How much force is required to produce an increase of \(0.2 \%\) in the length of a broses wire of diameter \(0.6 \mathrm{~mm}\) (Young's modulus for brass $=0.9 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)$ (A) Nearly \(17 \mathrm{~N}\) (B) Nearly \(34 \mathrm{~N}\) (C) Nearly \(51 \mathrm{~N}\) (D) Nearly \(68 \mathrm{~N}\)

Short Answer

Expert verified
The required force to produce an increase of 0.2% in the length of the brass wire is nearly 51 N. Therefore, the correct answer is: (C) Nearly 51 N

Step by step solution

01

List the given values

We are given the following values: - Diameter of the wire (d) = 0.6 mm - Young's modulus for brass (Y) = \(0.9 \times 10^{11} \, \mathrm{N/m^2}\) - Increase in length (strain) = 0.2% First, we need to convert the strain to a fraction by dividing it by 100. Strain (\(ε\)) = \(\frac{0.2}{100}\)
02

Calculate the cross-sectional area

We need to find the cross-sectional area (A) of the wire to calculate the required force. The wire is circular in shape, so the formula to find the area is: A = \(\frac{π (d^2)}{4}\)
03

Convert the diameter to meters

Diameter is given in millimeters and needs to be converted to meters for calculations. d = 0.6 mm = \(\frac{0.6}{1000}\, \text{m}\)
04

Calculate the cross-sectional area

Now, we can calculate the cross-sectional area using the formula and the converted diameter: A = \(\frac{π(0.6\times10^{-3})^2}{4}\) = \(2.826 \times 10^{-7}\, \text{m}^2\)
05

Apply the formula for stress, strain, and Young's modulus

Stress (σ) is defined as the force (F) applied per unit area (A) and is given by the formula: \(σ = \frac{F}{A}\) Strain (ε) is defined as the change in length divided by the original length and is given by the formula: \(ε = \frac{ΔL}{L_0}\) Young's modulus (Y) is a measure of material's stiffness and is given by the formula: \(Y = \frac{σ}{ε}\) Now, we'll substitute the values of stress and strain in terms of the force applied (F) and the cross-sectional area (A) on the wire to find the force: \(Y = \frac{\frac{F}{A}}{\frac{0.2}{100}}\)
06

Calculate the force applied

Rearranging the equation to find the force (F): F = Y * A * \(\frac{0.2}{100}\) Plugging in the values: F =\(0.9 \times 10^{11}\) * \(2.826 \times 10^{-7}\) * \( \frac{0.2}{100}\) F ≈ \(50.87\, \text{N}\) The required force to produce an increase of 0.2% in the length of the brass wire is nearly 51 N. Therefore, the correct answer is: (C) Nearly 51 N

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