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Chapter 7: Problem 875

The length of a wire is \(1.0 \mathrm{~m}\) and the area of cross-section is \(1.0 \times 10^{-2} \mathrm{~cm}^{2}\). If the work done for increase in length by \(0.2 \mathrm{~cm}\) is \(0.4\) joule. Then what is the young's modulus? Of material of the wire ? (A) \(2.0 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (B) \(4.0 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (C) \(2.0 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (D) \(4.0 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

Short Answer

Expert verified
The Young's modulus of the material of the wire is approximately \(2.0 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\).

Step by step solution

01

Find stress

The work done (W) is given as 0.4 J. We can use the formula for work: Work = Force × Distance. Let's find the force applied on the wire, F: 0.4 J = F × 0.2 cm Now, convert the distance from cm to meters: 0.2 cm = 0.002 m So, we get: 0.4 J = F × 0.002 m Solve for the force, F: F = 0.4 J / 0.002 m = 200 N Now, we need to find the stress. Stress is given by: Stress = Force / Area The area of cross-section is given as 1.0 × 10^{-2} cm². We need to convert it to square meters: 1.0 × 10^{-2} cm² = 1.0 × 10^{-6} m² Now, we can find the stress on the wire: Stress = 200 N / (1.0 × 10^{-6} m²) = 2.0 × 10^8 N/m²
02

Find strain

Now, we need to find the strain on the wire. Strain is given by: Strain = Change in length / Original length Change in length = 0.2 cm = 0.002 m (already converted to meters) Original length = 1.0 m Now, find the strain: Strain = 0.002 m / 1.0 m = 0.002
03

Find Young's modulus

Young's modulus (Y) is given by: Young's modulus = Stress / Strain Now, we can find the Young's modulus of the wire: Y = (2.0 × 10^8 N/m²) / 0.002 = 1.0 × 10^{11} N/m² From the given options, the closest value is: (C) \(2.0 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) So, the answer is (C).

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