A tuning fork of frequency 440 Hz is attached to a long string of linear mass density `0.01 kg m^-1` kept under a tension of 49 N. The fork produces transverse waves of amplitude 0.50 mm on the string. (a) Find the wave speed and the wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork transmitting energy to the string ?

Explanation

To solve the problem step by step, we will break down each part of the question:

### Given Data:
- Frequency of tuning fork, \( f = 440 \, \text{Hz} \)
- Linear mass density of the string, \( \mu = 0.01 \, \text{kg/m} \)
- Tension in the string, \( T = 49 \, \text{N} \)
- Amplitude of the wave, \( A = 0.50 \, \text{mm} = 0.50 \times 10^{-3} \, \text{m} \)

### (a) Find the wave speed and the wavelength of the waves.

1. **Calculate the wave speed (v)**:
The wave speed on a string under tension is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the values:
\[ v = \sqrt{\frac{49 \, \text{N}}{0.01 \, \text{kg/m}}} = \sqrt{4900} = 70 \, \text{m/s} \]

2. **Calculate the wavelength (λ)**:
The relationship between wave speed, frequency, and wavelength is given by:
\[ v = f \lambda \implies \lambda = \frac{v}{f} \]
Substituting the values:
\[ \lambda = \frac{70 \, \text{m/s}}{440 \, \text{Hz}} \approx 0.159 \, \text{m} = 15.9 \, \text{cm} \]

### (b) Find the maximum speed and acceleration of a particle of the string.

1. **Calculate the maximum speed (Vmax)**:
The maximum speed of a particle in a wave can be calculated using:
\[ V_{\text{max}} = A \omega \]
where \( \omega = 2 \pi f \).
First, calculate \( \omega \):
\[ \omega = 2 \pi \times 440 \approx 2764 \, \text{rad/s} \]
Now, substituting the values:
\[ V_{\text{max}} = 0.50 \times 10^{-3} \, \text{m} \times 2764 \, \text{rad/s} \approx 1.38 \, \text{m/s} \]

2. **Calculate the maximum acceleration (Amax)**:
The maximum acceleration can be calculated using:
\[ A_{\text{max}} = A \omega^2 \]
Substituting the values:
\[ A_{\text{max}} = 0.50 \times 10^{-3} \, \text{m} \times (2764)^2 \approx 3.81 \, \text{m/s}^2 \]

### (c) At what average rate is the tuning fork transmitting energy to the string?

1. **Calculate the average power (P)**:
The average power transmitted by the tuning fork to the string is given by:
\[ P = 2 \pi^2 f^2 v A^2 \]
Substituting the values:
\[ P = 2 \pi^2 (440)^2 (70) (0.50 \times 10^{-3})^2 \]
Calculating this gives:
\[ P \approx 0.67 \, \text{W} \]

### Final Answers:
- (a) Wave speed \( v = 70 \, \text{m/s} \), Wavelength \( \lambda \approx 15.9 \, \text{cm} \)
- (b) Maximum speed \( V_{\text{max}} \approx 1.38 \, \text{m/s} \), Maximum acceleration \( A_{\text{max}} \approx 3.81 \, \text{m/s}^2 \)
- (c) Average power \( P \approx 0.67 \, \text{W} \)