1) Introduction: Explain the theory behind this experiment in a paragraph between 150 and 250 words. (2 Points)
Suppose you are using external resources; include the reference. It would be best if you had any relevant formulas and explanations of each term. You may use the rich formula tools embedded here.
2) Hypothesis: In an If /Then statement, highlight the purpose of the experiment. (1 point)
Post-lab section:
3) Attach an image of your signed data sheet here. (3 Point3)
4) Attach your analysis here, including any table, chart, or plot image. (9 points)
This should include:
Table 1: 1 point
Table 2: 2 points
Table 3: 2 points
Table 4; 2 points
Table 5; 2 points
5) Attach the image of samples of your calculation here. (2 points)
6) In a paragraph between 100 and 150 words, explain what you Learn. What conclusion can you draw from the results of this lab assignment? (2 points)
7) In one sentence, compare the results of the experiment with your Hypothesis. Why? (1 point)
Sine Wave
A simple sine wave traveling along a stretched string at the positionx and timet can be represented by the equation: y1= A sin (ω t + kx) (traveling to the left). (1) The symbols in Eqn. (1) are as follows:
y1 displacement of the particles in the string.→
A amplitude of the vibration of the particles in y direction.→
ω angular frequency, related to the natural frequency→ f of the wave as: ω = 2π f . (2)
k wave number, related the wavelength→ λ of the wave as: λ = 2π / k . (3) The speed of the wave is given by: v = ω/k = λ f . (4) The right traveling wave is represented by the equation: y2 =−A sin(ω t − kx) (traveling to the right). (5)
Standing Wave
If the string is fixed at one end, the wave will be reflected back when it strikes the other end (Fig. 1). The reflected wave will then interfere with the original wave. In this situation we have a left traveling and a right traveling
wave interfering with each other. The resulting wave (assuming the amplitudes of these waves are small enough so that the elastic limit of the string is not exceeded) is a standing wave and is given by: y = y1 + y2= [2 A sin(kx)]cos(ω t) .(6) Nodes: The points of zero displacement
are called nodes. They are marked as “N” in Figure 1. Antinodes: The points of maximum
displacement are called antinodes. They are marked as “A” in Figure 1.
Resonance
At certain frequencies of oscillation, all the reflected waves are in phase, resulting in a very high amplitude standing wave. These frequencies are called resonance frequencies. In general, for a string fixed at both ends, resonance occurs
when the wavelength (λ ) satisfies the condition: λn = 2L /n , →L= nλn/2, n = 1,2,3,4 , … , (7)
whereL is the length of the string containing the standing wave. Then, Eqn. (4) becomes: f = v /λn . (8)
Speed Equations at resonance
For the wave traveling in the string, speed is given by: v = √F /μ , (9)
whereF is the tension (in units ofN ), andμ is the mass per unit length (line mass density) of the string (in units of kg /m ). Also from Eqn. (8), speed at resonance is: v = f λn . (10) Combining Eqn. (10) and Eqn. (9), we have:
F μ = v
2= λn 2 f 2, n=1,2,3, 4, … . (11)
Eqn. (11) can be rearranged in the following two ways:
Measurement #1: F f 2 = μλ2
2 , (n= 2only) . (12)
Measurement #2: Fn
λn 2 = μ f 2 , (n= 2,3, 4,5) . (13)
Here, μ = (string mass)/(string length) . (14)
Measurement #1
We vary the the frequencyf , while keeping the wavelengthλ fixed such that we have 2 segments of the standing wave in the string, and then plotF vs.f 2 . So we havef 2 along the x-axis andF along
the y-axis. We fit a straight line equation to the plot. Then the equation of this fitted straight line will be: F = m f 2 + b . (15) We can rewrite Eqn. (12), as: F = (μλ2
2) f 2 . (16) Comparing Eqn. (15) and Eqn. (16), we then have:
slope : m= μλ2 2 = F / f 2 . (17)
Since there are two segments, we have:λ = λ2 = L . (18) Then, calculated value for quantity in Eqn. (17): μλ2
2 =μ L2 .(19)
F
f 2
slope→μλ2 2
Measurement #2
We vary the the wavelength (# of segments)λn , while keeping the frequency fixed atf = 130Hz , and then plotFn vs.λn
2 . So we haveλn
2 along the x-axis andFn along the y-axis. We fit a straight line equation to the plot. Then the equation of this fitted straight line will be: Fn = mλn
2 + b . (20) We can rewrite Eqn. (13), as: Fn = (μ f 2)λn
2 . (21) From Eqn. (20), we then have:
slope : m= μ f 2= Fn/ λn 2 . (22)
Then, calculated value for quantity in Eqn. (22): μ f 2= μ(130)2 (23)
Fn
λn 2
slope→μ f 2
1
Provided data for Exp 12 and instructions for data analysis and lab report
1. Data to be used in both Measurements #1 and #2 in Exp 12
Table 1 Calculate string mass density ( ) and record it in Table 1
String length (m) String mass (kg) String mass density (kg/m)
6.60 0.00208
Table 2 Calculate wavelength of standing-wave 2, 3, 4, 5for n
n = and record them in Table 2.
