Let's walk through the detailed, step-by-step solution for calculating the frequency at which the tuning fork sets the wire into resonant vibration. Here are the steps involved:
### Step 1: Understanding Given Data and Problem Statement
- Length of the wire ([tex]\(L\)[/tex]): [tex]\(1.0\)[/tex] meters.
- Tension in the wire ([tex]\(T\)[/tex]): [tex]\(40.0\)[/tex] Newtons.
- Mass per unit length of the wire ([tex]\(\mu\)[/tex]): [tex]\(7.0 \times 10^{-3}\)[/tex] kg/m.
### Step 2: Determine the Velocity of the Wave in the Wire
The velocity ([tex]\(v\)[/tex]) of a wave in a string under tension can be calculated with the formula:
[tex]\[
v = \sqrt{\frac{T}{\mu}}
\][/tex]
where:
- [tex]\(T\)[/tex] is the tension in the string,
- [tex]\(\mu\)[/tex] is the mass per unit length of the wire.
Plugging in the values, we get:
[tex]\[
v = \sqrt{\frac{40.0}{7.0 \times 10^{-3}}} \approx 75.59289460184544 \, \text{m/s}
\][/tex]
### Step 3: Determine the Frequency of Resonant Vibration
At resonance, the wire vibrates at its fundamental frequency, which corresponds to the first harmonic. For the fundamental frequency, the wavelength ([tex]\(\lambda\)[/tex]) of the standing wave is twice the length of the wire:
[tex]\[
\lambda = 2L
\][/tex]
The frequency ([tex]\(f\)[/tex]) can be found from the relationship between the speed of the wave, its frequency, and wavelength:
[tex]\[
f = \frac{v}{\lambda}
\][/tex]
Substituting [tex]\(\lambda = 2L\)[/tex], we get:
[tex]\[
f = \frac{v}{2L}
\][/tex]
Now, we substitute the values for [tex]\(v\)[/tex] and [tex]\(L\)[/tex]:
[tex]\[
f = \frac{75.59289460184544}{2 \times 1.0} \approx 37.79644730092272 \, \text{Hz}
\][/tex]
### Conclusion
The velocity of the wave in the wire is approximately [tex]\(75.59 \, \text{m/s}\)[/tex], and the frequency of the tuning fork which sets the wire into resonant vibration is approximately [tex]\(37.80 \, \text{Hz}\)[/tex].
