Experiment 4- Standing Waves

The prupose of this experiment was to compare the final velcity of the wave when the wavelength and the mass of the string were made to be constant but the frequency and the tension of the spring were changed.


This experiment constisted two cases. For both cases, the mass and the length of the string were set to be constant. However, for the first case the tension was given the mass of0.2(kg), then the tension was changed to have the mass of 1/4 of the first case in the second case. Therefore, the tension of the string would be different in these two cases. In order to get the tension of the string, the equation, T=mg (T=tension(N),m=mass(kg),g=gravitational acceleration(ms²)) was applied. After that, the frequency had to be adjusted in order to obtain the desired loops of the wave (such as one loop, two loops, three loops,...etc). The same process would also be applied to the second case when the tension of the spring was changed.
After all the data were obtained, two equations for calculation the wave speed were both be applied for the results comparison. Theses two equations are :

v=fxλ       v= √(T/μ)

However, in order to find the wavelengh of the wave, the equation λ_n= 2L/n will be used to calculate the wavelengh. Here, L equals to the measured lengh of the string and n equals the numbers of the produced loops.
After all the results are obtained, the ratio of the wave speeds from two different equations need to be compared, and the ratio of the frequency for different hormonic will also be compared between case 1 and case 2.

The set up of this experiment

The function generator for adjusting the frequency

the wave driver which allows the spring to create loops

Weight was hung on the one end of the spring 

loops were created

Data:
chart1:

		Mass of the string			1.41 (g)	Length of the string			110 (cm)
Case one
# of loops	μ (m/lengh)		Tension (N)	Frequency(KHz)		λ [2L/n] (m)	v=f*λ(m/s)	v=√ (T/μ)(m/s)
1	0.00128 ±0.000005		1.96 ±0.005	0.015 ±0.0005		2.20 ±0.005	33.0 ±0.05	39.1 ±0.05
2	0.00128 ±0.000005		1.96 ±0.005	0.037 ±0.0005		1.10 ±0.005	40.7 ±0.05	39.1 ±0.05
3	0.00128 ±0.000005		1.96 ±0.005	0.056 ±0.0005		0.733 ±0.0005	41.1 ±0.05	39.1 ±0.05
4	0.00128 ±0.000005		1.96 ±0.005	0.076 ±0.0005		0.550 ±0.0005	41.8 ±0.05	39.1 ±0.05
5	0.00128 ±0.000005		1.96 ±0.005	0.095 ±0.0005		0.440 ±0.0005	41.8 ±0.05	39.1 ±0.05
6	0.00128 ±0.000005		1.96 ±0.005	0.114 ±0.0005		0.367 ±0.0005	41.8 ±0.05	39.1 ±0.05
7	0.00128 ±0.000005		1.96 ±0.005	0.133 ±0.0005		0.314 ±0.0005	41.8 ±0.05	39.1 ±0.05
Average wave speed (m/s)							40.28 ±0.05	39.1 ±0.05
Case two
1	0.00128 ±0.000005		0.49 ±0.005	0.009 ±0.0005		2.20 ±0.005	19.8 ±0.05	19.6 ±0.05
2	0.00128 ±0.000005		0.49 ±0.005	0.019 ±0.0005		1.10 ±0.005	20.9 ±0.05	19.6 ±0.05
3	0.00128 ±0.000005		0.49 ±0.005	0.028 ±0.0005		0.733 ±0.0005	20.5 ±0.05	19.6 ±0.05
4	0.00128 ±0.000005		0.49 ±0.005	0.038 ±0.0005		0.550 ±0.0005	20.9 ±0.05	19.6 ±0.05
5	0.00128 ±0.000005		0.49 ±0.005	0.048 ±0.0005		0.440 ±0.0005	21.1 ±0.05	19.6 ±0.05
6	0.00128 ±0.000005		0.49 ±0.005	0.058 ±0.0005		0.367 ±0.0005	21.3 ±0.05	19.6 ±0.05
7	0.00128 ±0.000005		0.49 ±0.005	0.070 ±0.0005		0.314 ±0.0005	22.0 ±0.05	19.6 ±0.05
Average wave speed (m/s)							21.0 ±0.05	19.6 ±0.05

The ratio between f200g and f50g
#of loops	1	2	3	4	5	6	7	average
f200g /f50g	1.7	1.9	2.0	2.0	2.0	2.0	1.9	1.9

Graphs:

 

figure 1: case one 

figure 2:　case two

Uncertainty:
Derivation 

v=f*λ=f*2L/n , where f and  L were measured:

v=√(T/μ)=√T/(mass/lengh), where mass and length were measured:

Results of uncertainty:

chart 2:

# of loops	Case1  v=f*λ(m/s)	Case 1 v=√ (T/μ)(m/s)	Case2  v=f*λ(m/s)	Case2 v=√ (T/μ)(m/s)
1	33.0 ±0.015977	19.8 ±0.194	39.1 ±0.009054	19.6 ±0.0972
2	40.7 ±0.037793	20.9 ±0.194	39.1 0.021425	19.6 ±0.0972
3	41.1 ±0.056388	20.5 ±0.194	39.1 0.02935	19.6 ±0.0972
4	41.8 ±0.076286	20.9 ±0.194	39.1 0.039006	19.6 ±0.0972
5	41.8 ±0.095159	21.1 ±0.194	39.1 0.0488	19.6 ±0.0972
6	41.8 ±0.122197	21.3 ±0.194	39.1 0.058664	19.6 ±0.0972
7	41.8 ±0.137033	22.0 ±0.194	39.1 0.070859	19.6 ±0.0972

Percent Error (%)

Case one:
40.28±0.05=40.23~40.33

21.0±0.05=20.95~21.05

Case two:

39.1±0.05=39.05~39.15

19.6±0.05=19.55~19.65

Case one:
v=f*λ
(42.979-40.28)/42.979x100%=6.28%

 v=√ (T/μ)(m/s)
(27.079-21.0)/27.079x100%=22.45%

Case two:
v=f*λ
(42.979-39.1)/42.979x100%=9.04%

 v=√ (T/μ)(m/s)
(27.079-19.6)/27.079x100%=27.62%

Results:

velocity obtained by the slope:

200g:   42.979(m/s)

50g:   27.079(m/s)

Conclusion:

First the first case of this experiment will be analyzed. Reasonably, the slope of the frequency v.s. 1/λ should be equal to the wave speed. Two equations were applied to calculate the wave speed. The results obtained by applying the equation v=fxλ has smaller error compared to the results obtained from applying equation v= √(T/μ). Since both of the equations have the variable, the length of the wave, so the factor which caused this error difference would be the inaccurate measure of the spring's mass. When the mass was measured, the obtained mass was the whole spring. However, only partial spring was used to created loops. Therefore, the mass of the used spring contributed to the huge error difference. On the other side, the results obtained by equation v=fxλ in both cases have error within 10%. Therefore, the experiment showed that the wave speed is determined by the mass, length, and also most importantly, the frequency.