave speed. This theory proved right as the the values received in part 1
for average wave speed were 33.94 m/s for white elastic string 1 and 119.58 m/s for thin
string 2.
Part 2:
8.
Make a graph of
f
2
vs mass, m. Is the graph linear? If so, what is the slope?
-
The graph has a very linear trendline. The slope is also 42,518 which is equal to the
equation 4g /
u
L
2
.
10.
From the slope of your graph and your measured value for L, calculate an experimental
value for the linear mass density,
u
, of the string. (Check with your instructor to be sure
you are going about this correctly). How does this compare to the density you determine for
this string from part 1? What is the percent difference?
-
The calculated linear mass density calculated for the thin String 2 yielded 3.04 x10
-4
kg/m
and the experimental value of linear mass density was calculated at 2.61 x 10
-4
kg/m. The
percent difference between these values is only 26.7%, indicating that the values may not
be exactly the same but are in close to similar range.
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