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Chapter 7: Problem 844

On increasing the length by \(0.5 \mathrm{~mm}\) in a steel wire of length $2 \mathrm{~m}\( and area of cross-section \)2 \mathrm{~mm}^{2}$ the force required is..... \(\left[\mathrm{Y}\right.\) for steel $\left.=2.2 \times 10^{11} \mathrm{~N} / \mathrm{m}\right]$ (A) \(1.1 \times 10^{5} \mathrm{~N}\) (B) \(1.1 \times 10^{4} \mathrm{~N}\) (C) \(1.1 \times 10^{3} \mathrm{~N}\) (D) \(1.1 \times 10^{2} \mathrm{~N}\)

Short Answer

Expert verified
The required force to increase the length of the steel wire by 0.5 mm is \(2.2 \times 10^2 N\).

Step by step solution

01

Understand the problem

We are given the original length, change in length, area of cross-section and Young's modulus of steel. We need to find the force required to cause the given increase in length.
02

Write the formula relating stress, strain, and Young's modulus

The formula relating stress, strain, and Young's modulus is: \(Stress = Young's~modulus × Strain\) where \(Stress = \frac{Force}{Area}\) and \(Strain = \frac{Change~in~length}{Original~length}\).
03

Substitute the given values

Substitute the given values in the formula: \(Force = Stress × Area = (Young's~modulus × Strain) × Area\) \(Force = (2.2 × 10^{11} N/m~ × \frac{0.5 × 10^{-3} m}{2 m}) × 2 × 10^{-6} m^2\)
04

Calculate the required force

Now, calculate the force: \(Force = (2.2 × 10^{11} × \frac{0.5 × 10^{-3}}{2}) × 2 × 10^{-6}\) \(Force = (2.2 × 10^{11} × 0.25 × 10^{-3}) × 2 × 10^{-6}\) \(Force = 1.1 × 10^{8} × 10^{-3} × 2 × 10^{-6}\) \(Force = 1.1 × 10^{5} × 2 × 10^{-6}\) \(Force = 1.1 × 2 × 10^{5-6}\) \(Force = 2.2 × 10^{5-6}\) \(Force = 2.2 × 10^{-1}\) \(Force = 2.2 × 10^{2} N\) So, the required force to increase the length of the steel wire by 0.5 mm is \(2.2 \times 10^2 N\) which corresponds to the answer choice (D).

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Most popular questions from this chapter

A rubber ball when taken to the bottom of a \(100 \mathrm{~m}\) deep take decrease in volume by \(1 \%\) Hence, the bulk modulus of rubber is $\ldots \ldots \ldots . . .\left[\mathrm{g}=10\left(\mathrm{~m} / \mathrm{s}^{2}\right)\right]$ (A) \(10^{6} \mathrm{~Pa}\) (B) \(10^{8} \mathrm{~Pa}\) (C) \(10^{7} \mathrm{~Pa}\) (D) \(10^{9} \mathrm{~Pa}\)

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