Purpose: To investigate the properties of standing waves on a string. To determine the relationship between the number of standing wave segments and the tension in the string.
Discussion: In this experiment a mass M is attached to the end of the string which runs over the pulley. Assuming there is no friction in the pulley, the tension in the string, T, is given by T = Mg. The other end of the string is attached to a vibrating plate. Standing wave patterns with nodes at the two ends can be produced by varying M. The wavelength l is related to the length of the string L and the number of antinodes n by nl/2 = L. Recalling that l = v/f, where v =(T/m)1/2 is the speed of the wave and m is the "linear density" (mass per length) of the string. Putting all this together we arrive at the relationship
T = 4L2f 2m/n2.
This relationship predicts that T is proportional to 1/n2, and that the constant of proportionality is 4L2 f2 m.
Note: The frequency of the vibrating plate is actually 120 Hz, not 60 Hz as is marked on the motor.
Procedure:
1. Measure the length L of your string from the end of the vibrating plate to the pulley.
2. Calculate the linear density m of your type of string from the given mass and length of a sample of that string.
3. Increase M until you obtain a standing wave pattern with two antinodes (n=2). It would be useful to produce one antinode, but the required tension may be too large for the string.
4. Change (i. e. decrease) M from step 3 until you obtain standing wave patterns with successive values of n. Produce as many wave patterns as possible (n = 8 or more). In each case, make small adjustments to M until you produce the best possible pattern.
5. Make a graph of T vs. 1/n2. Draw in your "best fit line" and calculate the slope. This may be a little awkward because of the uneven spacing of the points.
6. From the slope calculated in step 5 and the theoretical expression for the slope, calculate the linear density of the string. Compare this value to the value calculated in step 2 (i. e. calculate the percentage error). Notice that the point obtained for n=2 is very far from the other points. If this point does not line up with the others, and you feel that it is having a bad effect on your results, draw another "best fit line" which ignores this point.
7. Repeat steps 1-6 for a different string.