To solve the problem step by step, we will break it down into two parts: (a) calculating the average power transmitted across a given point on the string, and (b) finding the total energy associated with the wave in a 2.0 m long portion of the string.
### Part (a): Average Power Transmitted
1. **Identify the Given Values**:
- Frequency \( f = 200 \, \text{Hz} \)
- Amplitude \( A = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Linear mass density \( \mu = 6 \, \text{g/m} = 6 \times 10^{-3} \, \text{kg/m} \)
- Tension \( T = 60 \, \text{N} \)
2. **Calculate the Wave Velocity**:
The velocity \( v \) of the wave on the string can be calculated using the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
Substituting the values:
\[
v = \sqrt{\frac{60 \, \text{N}}{6 \times 10^{-3} \, \text{kg/m}}} = \sqrt{10000} = 100 \, \text{m/s}
\]
3. **Calculate the Angular Frequency**:
The angular frequency \( \omega \) is given by:
\[
\omega = 2\pi f
\]
Substituting the frequency:
\[
\omega = 2\pi \times 200 = 400\pi \, \text{rad/s}
\]
4. **Calculate the Average Power**:
The average power \( P \) transmitted by the wave can be calculated using the formula:
\[
P = \frac{1}{2} \mu \omega^2 A^2 v
\]
Substituting the values:
\[
P = \frac{1}{2} \times (6 \times 10^{-3}) \times (400\pi)^2 \times (1 \times 10^{-3})^2 \times 100
\]
First, calculate \( (400\pi)^2 \):
\[
(400\pi)^2 = 160000\pi^2
\]
Now substituting back:
\[
P = \frac{1}{2} \times 6 \times 10^{-3} \times 160000\pi^2 \times 10^{-6} \times 100
\]
Simplifying:
\[
P = 3 \times 10^{-3} \times 160000\pi^2 \times 10^{-4}
\]
\[
P = 480\pi^2 \times 10^{-7} \approx 0.48 \, \text{W}
\]
### Part (b): Total Energy Associated with the Wave
1. **Calculate the Time for the Wave to Travel 2 m**:
The time \( t \) it takes for the wave to travel a distance \( L \) is given by:
\[
t = \frac{L}{v}
\]
Substituting the values:
\[
t = \frac{2 \, \text{m}}{100 \, \text{m/s}} = 0.02 \, \text{s}
\]
2. **Calculate the Total Energy**:
The total energy \( E \) associated with the wave in a length \( L \) can be calculated using:
\[
E = P \times t
\]
Substituting the values:
\[
E = 0.48 \, \text{W} \times 0.02 \, \text{s} = 0.0096 \, \text{J} = 9.6 \, \text{mJ}
\]
### Final Answers:
- (a) Average Power Transmitted: \( 0.48 \, \text{W} \)
- (b) Total Energy in 2 m: \( 9.6 \, \text{mJ} \)
