Wave Motion Lab
Title: Wave Motion Lab
Date: 4/30/15
Partner: Steph Kinsella
Date: 4/30/15
Partner: Steph Kinsella
Purpose
The purpose of this lab is to determine the velocities of standing waves in a vibrating string.
Theory
A standing wave (shown at right) is formed when two identical waves travel through the same medium in opposite directions. The top wave in the image is the fundamental wave, with length equal to half a wavelength. The second wave is known as the first harmonic wave, and its length is equal to one wavelength. The final wave is the second harmonic, and its length is equal to three halves of its wavelength. Rearranging this pattern, we can state:
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From this, we can easily determine the velocity of the wave.
A second method of determining the velocity of the wave has to do with the proportionality of the velocity to the tension and the linear density of the material. The derivation is shown below, where subscripts s refer to the mass of the string and subscripts h refer to the hanging mass.
Using these two derivations, we can calculate velocity using constant and measured values and compare.
Experimental Technique
The apparatus should be set up as shown below. Use a string about the length of the table and knot it to the string vibrator at one end. At the other end, clamp a pulley to the table and hang the string and mass over the pulley. Measure this distance from the knot to the center of the pulley; this will be the value L. Add mass until you reach a steady standing wave of maximum amplitude, as shown on the far right. Begin with a wave that has a high number of antinodes n. Measure the mass hanging from the string and mark the spot above the axel of the pulley with a marker. Continue to add small amounts of weight until the amplitude of the wave changes; this will be the uncertainty of the measurement. Repeat this process for the next lowest node until you reach the lowest number from which your pulley can support the mass.
Once complete, cut your string at the first mark you made and measure the mass of the string. Repeat until you have cut all of the marks.
Once complete, cut your string at the first mark you made and measure the mass of the string. Repeat until you have cut all of the marks.
Data
The frequency of the string vibrator is 60 Hz.
The length of the string from knot to pulley is 2.37 m.
The following data was recorded from the hanging mass portion of the experiment.
The length of the string from knot to pulley is 2.37 m.
The following data was recorded from the hanging mass portion of the experiment.
Analysis
Using the constant equation, velocities were calculated as shown below. The table at right shows the velocities for all values n.
Using the derived tension equation for the measured values, the velocities were calculated again.
Percent difference between the calculations was then analyzed, as shown below.
The data is summarized in the following table.
Conclusion
The purpose of this lab was to use two methods to determine the velocity of a standing wave in a vibrating string. One method involved a calculation that included only constant values, while the other included multiple measurements. Because the percent differences between the measurements throughout the various wave numbers n ranged from about 1-3%, it can be concluded that the velocities were determined accurately.
A possible source of error in this lab could involve the placement of the node near the string vibrator. While trying to determine the "best" wave at each number n, sometimes the first node moved away from the knot at the vibrator, which could skew both the hanging mass and mass of string measurements. The node should consistently be exactly at the knot.
A possible source of error in this lab could involve the placement of the node near the string vibrator. While trying to determine the "best" wave at each number n, sometimes the first node moved away from the knot at the vibrator, which could skew both the hanging mass and mass of string measurements. The node should consistently be exactly at the knot.
References
Big Standing Wave - Small Effort! (n.d.). Retrieved May 5, 2015, from http://www.arborsci.com/cool/big-standing-wave-small-effort
Giancoli, D. (2009). Physics for Scientists and Engineers (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Lahs Physics (n.d.). Retrieved October 6, 2014, from www.lahsphysics.weebly.com
Giancoli, D. (2009). Physics for Scientists and Engineers (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Lahs Physics (n.d.). Retrieved October 6, 2014, from www.lahsphysics.weebly.com