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Tension in a sting, the length density of the string, the length of the string, and the standing waves
Typology: Lab Reports
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Qty Items Parts Number 1 String Vibrator WA- 9857 1 Mass and Hanger Set ME- 8967 1 Pulley ME-9448B 1 Universal Table Clamp ME-9376B 1 Small Rod ME- 8988 2 Patch Cords 1 String
The purpose of this activity is to examine the relationships between the tension in a sting, the length density of the string, the length of the string, and the standing waves that can form on the string.
A standing wave is a wave where the overall pattern does not appear to move. For standing wave to form on a string, the basic condition that must be met is that both ends of the string must be fixed in place, never moving themselves. This means that for a string stretched out horizontally, the vertical displacement of each of the two ends of the string must always be zero! In a standing wave, the locations of the string that never move are called nodes, while the locations on the string where the vertical displacement will reach its greatest value are called antinodes. Using the condition that there must be a node located at each end of the string for a standing wave to form on it, let us construct a relationship between the length of the string and the size of the waves that can form standing waves on that string. Let the length of the string be ๐ฟ, then the largest wavelength that will allow for our condition to be met is a wave that is twice as long as the string itself, ๐ = 2 ๐ฟ which will result in a standing wave with 1 loop to form on the sting. rev 05 /201 9
The speed of a wave ๐ฃ is given by the product of its wavelength ๐, and its frequency ๐. ๐ฃ = ๐๐ Inserting 2 ๐ฟ in for the wavelength, and then solving for the frequency gives. ๐ 1 =
This equation gives us the fundamental frequency, or the first frequency that will cause a standing wave to form on the string. This frequency is also called the first harmonic. Now let us repeat this process to find the second largest wavelength that will form a standing wave on the string of length ๐ฟ. The second largest wavelength that will meet the condition of nodes being at both ends of the string is one where its wavelength is equal to the length of the string itself ๐ = ๐ฟ, which will cause a standing wave with 2 loops to form on the string. Inserting this into the equation for the speed of a wave, and solving it for the frequency we obtain: ๐ 2 =
which is the second frequency that will form a standing wave on this string, aka the second harmonic, aka the first overtone. Repeating this process one more time to find the third largest wave that will form a standing wave on the string of length ๐ฟ. The wavelength of this wave will be related to the length of the string by ๐ = 2 ๐ฟ 3 , which will cause a standing wave with 3 loops to form on the string. Inserting this into the equation for the speed of a wave, and solving for the frequency gives, ๐ 3 =
๏ท Click on the push pin icon near the top right of the signal generator window to rescale the main window so you can keep the signal generator open while you perform the experiment.
๏ท In the signal generator window, you will have to increase the amplitude of the wave to better see the standing wave patterns for the higher frequencies.
1 ๐^2