To determine how much the steel guitar string is stretched, we can apply Hooke's Law for elastic deformation. The formula is:

$\Delta L = \frac{F L_0}{A Y}$

where:

• $\Delta L$ is the change in length (the amount the string is stretched),
• $F$ is the applied tension (force) on the string,
• $L_0$ is the original length of the string,
• $A$ is the cross-sectional area of the string,
• $Y$ is Young's modulus of the material (steel in this case).

Step 1: Values and Constants

Given:

• $F = 2000 \, \text{N}$ (tension),
• $L_0 = 80 \, \text{cm} = 0.80 \, \text{m}$ (initial length of the string),
• Diameter of the string = $1.0 \, \text{mm} = 1.0 \times 10^{-3} \, \text{m}$,
• $Y$ for steel ≈ $2.0 \times 10^{11} \, \text{N/m}^2$.

Step 2: Cross-sectional Area

The string is cylindrical, so the cross-sectional area $A$ is the area of a circle:

$A = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{1.0 \times 10^{-3}}{2}\right)^2$

Step 3: Calculation

Let’s calculate how much the string is stretched.The steel guitar string is stretched by approximately $0.0102 \, \text{m}$, or 1.02 cm.

Do you need more details or have any questions about the process? Here are some related questions to deepen your understanding:

1. How does the tension in a string affect its frequency of vibration?
2. What is Young's modulus, and how does it relate to material stiffness?
3. How would the stretch change if the string were made of a different material?
4. How does the diameter of the string impact the amount it stretches under tension?
5. What would happen to the stretch if the length of the string were doubled?

Tip: The more a material stretches under a given force, the less stiff it is, which is characterized by a lower Young's modulus.