Experiment 4: Standing Waves

Experiment 4: Standing Waves

In this lab the goal was to observe standing waves while they were in resonance. We would vary the conditions to observe different wavelengths(λ). The wavelength would change when the frequency changed as we kept the mass and the length of the string to remain constant. We did this for two trials. In the second trial we varied the mass and then repeated the same as we did for the first part of the experiment

To begin this experiment we had mass hanging from a string and our string was connected to a frequency generator. After setting up our apparatus we collected various data from our two trials. We began collecting data once our string reached its fundamental frequency. We began collecting data until we hit a maximum of 100 Hz.

 String mass per until length is 1.25g/m

Part 1

Mass 198.7g ± 0.05

Frequency (Hz)	15.52 ± 0.005	31.52 ± 0.005	47.41 ± 0.005	62.21 ± 0.005	78.76 ± 0.005	91.38 ± 0.005
Wavelength λ  (m)	2.52 ± 0.005	1.242 ± 0.005	0.82 ± 0.005	0.62 ± 0.005	0.51 ± 0.005	0.41 ± 0.005
Δx (m)	1.26 ± 0.005	0.621 ± 0.005	0.41 ± 0.005	0.31 ± 0.005	0.255 ± 0.005	0.205 ± 0.005
Number of Nodes	2	3	4	5	6	7

Part 2

Mass 50.0g

Frequency (Hz)	8.21 ± 0.005	16.62 ± 0.005	25.34 ± 0.005	34.02 ± 0.005	46.06 ± 0.005	63.83 ± 0.005
Wavelength λ (m)	2.52 ± 0.005	1.242 ± 0.005	0.82 ± 0.005	0.62 ± 0.005	0.51 ± 0.005	0.41 ± 0.005
Δx (m)	1.26 ± 0.005	0.621 ± 0.005	0.41 ± 0.005	0.31 ± 0.005	0.255 ± 0.005	0.205 ± 0.005
Number of Nodes	2	3	4	5	6	7

In the data table the frequency represents the frequency the generator was turned to. The wavelength was how long the wave was. Δx represents half of one wavelength. The number nodes corresponds to how many nodes were produced. 

In the first part of the experiment we used 198.7g of mass attached to our pulley. 

In the second part of the experiment we only used 50.0 g of mass attached to our pulley.

The amount of nodes were produced at lower frequencies in the second part of the experiment because the mass was less.

Data Analysis 

Here are plots of each part of the experiment. These graphs are plotted as Frequency vs. 1/λ. The slope of the line is equal to the wave speed. The wave speed can be calculated with this formula 

V= sqrt(T/μ) where T is the tension and μ is the mass per unit length. 

The theoretical velocity using this formula in Part 1 is 39.47 m/s.

the theoretical velocity in Part 2 is 19.80 m/s.

The ratio of the waves speeds from part 1 is 3.86% where the ratio is Δv/Experimental.

This error is not significant.

The ratio of the wave speeds from part 2 is 25.8%. However on part 2 the error is significant. I think this error may have occurred because the length of the string might have changed when we were
the mass.

The calculated theoretical frequencies for part 1 are

15.66 Hz, 31.32 Hz, 46.99 Hz, 62.65 Hz, 78.31 Hz, 93.98 Hz. 

The values are not equal but the percent error for all the calculations is insignificant because they are all under 5%. 

The calculated theoretical frequencies for part 2 are

7.85 Hz, 15.7 Hz, 23.6 Hz, 31.4 Hz, 39.3 Hz, 47.1 Hz

The ratios of the frequencies is Δf/Experimental.

Part 1

0.90%, 0.63%, 0.89%, 0.71%, 0.70%, 2.85%

Part 2

4.45%, 5.73%, 7.20%, 8.28%, 15.20%, 35.46%

Conclusion

The ratios from each part are different, there is much more error in the second because the wavelength was changed and was not remeasured. There is not a pattern here, the ratios from each part vary too much for one to be noticed.

The uncertainties were determined by taking half of the next increment of measure. The main source of error that changed was that the length of the string was changed in the second part of the experiment and was not noted which caused a massive amount of percent error.