Experiment 6: Length of the pipe

Objective: The purpose of this lab is to find out the length of the hollow pipe without using ruler. The concept of sound wave will be used. Students will spin the pipe and record the sound pressure respect to time using logger pro. This procedure will be repeated second time with a faster spinning speed and a greater frequency.

                                Figure 1: Procedure, data, and calculations

Data

Data table 1: calculated wavelength and frequency

Spinning Speed	ω(rad/s)	ƒ(Hz)	λ(m)
slow	3859 ± 0.4736	614.18 ± 0.07538	0.558 ± 0.0000002
Fast	5068 ± 1.016	806.60 ± 0.1617	0.425 ± 0.000000249

V= ƒ*λ

(V=343m/s)

Data Table 2: length of the pipe

Spinning Speed	Value of n	Exp. length(m)	Actual length(m)	% Error(%)
slow	3	0.837 ± 0.000102	0.80 ± 0.05	4.6
Fast	4	0.850 ± 0.000169		6.3

% Error= (experimental length-actual length)/actual length*100%

Conclusion:

The purpose of this experiment was to find out the length of an opened hollow pipe using sound waves. The pipe was first spinned in a slower speed, and then in a faster speed. If the harmonics of the slower speed was n, that of the faster speed should be n+1. This relationship was used in calculating the length of the pipe by using f_n=(nv)/(2L). n for the slower speed trial is calculated to be 3, so that for the faster speed trial should be 4. The length was calculated to be 0.837m and 0.850m respectively for the slow and fast speed. The percentage errors for these two trials were acceptable. Sources of error could be the stretching pipe. Both of the experimental lengths had larger results than the actual length. This could be caused by the pipe stretched during spinning.

Experiment 5: Introduction to Sound

Objective: The purpose of this lab is to learn the properties of Sound waves. Students will use LabPro and microphone to make four different recordings. These include two "AAAA" Sound from two different people, one tuning fork sound from striking on a soft object, and one soft tuning fork sound. Student will obtain four graphs and analyze the graphs based on the questions provided in the lab manual.

 Figure 1: The sound wave made by a student with an "AAA" sound.

1. "AAA" Sound
a. This  is a periodic wave. From the graph, students can observe the wave has repeating pattern. When students fit the graph to sine function, they can find the wavelength, frequency, and amplitude.
b. About 2 waves are shown in the sample. One complete wave is from one high peak to the second high peak. In the graph there are three peaks, so there are two waves.
c. The probe collected data as fast as the frames separated on TV.
d. There are two waves in 0.02s. The period of these waves should be 0.01s/wave.
e. The frequency should be 1/T=1/0.01=100Hz
f. lambda=v/f=340/100=3.4m. This can be the length of a table in the classroom.
g. w=2pi*f=2*3.14*100=628rad/s
     A=v/w=340/628=0.541m
h. The number of waves would change and be increased. The time it takes to collect the data will increase too. Period, frequency, wavelength, and amplitude wont change.


 
   Figure 1: The sound wave made by the second student with an "AAA" sound.


2.  There are two waves. The period of these waves is 0.009. Frequency is 111.1Hz. Wavelength is v/f=3.06m. The amplitude of waves for this person is shorter. It is v/w=0.487m.



Figure 3: The sound wave made from a tuning fork by striking it on a soft object.

3. The waves from tuning fork is a sinusoidal wave. There are more waves with less variation. The period is smaller, it is 0.0025 s/wave. The frequency is 400 Hz. The wavelength is 0.85m/s. The amplitude is 0.135.


  



 Figure 4: The soft tuning fork sound.


4. The student hit the tuning fork softer and put the microphone away from the tuning fork. The amplitude is shorter. All others stay the same.

Experiment 4: Standing Waves

Objective: The purpose of this experiment is to study the characteristics of standing waves generated by an external force.

Students will attach a string to the Pasco Variable Frequency Wave Drive. Waves with different wavelength will be generated by the drive. There are two cases in this experiment, case 1 will use a 200g weight as a tension source; case 2 will use a 50g weight. 7 harmonics are measured for each case. Frequency and wavelength of the wave will be changed. The wave speed will be calculated. The data collected from the experiment will be compared to the results calculated by equations. Also, the influence of tension of frequency and wave speeds will be discussed. 

Figure 1:Set up for Case 1 (200g weight)

Figure 2: Wave is generated by the driver.

 

Figure 3:Standing waves were generated. Loop with different wavelength and frequency was formed.

Data:

Mass of the string: 0.00256kg±0.000005

Length of the string: 1.20m±0.005

Tension of the string in case 1: 0.2*9.81=1.96N±0.000049

Tension of the string in case 2: 0.05 *9.81=0.49N±0.000049

µ: (0.00256/2.2)=0.00117kg/m±0.000107

Analysis of the Data

1.      The wavelength of the wave, the frequency, and the value of n:

 Table 1: for 200g of weight (case 1)

n (Loop #)	λ (m)	ƒ1 (Hz)
1	2.40 ± 0.10	18 ± 0.5
2	1.20 ± 0.05	37 ± 0.5
3	0.800 ± 0.033	56 ± 0.5
4	0.600 ± 0.025	76 ± 0.5
5	0.480 ± 0.020	95 ± 0.5
6	0.400 ± 0.017	114 ± 0.5
7	0.343 ± 0.014	133 ± 0.5

