The speed v of a wave on a stretched string depends upon the tension F and the linear mass density μ (mass/length) according to
We will study wave behavior using standing waves produced by stretching a string between a mechanical oscillator and a pulley. Frequencies can be varied using DataStudio. The arrangement is shown below. The overall length of the string should be a little over 2 m. Use your spreadsheet to record and analyze the data.
1. The pattern shown
here is composed of three loops. Place a 200-g mass (maximum) on the hanger and
find the values of the frequency that will maximize the amplitude of the
standing waves for 2, 3, 4, and 5 loops. The distance between the nodes in these
standing waves is one half of the wavelength. Measure this distance. Use
equation (1) to find the wave speed v by plotting frequency vs. (wavelength)-1.
2. Now consider equation (2). A plot of the tension versus the square of the speed should give a straight line whose intercept is 0 and whose slope is μ. Use masses of 100, 150, ..., 300 gms. Keep the number of nodes equal to 3. Use your spreadsheet to obtain this plot; a linear regression for the slope and intercept (the linest command in Excel), to 90% confidence; and the best fit line for the data. Obtain a copy of the plot and the spreadsheet for your notebook. How linear is the data plotted in this manner? Explain.
Your instructor will give you a 3-meter length of string to weigh. To 90% confidence, how well does the slope of your graph agree with the value of μ obtained by finding the actual mass per unit length? Explain any discrepancies.