Laboratory #4 Standing Waves

In an elastic medium such as a string of linear density µ (mass per unit length), the speed a wave is a function of the tension F in the medium. The more dense, the slower it moves (think of the thick piano bass strings). The more tension, the faster it moves (when you tighten a guitar string, the tension goes up and the frequency gets higher). In this experiment you will determine the speed using a method common to the investigation of many types of wave phenomena: the formation of standing waves. This experiment is an important example of a “quantized” system and we will relate our results to the allowed energies in atomic systems in the coming weeks.

Periodic waves in a uniform elastic medium travel with a speed v that is related to the frequency f of the wave source and the distance l between successive identical points on the medium (wavelength). The speed is given by:

u = fl

When a “resonant” condition is satisfied, the frequency of the wave source equals one of the natural vibration frequencies of the medium. In this case the distance along the medium between points of destructive interference (nodes) is l /2, one-half the wavelength. It should be noted that the distance between the point where the string is connected to the wave source and the first node beyond that point, is not l /2, since the point of contact is not a node, but oscillates with non-zero amplitude. The situation is shown in the following sketch.

/
(Photo courtesy of PASCO Scientific
Company. Reprinted with permission.)

Materials

Mechanical wave driver / Function generator
Pulley / Weights and weight hanger
Rods and table clamps / String
Meter stick with jaws
The apparatus should be set up so that the string is about two meters long, from the slotted plug in the top of the wave driver to the point of contact with the pulley. (Illustration courtesy of PASCO Scientific Company. Reprinted with permission.) /

Procedure

1. Open the Excel Data Table: Lab 4 Data1.xls

Record the Hanger Mass and the linear mass density for your setup.

2. With the amplitude control turned to zero, turn on the wave source. Used the range selection button to produce a frequency of approximately 50 Hz (50 oscillations/second). Turn up the amplitude until you can observe the oscillations of the vibrator post. Increase the frequency and find the value for which the string vibrates in several well defined segments. The amplitude should become a maximum at this “resonant” frequency. Continue to increase the frequency until a new resonance is reached and the string vibrates in one additional segment. Remember, the distance between two nodes is l /2. Record the appropriate values in the data table. Repeat this process for higher frequencies, so that you have at least five data points. You can reduce the uncertainty in your measurement of the wavelength by measuring the distance from the first to the last node and then dividing by the appropriate integer to get the wavelength.

Analysis

Part 1

Complete the data table by finding the velocity for each case in column four. Is the velocity a constant (or nearly so) for all values of period? How do the numbers compare to the calculated value? Use any analysis you choose to quote a “best” value.

Another approach to extracting a best value is the “Method of Least Squares” This is the approach taken in Parts 2 and 3.

Part 2

Construct an x-y scatter graph for you data with wavelength on the vertical axis and period on the horizontal axis. Construct a Trendline for your data. How does the slope compare to the calculated value for the velocity?

Unfortunately, the Trendline analysis in Excel does not report the error associated with the slope and intercept of the best line. Continue with Part 3. You will find an Excel spreadsheet that computes the slope and intercept along with the associated errors.

Part 3

Excel does not estimate the uncertainty in the velocity. Copy and paste your data into the file: Lab 4 Data2.xls. When you paste your data select “Paste Special” and choose values! This file will perform the same linear fit but will also report the uncertainty in the value for the velocity.

Lab Report due in one week

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