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.

Question

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.

Answer 1

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.