Experiment 4: Standing Waves

Introduction: The purpose of this experiment is to observe the relationship between wavelength, frequency, and wave velocity at resonant conditions on a standing wave on string.

Procedure: We attached an oscillator to a frequency generator.

We then attached a length of string to a frequency generator via a clamp.

We put the other end of the string over a pulley and attache a 200g mass to it.

We changed the frequency of the frequency generator until we got 2 nodes and measured the distance between the nodes.

We did this for 3 nodes and 4 nodes up until 12 nodes and recorded the lengths between each.

We then replaced the 200g mass for 1/4 of it or 50g and repeated the entire process, only recording 7 different node distances rather than 11.

Data:

Case 1: 200g mass

Number of nodes	Frequency (Hz)	Distance d between nodes (m)	λ=2d (m)	1/λ	n=2L/λ (L=1.97)
2	22	1.97	3.94	0.253807106598985	1
3	44	0.98	1.96	0.510204081632653	2.01
4	59	0.652	1.304	0.766871165644172	3.02
5	77	0.44	0.88	1.13636363636364	4.48
6	97	0.393	0.786	1.27226463104326	5.01
7	120	0.33	0.66	1.51515151515152	5.97
8	130	0.295	0.59	1.69491525423729	6.68
9	170	0.245	0.49	2.04081632653061	8.04
10	190	0.215	0.43	2.32558139534884	9.16
11	200	0.185	0.37	2.7027027027027	10.6
12	215	0.175	0.35	2.85714285714286	11.3

  

 Case 2: 50g mass

Number of nodes	Frequency (Hz)	Distance d between nodes (m)	λ=2d (m)	1/λ	n=2L/λ (L=1.887)
2	10	1.887	3.774	0.264970853206147	1
4	29	0.662	1.324	0.755287009063444	2.85
5	34	0.481	0.962	1.03950103950104	3.92
6	50	0.395	0.79	1.26582278481013	4.78
7	59	0.33	0.66	1.51515151515152	5.72
8	70	0.277	0.554	1.80505415162455	6.81
14	130	0.115	0.23	4.34782608695652	16.4

Mass of String: 0.99g +/– 0.01g

Experimental mass of 200g mass: 199.9g +/– 0.01g

Experimental mass of 50g mass: 49.89g +/– 0.01g

String Length (oscillator to mass): 2.66m +/– 0.1m

Mass per unit length: 3.72 x 10^-4 kg/m

Analysis: Here is a graph of the inverse of wavelength versus frequency. The slope should represent the wave velocity.

Using the equation velocity=sqrt(Tension/mass per unit length), we get velocity=72.62 meters per second. That yields a percentage error of 4.91%. There exist uncertainty in both the equation value and the graph value, however I assumed the equation value to be the true value here. There is a small uncertainty that comes from the the measurement of the length and mass of the string as well as the fact that the string and pulley are not ideal. This is very small and that is why the equation can be assumed more accurate than the graph data. The graph data relies on the distance between nodes which was more difficult to measure. For one, we could not bring the meter stick too close to the string while it was vibrating. Also as the nodes became more numerous, it became difficult to pinpoint the point of stillness on the vibrating string. These measurements were less accurate and will most likely account for this error. The final contribution is the uncertainty of the frequency generator, given the lack of precise markings on the knob.

Here is the same graph for case 2

The ratio case 1/case 2 for velocity from the graphs is 2.64. The ratio of velocity from the equation is 2.00. The ratios are not equal, but similar. The difference here is probably due to the inability to measure the distance between nodes accurately as well as the uncertainty of the knob on the frequency generator. The values are close so it can be assumed that the measurements were close to the true measurements.

The resonant frequency times n does not equal the set frequency for any value in case 1, but they are relatively close. This can be caused by any of three things: the wave velocity used is inaccurate, the value for n is incorrect, or the frequency recorded from the knob is wrong. It is probably a mixture of all but given how the velocity is derived, it is one of  most influential factors.  The other is the value for n. It is certainly incorrect because n is always one number less than the number of nodes.

The frequency for case 1 is just about double that of case 2 for the same harmonic. This makes sense because we used 4 times the tension in case 1 compared to case 2, and the formula for velocity includes tension under a radical.