A 0.0150 kg, 2.09 m long wire is fixed at both ends and vibrates in its simplest mode under a tension of 214 N. When a tuning fork is placed near the wire, a beat frequency of 5.20 Hz is heard. What are the possible frequencies of the tuning fork?
I solved for speed using square root (T/u). Then, I used the speed with the formula f = v/2L. Finally, I used fb = f1 - f2 to find the other frequency. I didn't get the correct answer.
To find the possible frequencies of the tuning fork, we can use the formula:
f = v/2L
where:
- f is the frequency of the wave on the wire,
- v is the speed of the wave on the wire,
- L is the length of the wire.
First, let's calculate the speed of the wave on the wire. The formula to calculate the speed of a wave on a string is:
v = sqrt(T/u)
where:
- T is the tension in the wire,
- u is the linear mass density of the wire.
The linear mass density (u) is the mass per unit length of the wire and can be calculated using the formula:
u = m/L
where:
- m is the mass of the wire,
- L is the length of the wire.
Given values:
m = 0.0150 kg (mass of the wire)
L = 2.09 m (length of the wire)
T = 214 N (tension in the wire)
First, calculate the linear mass density (u):
u = m/L = 0.0150 kg / 2.09 m = 0.00718 kg/m
Next, calculate the speed (v) of the wave on the wire:
v = sqrt(T/u) = sqrt(214 N / 0.00718 kg/m) ≈ 52.31 m/s²
Now, we can use the formula f = v/2L to calculate the frequency (f) of the wave on the wire:
f = 52.31 m/s² / (2 * 2.09 m) ≈ 12.54 Hz
So, the frequency of the wave on the wire is approximately 12.54 Hz.
To find the frequencies of the tuning fork, we know that the beat frequency (fb) is given as 5.20 Hz. The beat frequency is the difference between the frequencies of two waves. Let's assume one frequency as f1 and the other as f2.
fb = |f1 - f2| = 5.20 Hz
To find the two possible frequencies, we can set up the equation:
f1 - f2 = 5.20 Hz
Since f1 = 12.54 Hz (frequency of the wave on the wire), we can rearrange the equation to solve for f2:
f2 = f1 - 5.20 Hz = 12.54 Hz - 5.20 Hz = 7.34 Hz
So, the two possible frequencies of the tuning fork are approximately 12.54 Hz and 7.34 Hz.