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 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.
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