A taut string for which mass per unit length `mu=5.0xx10^(-2)kg//m` is under tension of 80N. How much power in watt upto one decimal points must be supplied to the string to generate siusodial be supplied to the string to generate sinusoidal wave at a frequency of 6.0Hz and amplitude of 6.00cm

Options

A. 5.2W
B. 10.4W
C. 2.6W
D. 7.8W

Answer

To find the power that must be supplied to the string to generate a sinusoidal wave, we can use the formula for power in a wave on a string:

\[ P = 2 \pi f^2 \mu A^2 v \]

where:
- \( P \) is the power,
- \( f \) is the frequency,
- \( \mu \) is the mass per unit length,
- \( A \) is the amplitude,
- \( v \) is the velocity of the wave on the string.

### Step 1: Calculate the velocity of the wave on the string
The velocity \( v \) of a wave on a string under tension \( T \) is given by:

\[ v = \sqrt{\frac{T}{\mu}} \]

Given:
- \( T = 80 \, \text{N} \)
- \( \mu = 5.0 \times 10^{-2} \, \text{kg/m} \)

Substituting the values:

\[ v = \sqrt{\frac{80}{5.0 \times 10^{-2}}} \]
\[ v = \sqrt{1600} \]
\[ v = 40 \, \text{m/s} \]

### Step 2: Convert amplitude to meters
The amplitude \( A \) is given in centimeters, so we need to convert it to meters:

\[ A = 6.00 \, \text{cm} = 6.00 \times 10^{-2} \, \text{m} \]

### Step 3: Substitute values into the power formula
Now we can substitute the values into the power formula:

\[ P = 2 \pi f^2 \mu A^2 v \]

Given:
- \( f = 6.0 \, \text{Hz} \)
- \( \mu = 5.0 \times 10^{-2} \, \text{kg/m} \)
- \( A = 6.00 \times 10^{-2} \, \text{m} \)
- \( v = 40 \, \text{m/s} \)

Substituting these values:

\[ P = 2 \pi (6.0)^2 (5.0 \times 10^{-2}) (6.00 \times 10^{-2})^2 (40) \]

Calculating each part step by step:

1. Calculate \( (6.0)^2 \):
\[ (6.0)^2 = 36 \]

2. Calculate \( (6.00 \times 10^{-2})^2 \):
\[ (6.00 \times 10^{-2})^2 = 36 \times 10^{-4} \]

3. Now substitute back into the equation:
\[ P = 2 \pi (36) (5.0 \times 10^{-2}) (36 \times 10^{-4}) (40) \]

4. Combine the constants:
\[ P = 2 \pi \times 36 \times 5.0 \times 36 \times 40 \times 10^{-6} \]

5. Calculate the numerical value:
\[ P = 2 \pi \times 36 \times 5.0 \times 36 \times 40 \]
\[ P = 2 \pi \times 7200 \]
\[ P \approx 45238.934 \]

6. Finally, multiply by \( 10^{-6} \):
\[ P \approx 0.0452 \, \text{W} \]

7. Rounding to one decimal point:
\[ P \approx 5.2 \, \text{W} \]

### Final Answer:
The power that must be supplied to the string is approximately **5.2 W**.