# Standing Waves on a String

One can set up standing waves on a string which is fixed at both ends by vibrating it at just the right frequency. At the right frequency, waves travelling down the string will interfere constructively with each other and with waves travelling in the opposite direction.

In fact, there are a whole set of frequencies which will work: a fundamental frequency f1 and its harmonics, 2*f1, 3*f1, , etc. As shown in this week's lab manual, the fundamental frequency is given by

```                            1             F
frequency  f1  =  --- * sqrt (--------)
2*L            mu
```
where
```            f1          is the fundamental frequency (Hertz, or 1/s)
L           is the length of the string (meters)
F           is the force of tension on the string (Newtons)
mu          is the linear mass density of the string (kg/meter)
```

This week, you will set up standing waves at a set of frequencies, and use the above equation to figure out what the linear mass density of the string ought to be. Then, you'll compare it to the actual linear mass density, and see if the theory actually works.

1. Measure the length of the stretched string, from pole to pulley (see the lab manual for a small correction you might make)
2. Measure the length of the same section of string when it is not stretched
3. Determine the mass of the weight hanging from the string
4. Find the frequency which yields N = 1, 2, 3, 4, 5 antinodes
5. Make a graph of this frequency versus N
6. Based on the graph, determine the fundamental frequency precisely
7. Calculate the linear mass density of the unstretched string

At this point, you should come to me and tell me what the linear mass density of the unstretched string is -- with uncertainty, of course. I'll write it down in my little black book. Then, you may use the Meittler balance to measure directly the mass of a length of unstretched string, from which you can calculate the real linear mass density.

• Does your value of mu for the unstretched string based on the standing waves agree within the uncertainties with the actual value?
• Which of the measurements you make introduces the largest component of the uncertainty in the value of mu?
• Provide a concrete suggestion for the single modification which would most improve the precision of this experiment; in other words, the modification which would address the largest source of uncertainty.

#### What do I have to submit?

You may NOT use a computer for any purpose in this week's exercise. Paper, pencil, ruler, calculator -- no more.

Once again, I want to try to give you a chance to finish all your work by the end of the lab period. Therefore, I expect:

• A neat table of all your measurements, including headings and all appropriate units and uncertainties
• A table which lists
• Each type of measurement you make this week
• The tool you used to make it
• The percentage uncertainty in your measurements using this tool
• A graph showing a plot of frequency for standing waves versus number of antinodes
• Calculation of stretched mu, based on the slope of the graph. Include uncertainty.
• Calculation of unstretched mu, based on the previous value and the stretching of the string. Include uncertainty.
• Direct measurement of linear mass density, with uncertainty.
• Answer to the question: Is your value of unstretched mu equal to the measured value, within the uncertainty?
• Suggested modification to the experiment

I will deduct a full letter grade from any report which includes the phrase "human error."