Standing waves are set up on two strings fixed at each end, as shown in the drawing. The two strings have the same tension and mass per unit length, but they differ in length by 0.55 cm. The waves on the shorter string propagate with a speed of 41.8 m/s, and the fundamental frequency of the shorter string is 234 Hz. Determine the beat frequency produced by the two standing waves.
To determine the beat frequency produced by the two standing waves, we need to calculate the frequency of the longer string and then find the difference between the frequencies of the two strings.
The fundamental frequency of a string can be calculated using the formula:
f = v / (2L)
where f is the frequency, v is the wave speed, and L is the length of the string.
Given that the speed of the waves on the shorter string is 41.8 m/s and the fundamental frequency of the shorter string is 234 Hz, we can rearrange the formula to solve for the length of the shorter string:
L_short = v / (2f_short)
L_short = 41.8 m/s / (2 * 234 Hz)
L_short ≈ 0.089 m
We are also given that the length of the longer string is 0.55 cm longer than the shorter string, or 0.0055 m longer.
L_long = L_short + 0.0055 m
L_long = 0.089 m + 0.0055 m
L_long ≈ 0.0945 m
Now we can calculate the frequency of the longer string using the same formula:
f_long = v / (2L_long)
f_long = 41.8 m/s / (2 * 0.0945 m)
f_long ≈ 221.8 Hz
Finally, we can find the beat frequency by subtracting the frequency of the shorter string from the frequency of the longer string:
beat frequency = |f_long - f_short|
beat frequency = |221.8 Hz - 234 Hz|
beat frequency ≈ 12.2 Hz
Therefore, the beat frequency produced by the two standing waves is approximately 12.2 Hz.