37. A string 4.0m long is tied at both ends to fixed positions. A pulse in the string takes 0.10s to start from one end and reach the other end.
a) Find three frequencies that will produce standing waves in the string.
b) Will a frequency of 23 Hz produce a standing wave in the string? Explain your reasoning.
a) To find the three frequencies that will produce standing waves in the string, we can use the formula for the fundamental frequency of a string:
f = (1/2L) * c
where f is the frequency, L is the length of the string, and c is the speed of the wave.
Given that the length of the string is 4.0m and the time it takes for the pulse to travel is 0.10s, we can calculate the speed of the wave using the formula:
c = L / t
where t is the time taken for the pulse to travel.
Substituting the given values:
c = 4.0m / 0.10s = 40 m/s
Now we can calculate the three frequencies using the formula for the fundamental frequency:
f1 = (1/2L) * c
= (1/2 * 4.0m) * 40 m/s
= 20 Hz
For the second frequency, we need to consider the first overtone, which is twice the fundamental frequency:
f2 = 2 * f1
= 2 * 20 Hz
= 40 Hz
And for the third frequency, we consider the second overtone, which is three times the fundamental frequency:
f3 = 3 * f1
= 3 * 20 Hz
= 60 Hz
Therefore, the three frequencies that will produce standing waves in the string are 20 Hz, 40 Hz, and 60 Hz.
b) To determine if a frequency of 23 Hz will produce a standing wave in the string, we need to compare it to the calculated frequencies from part a). Since 23 Hz is not equal to any of the calculated frequencies (20 Hz, 40 Hz, and 60 Hz), it will not produce a standing wave in the string. In order to produce a standing wave, the frequency must match one of the resonant frequencies of the string.
a) To find the frequencies that will produce standing waves in the string, we can use the formula:
f = n * v / (2L)
where:
f is the frequency of the standing wave,
n is the harmonic number (1, 2, 3, ...),
v is the velocity of the wave, and
L is the length of the string.
Given:
L = 4.0 m
v = L / t
t = 0.10 s
Calculating the velocity of the wave:
v = L / t
v = 4.0 m / 0.10 s
v = 40 m/s
Substituting the values into the formula, we can find the three frequencies:
For n = 1:
f1 = 1 * 40 m/s / (2 * 4.0 m)
f1 = 10 Hz
For n = 2:
f2 = 2 * 40 m/s / (2 * 4.0 m)
f2 = 20 Hz
For n = 3:
f3 = 3 * 40 m/s / (2 * 4.0 m)
f3 = 30 Hz
Therefore, the three frequencies that will produce standing waves in the string are 10 Hz, 20 Hz, and 30 Hz.
b) To determine whether a frequency of 23 Hz will produce a standing wave in the string, we can compare it to the calculated frequencies above.
Since 23 Hz is not equal to any of the three frequencies we calculated (10 Hz, 20 Hz, and 30 Hz), it will not produce a standing wave in the string. In order for a standing wave to form, the frequency must match one of the natural frequencies of the string, which in this case are 10 Hz, 20 Hz, and 30 Hz.
To find the frequencies that will produce standing waves in the string, we need to use the formula for the fundamental frequency of a string fixed at both ends:
f₁ = (1/2L) * v
where f₁ is the fundamental frequency, L is the length of the string, and v is the speed of the wave.
In this case, the length of the string is 4.0 m. To find the speed of the wave, we can use the formula:
v = d/t
where v is the speed, d is the distance traveled by the wave, and t is the time taken.
In this case, the distance traveled by the wave is 4.0 m (since it goes from one end to the other end), and the time taken is 0.10 s. Therefore, the speed of the wave is:
v = d/t = 4.0 m / 0.10 s = 40 m/s
Now, we can substitute the values into the formula for the fundamental frequency:
f₁ = (1/2L) * v = (1/2 * 4.0 m) * 40 m/s = 20 Hz
The fundamental frequency of the string is 20 Hz.
To find the three frequencies that will produce standing waves, we need to consider the harmonics of the fundamental frequency. The harmonics are integer multiples of the fundamental frequency.
The first harmonic (also known as the second overtone) is twice the frequency of the fundamental frequency. Therefore, the first harmonic is:
f₂ = 2 * f₁ = 2 * 20 Hz = 40 Hz
The second harmonic (also known as the third overtone) is three times the frequency of the fundamental frequency. Therefore, the second harmonic is:
f₃ = 3 * f₁ = 3 * 20 Hz = 60 Hz
The third harmonic (also known as the fourth overtone) is four times the frequency of the fundamental frequency. Therefore, the third harmonic is:
f₄ = 4 * f₁ = 4 * 20 Hz = 80 Hz
So, the three frequencies that will produce standing waves in the string are 20 Hz, 40 Hz, and 60 Hz.
Now, let's consider whether a frequency of 23 Hz will produce a standing wave in the string. Since 23 Hz is not equal to any of the three frequencies we found above (20 Hz, 40 Hz, or 60 Hz), it means that 23 Hz does not correspond to any of the standing wave frequencies of the string. Therefore, a frequency of 23 Hz will not produce a standing wave in the string.