( )L m 2
( )m 3
( )m 4
( )m 5
( )m
1.00
n is the # of segments of standing wave.
L is the length of the string between two knots: one is tied to the blade on the string vibrator and the
other knot is on the loop linked to the hook on the force sensor.
2. Provided data from Measurement #1 are given in Table 3
Table 3 Provided data of frequency f (Hz) and tension F (N) from Measurement #1 f (Hz) f 2 (Hz)2 F (N)
60 0.894
85 2.025
104 3.184
120 4.364
134 5.499
147 6.891
3. Instruction for data analysis for Measurement #1 in Exp 12 (a) Calculate f 2 (Hz)2 and record them in Table 3.
(b) Open an excel file, copy all the data in Table 3 and paste them in columns A, B, C of the Excel file.
(c) Plot the data F (N) vs f 2 (Hz)2 in the Excel file (with the chart title, x-axis title and y-axis title labeled).
(d) Fit the data with a linear function y = mx + b where m and b are the fitting parameters of slope and
intercept respectively. The fitting parameters must be displayed in the plot.
You can search on google to find out how to plot and fit data in Excel.
(e) Save the Excel file as “Data analysis for measurement #1 in Exp 12”.
(f) From the fitting parameters to determine the ratio of 2
/F f and record it in Table 4.
(g) Use the value of string mass density ( ) in Table 1 and wavelength ( 2
) in Table 2 to calculate
2
2 (with units) and record it in Table 4.
(h) Calculate the percentage error between 2
/F f and 2
2 , record it in Table 4.
Table 4 Comparison of 2
/F f with 2
2 .
2 /F f Calculated
2
2 % error
2
4. Provided data from Measurement #2 are given in Table 5
Table 5 Provided data of frequency f and tension , 2, 3, 4, 5 n
F n =
f (Hz) # of segments n
(m) 2
n (m2) n
F (N)
130 2 5.282
130 3 2.083
130 4 0.959
130 5 0.437
5. Instruction for data analysis for Measurement #2 in Exp 12
(a) Copy n
for 2, 3, 4, 5n = from Table 2 and record them in Table 5.
(b) Calculate 2
n and record them in Table 5.
(c) Copy the data of 2
n and
n F from Table 5, paste them in columns A and B of a new Excel file.
(d) Plot the data n
F (N) vs 2
n (m2) in the Excel file (with the chart title, x-axis title and y-axis title labeled).
(e) Fit the data with a linear function y = mx + b where m and b are the fitting parameters: slope and intercept
respectively. The fitting parameters must be displayed in the plot.
(f) Save the Excel file as “Data analysis for measurement #2 in Exp 12”.
(g) From the fitting parameters to determine the ratio of 2
/ n n
F and record it in Table 6.
(h) Use the value of string mass density ( ) in Table 1 and the value of f in Table 5 to calculate 2
f (with
units) and record it in Table 6.
(i) Calculate the percentage error between 2
/ n n
F and 2
f , record it in Table 6.
Table 6 Comparison of 2
/ n n
F with 2
f
2 /
n n F
2 f % error
6. Instructions for lab report of Exp 12 (a) Two Excel files “Data analysis for measurement #1 in Exp 12” and “Data analysis for measurement #2 in
Exp 12” with required plot and fitting parameters must be included in your Exp 12 report.
(a) Tables 1 to 6 with all the analyzed data must be included in your Exp 12 lab report.
(b) The required other contents and format for your lab report can be found in the syllabus.
(c) Calculated quantities listed in Table 7 must be included in your Exp 12 report.
Use the following two methods to calculate the wave speeds in the string with 2, 3, 4 and 5 segments
in the standing waveforms, and record your results in Table 7.
Method #1: use the data of frequency f and wavelengths n
in Table 5.
Method #2: use the data of tension n
F in Table 5, and the string mass density in Table 1.
Table 7 Wave speed in string with different # of segments in the standing waveforms
n (# of segments) 2 3 4 5
Speed of wave (m/s) from f & n
Speed of wave (m/s) from n
F &
Percentage error %
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