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Exploring Standing Waves on a
String
Standing Waves Lab Online
Purpose
The purpose of this activity is to examine the relationships between the tension in a string, the linear density of the string, the length of the string, and the standing waves that can form on the string.
Theory
A standing wave is a wave where the overall pattern does not appear to move. For a standing wave to form on a string, both ends of the string must be fixed in place, with the vertical displacement of each end always being zero. The locations on the string where the vertical displacement is zero are called nodes, while the locations where the vertical displacement is greatest are called antinodes.
The largest wavelength that can form a standing wave on a string of length L is 2L, resulting in a standing wave with 1 loop. The frequency of this first harmonic is given by:
$f_1 = \frac{v}{2L}$
where v is the speed of the wave on the string, given by $v = \sqrt{\frac{T} {\mu}}$, with T being the tension in the string and $\mu$ being the linear density of the string.
The second largest wavelength that can form a standing wave is L, resulting in a standing wave with 2 loops. The frequency of this second harmonic is:
$f_2 = \frac{v}{L} = 2f_1$
Similarly, the third largest wavelength that can form a standing wave is $ \frac{2L}{3}$, resulting in a standing wave with 3 loops. The frequency of this third harmonic is:
$f_3 = \frac{3v}{2L} = 3f_1$
This pattern continues, with the $n^{th}$ harmonic having a frequency of:
$f_n = n \cdot f_1 = \frac{nv}{2L}$
Setup
- Go to the website: https://ophysics.com/w8.html
Observe the simulation of a string with standing waves.
Procedure: Part 1
Set the linear density $\mu = 5 \cdot 10^{-3} \frac{kg}{m}$, the tension $T = 100 N$, and the string length $L = 4.0 m$. Find the lowest frequency that forms a standing wave on the string and record it in Table 1, along with the number of antinodes. Find the next 4 frequencies that form standing waves on the string and record them in Table 1.
Procedure: Part 2
Set the frequency to $f = 60.0 Hz$ and the linear density to $\mu = 3 \cdot 10^{-3} \frac{kg}{m}$. Adjust the tension until a standing wave with $n = 8$ loops is formed, and record the tension in Table 2. Calculate the theoretical wave speed $v_T$ using the equation $v = \sqrt{\frac{T}{\mu}}$. Calculate the experimental wave speed $v_E$ using the equation $v = \lambda f$, where $\lambda = \frac{L}{n/2}$. Calculate the percent error between the theoretical and experimental wave speeds. Repeat steps 2-5 for $n = 7, 6, 5, 4$.
Analysis of Standing Waves Lab
Calculate the theoretical fundamental frequency $f_1$ using the equation $f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$. Calculate the average experimental value of the fundamental frequency $f_1$ using the data from Part 1. Calculate the percent error between the theoretical and average experimental values of the fundamental frequency. Graph the tension T vs. the harmonic number n, and describe the shape of the graph. Use algebra to show that the tension can be written as $T = \frac{4\mu f^2 L^2}{n^2}$. Graph the tension T vs. $\frac{1}{n^2}$, and use the slope of the graph to calculate the linear density of the string. Compare the calculated value to the given value, and calculate the percent difference. Describe how the frequency of vibration of a standing wave relates to the number of segments of the standing wave.