Table 2: for 50g of weight (case 2)

n (Loop #)	λ (m)	ƒ2 (Hz)
1	2.40 ± 0.10	8 ± 0.5
2	1.20 ± 0.05	19 ± 0.5
3	0.800 ± 0.033	28 ± 0.5
4	0.600 ± 0.025	38 ± 0.5
5	0.480 ± 0.020	48 ± 0.5
6	0.400 ± 0.017	58 ± 0.5
7	0.343 ± 0.014	70 ± 0.5

     

2. Graph1:  Frequency vs. 1/wavelength for case 1 (200g weight)

 

Wave speed should be equal to the slope of the line, 46.129m/s

Wave speed of a transverse wave in a string:

                v = √(T/μ)

v=40.9m/s±1.87

Percent error= (46.129-40.9)/40.9*100%=12.8%

3.      Graph of frequency vs. 1/wavelength for 50g of weight

Wave speed should be equal to the slope of the line, 24.35m/s

Wave speed of a transverse wave in a string:

                v = √(T/μ)

v=20.5m/s±0.936

Percent error= (24.35-20.5)/20.5*100%=18.8%

4.     Table 3: Ratios of wave speeds for case 1 compare to 2 using different methods:

Methods	v1/v2
Graph	1.892
Calculations	2.00

These ratios are similar to each other. 

5.      Table 4: Measured frequency compared to nf_1

ƒ1 (Hz)	ƒn = nƒ1 (Hz)
18 ± 0.5	18 ± 0.5
37 ± 0.5	36 ± 1.0
56 ± 0.5	54 ± 1.5
76 ± 0.5	72 ± 2.0
95 ± 0.5	90 ± 2.5
114 ± 0.5	108 ± 3.0
133 ± 0.5	126 ± 3.5

 With a higher harmonic, the differences between measured frequencies for case 1 and nf1 become larger. Started from the fourth harmonic, the measured frequencies do not fall within the uncertainties of nf1. The reason for the increased in differences is that waves with shorter wavelength are harder to see, so the frequencies will be increased to obtain more visible loops. Therefore, the frequencies increase.

6..      Ratio of the frequency for case 1 compared to case 2.

n (loops #)	1	2	3	4	5	6	7
ƒ1 / ƒ2	2.3	1.9	2.0	2.0	2.0	2.0	1.9

  The ratio is similar for all harmonic. Ratio of the frequency for case 1 compared to case 2 is 2.

 

Calculations of uncertainty

1)  λ = 2L/n

uλ = √(dλ/dL x uL)² = √(2/n x uL)²

     = √(2/1 x 0.05)² = 0.10 m

        2)  T = mg

uT = √(dT/dm x um)² = √(g x um)²

     = √(9.8 x 0.000005)² = 0.000049 N

       3)  μ = m/L

uμ = √[(∂μ/∂m x um)² + (∂μ/∂L x uL)²] = √[1/L x um)² + (-m/L² x uL)²]

      = √[1/1.20 x 0.000005)² + (-0.00256/1.2² x 0.05)²] = 0.000107kg/m

      4)  v = √(T/μ)

uv = √[(∂v/∂T x uT)² + (∂v/∂μ x uμ)²] = √[(0.5x 1/√μ x 1/√T x uT)² + (-0.5x √T x μ^-3/2 x uμ)²]

            = √[(0.5x 1/√0.00117x 1/√1.96 x  0.000049)² + (-0.5x √1.96 x 0.00117^-3/2 x 0.000107)²]

= √(0.000000262 + 3.5)

            =1.87m/s

Conclusion

The purpose of this experiment was to study the properties of standing waves. The wavelength, number of loops, and frequency of each harmonic were recorded for both case. The measured values were compared to the values calculated using the equations. Two graphs of frequency vs. 1/λ were made，the slopes of the graph were the speed of the waves. This wave speed was compared to the wave speed calculated using the equation v = √(T/μ).The percent error for case 1 is 12.8%, for case two was 18.8%. The relationship of the tension and velocity could be examined by comparing the ratio of wave speeds of case with case 2. The ratios from the graphs or from the calculations were about 2. This means that the when the tension decreased by a factor of 4, the velocity would decreased by a factor 2. This can be proved by the data we collected or by equation v = √(T/μ). Also, the relationship between velocity and frequency could be observed by the ratio of the frequency of the harmonic for case 1 compared to case 2. The ratios were approximately 2 for all the harmonics. This means that the frequency was directly proportional to the velocity. When the velocity was decreased by a factor 2, frequency was decreased by a factor of 2. This could be proved by the data collected or the equation ƒ = v/λ.

There were experimental sources of error that contribute to the inaccuracy of the results. Firstly, when the harmonic became higher, the loop was more difficult to be seen by naked eyes, this could cause inaccuracy on adjusting the correct frequency. Students should use a longer string, so the loops would be easier to see. Secondly, the wavelength of the wave was measured from the point the string tied to the clamp instead of from the driver. The loops became more difficult to see with disturbance from the driver. Students should set the driver as the starting point of the wavelength. Also, the length of the string is measured by a ruler instead of more advanced equipment. Lastly, since the 50g hanger was so light that it kept vibrating when waves were traveled along the string. The vibration could influence the tension and reduce the reliability of the experiment.