1. Use wave speed on a stretched string

For a transverse wave on a string, $v = \sqrt{\frac{T}{\mu}}$ where:

• $v = 60\,\text{m s}^{-1}$
• $T$ = tension
• $\mu$ = mass per unit length

2. Find linear mass density

Given:

• Length $L = 50\,\text{cm} = 0.50\,\text{m}$
• Mass $m = 10\,\text{g} = 0.010\,\text{kg}$

So, $\mu = \frac{m}{L} = \frac{0.010}{0.50} = 0.02\,\text{kg m}^{-1}$

3. Find the tension in the string

Using $T = \mu v^2$ $T = 0.02 \times (60)^2 = 0.02 \times 3600 = 72\,\text{N}$

4. Use Young's modulus relation

Young's modulus is $Y = \frac{\text{stress}}{\text{strain}} = \frac{T/A}{\Delta L/L}$

Hence extension, $\Delta L = \frac{TL}{AY}$

Given:

• $A = 2.0\,\text{mm}^2 = 2.0 \times 10^{-6}\,\text{m}^2$
• $Y = 1.2 \times 10^{11}\,\text{N m}^{-2}$
• $L = 0.50\,\text{m}$
• $T = 72\,\text{N}$

Substitute: $\Delta L = \frac{72 \times 0.50}{(2.0 \times 10^{-6})(1.2 \times 10^{11})}$

$\Delta L = \frac{36}{2.4 \times 10^5}$

$\Delta L = 15 \times 10^{-5}\,\text{m}$

So, $x = 15$

5. Comparison with stored answer

Stored correct answer = $15$

This matches our derived answer.