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Chapter 7: Problem 840

The dimensions of four wires of the same material are given below, in which wire the increase in length will be maximum when the same strain is applied. (A) Length \(100 \mathrm{~cm}\), Diameter \(1 \mathrm{~mm}\) (B) Length \(200 \mathrm{~cm}\), Diameter \(2 \mathrm{~mm}\) (C) Length \(300 \mathrm{~cm}\), Diameter \(3 \mathrm{~mm}\) (D) Length \(50 \mathrm{~cm}\), Diameter \(0.5 \mathrm{~mm}\)

Short Answer

Expert verified
The wire with the maximum increase in length when the same strain is applied is Wire C with a length of \(300 \mathrm{cm}\) and a diameter of \(3 \mathrm{mm}\).

Step by step solution

01

Recall the relation between strain, stress, and Young's modulus

The strain (\(\varepsilon\)) is a dimensionless quantity that describes the amount of deformation experienced by a material when subjected to a stress (\(\sigma\)). Young's modulus (Y) is a property of the material that characterizes its ability to deform under stress. The relationship between these quantities is given by the following equation: \[\sigma = Y \varepsilon\]
02

Define vars and constants

Let's denote the initial length, cross-sectional area, Young's modulus, and stress of the wire as \(L_i\), \(A_i\), \(Y\), and \(\sigma_i\) respectively. The elongation of the wire, \(\Delta L_i\) is the amount by which the wire length increases when subjected to stress, and can be calculated as follows: \[\Delta L_i = \frac{\sigma_i L_i}{Y}\] We are told that the same strain is applied to all the wires, which means the same stress is applied as well, since the wires are made from the same material and have the same Young's modulus.
03

Calculate the cross-sectional area of each wire

To find the cross-sectional area of each wire, we can use the following formula for the area of the circle since the wires are cylindrical: \[A = \pi (r^2)\] For each wire, calculate their cross-sectional areas using their respective diameters (which is twice the radius): - Wire A: \(A_a = \pi (0.5 \times 10^{-3} \mathrm{m})^2\) - Wire B: \(A_b = \pi (1 \times 10^{-3} \mathrm{m})^2\) - Wire C: \(A_c = \pi (1.5 \times 10^{-3} \mathrm{m})^2\) - Wire D: \(A_d = \pi (0.25 \times 10^{-3} \mathrm{m})^2\)
04

Find the elongation of each wire

Now that we have found the cross-sectional area of each wire, we can calculate the elongation of each wire, using the relationship established in Step 2. Remember that the same stress is applied to each wire, the Young's modulus is the same, and we have different lengths and areas for each wire: - Wire A: \(\Delta L_a = \frac{\sigma L_a}{Y} = \frac{\sigma (100 \times 10^{-2} \mathrm{m})}{Y}\) - Wire B: \(\Delta L_b = \frac{\sigma L_b}{Y} = \frac{\sigma (200 \times 10^{-2} \mathrm{m})}{Y}\) - Wire C: \(\Delta L_c = \frac{\sigma L_c}{Y} = \frac{\sigma (300 \times 10^{-2} \mathrm{m})}{Y}\) - Wire D: \(\Delta L_d = \frac{\sigma L_d}{Y} = \frac{\sigma (50 \times 10^{-2} \mathrm{m})}{Y}\)
05

Determine the wire with maximum elongation

Since the goal is to find the wire with the greatest increase in length when subjected to the same stress, we need to compare the elongations computed in Step 4. The Young's modulus and applied stress will be the same for all wires, so we can simplify by comparing the lengths of the wires instead: Wire A: \(100 \mathrm{cm}\) Wire B: \(200 \mathrm{cm}\) Wire C: \(300 \mathrm{cm}\) Wire D: \(50 \mathrm{cm}\) The wire with the highest initial length will experience the maximum elongation under the same stress. Thus, the wire with the maximum increase in length when the same strain is applied is the wire with the longest initial length: \( \boxed{\text{Wire C}} \)

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Most popular questions from this chapter

When \(100 \mathrm{~N}\) tensile force is applied to a rod of $10^{-6} \mathrm{~m}^{2}\( cross-sectional area, its length increases by \)1 \%$ so young's modulus of material is \(\ldots \ldots \ldots \ldots\) (A) \(10^{12} \mathrm{~Pa}\) (B) \(10^{11} \mathrm{~Pa}\) (C) \(10^{10} \mathrm{~Pa}\) (D) \(10^{2} \mathrm{~Pa}\)

If the young's modulus of the material is 3 times its modulus of rigidity. Then what will be its volume elasticity? (A) zero (B) infinity (C) \(2 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (D) \(3 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

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