One end of a piano wire is wrapped around a cylindrical tuning peg and the other end is fixed in place. The tuning peg is turned so as to stretch the wire. The piano wire is made from steel (Y= 2.0x1011 N/m²). It has a radius of 0.86 mm and an unstrained length of 0.74 m. The radius of the tuning peg is 2.0 mm. Initially, there is no tension in the wire, but when the tuning peg is turned, tension develops. Find the tension in the wire when the tuning peg is turned through two revolutions.
One end of a piano wire is wrapped around a cylindrical tuning peg and the other end is fixed in place. The tuning peg is turned so as to stretch the wire. The piano wire is made from steel (Y= 2.0x1011 N/m²). It has a radius of 0.86 mm and an unstrained length of 0.74 m. The radius of the tuning peg is 2.0 mm. Initially, there is no tension in the wire, but when the tuning peg is turned, tension develops. Find the tension in the wire when the tuning peg is turned through two revolutions.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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The first image is a practice problem I am working out. The second image is showing how I've worked it out so far. But when I try to do the math to find what T=, I can't get it right on my calculator.
Transcribed Image Text:**Problem Statement:**
One end of a piano wire is wrapped around a cylindrical tuning peg, and the other end is fixed in place. The tuning peg is turned to stretch the wire. The piano wire is made from steel with a Young's modulus \(Y = 2.0 \times 10^{11} \, \text{N/m}^2\). The wire has a radius of 0.86 mm and an unstrained length of 0.74 m. The radius of the tuning peg is 2.0 mm. Initially, there is no tension in the wire, but tension develops when the tuning peg is turned. Calculate the tension in the wire when the tuning peg is turned through two revolutions.
![### Transcription of Handwritten Notes on Young's Modulus and Tension Calculation
---
#### Step 1: Given
- \( Y = 2 \times 10^{11} \, \text{N/m}^2 \)
- \( R = 0.86 \, \text{mm} = 8.6 \times 10^{-4} \, \text{m} \)
- \( L = 0.74 \, \text{m} \)
- \( r = 2.0 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- \( n = 2 \, (\text{revolutions}) \)
#### Step 2:
**Change in length is given by**
\[
\Delta L = 2 (2 \pi R)
\]
**Substituting values**
\[
\Delta L = 2 (2 \pi \times 8.6 \times 10^{-4})
\]
\[
\Delta L = 6 \pi \times 10^{-3} \, \text{m}
\]
**Young's Modulus is given by**
\[
Y = \frac{TL}{(\Delta L)A}
\]
**Therefore,**
\[
T = \frac{Y A (\Delta L)}{L}
\]
**Substituting values in the equation for tension \( T \):**
\[
T = \frac{2 \times 10^{11} \times \pi \times (8.6 \times 10^{-4})^2 \times (6 \times 3.14 \times 10^{-3})}{0.74}
\]
**Conclusion:**
The tension in the wire is calculated using Young's Modulus and the given dimensions and parameters.
---
This transcription serves as a detailed walkthrough of calculating the tension in a wire by using the relationship between Young's Modulus, change in length, and the wire's dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd261088c-f753-45e4-ae14-7adbaf3c257c%2F23e5c05e-cc38-4d43-ad0c-203ef571233a%2Fmp9mqqe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription of Handwritten Notes on Young's Modulus and Tension Calculation
---
#### Step 1: Given
- \( Y = 2 \times 10^{11} \, \text{N/m}^2 \)
- \( R = 0.86 \, \text{mm} = 8.6 \times 10^{-4} \, \text{m} \)
- \( L = 0.74 \, \text{m} \)
- \( r = 2.0 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- \( n = 2 \, (\text{revolutions}) \)
#### Step 2:
**Change in length is given by**
\[
\Delta L = 2 (2 \pi R)
\]
**Substituting values**
\[
\Delta L = 2 (2 \pi \times 8.6 \times 10^{-4})
\]
\[
\Delta L = 6 \pi \times 10^{-3} \, \text{m}
\]
**Young's Modulus is given by**
\[
Y = \frac{TL}{(\Delta L)A}
\]
**Therefore,**
\[
T = \frac{Y A (\Delta L)}{L}
\]
**Substituting values in the equation for tension \( T \):**
\[
T = \frac{2 \times 10^{11} \times \pi \times (8.6 \times 10^{-4})^2 \times (6 \times 3.14 \times 10^{-3})}{0.74}
\]
**Conclusion:**
The tension in the wire is calculated using Young's Modulus and the given dimensions and parameters.
---
This transcription serves as a detailed walkthrough of calculating the tension in a wire by using the relationship between Young's Modulus, change in length, and the wire's dimensions.
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