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Chapter 7: Problem 853

\(\mathrm{A}\) and \(\mathrm{B}\) are two wires. The radius of \(\mathrm{A}\) is twice that of \(\mathrm{B}\). They are stretched by the same load. Then what is the stress on \(\mathrm{B}\) ? (A) Equal to that on \(\mathrm{A}\) (B) Four times that on \(\mathrm{A}\) (C) Two times that on \(\mathrm{A}\) (D) Half that on \(\mathrm{A}\)

Short Answer

Expert verified
The stress on B is 4 times the stress on A. The correct answer is (B) Four times that on A.

Step by step solution

01

Recall the formula for stress

To find the stress on the two wires, we need to recall the formula for stress. Stress is the force per unit area, which can be represented as: \[ \text{Stress} = \frac{\text{Force}}{\text{Area}} \]
02

Given a relationship between the radii of A and B

We are given that the radius of wire A is twice the radius of wire B. Let the radius of wire B be r. Then, the radius of wire A will be 2r.
03

Find the cross-sectional area of A and B

To find the stress, we need the cross-sectional area of the wires. The cross-sectional area of a cylinder (in this case, our wires) is given by the formula: \[ \text{Area} = \pi r^2 \] For wire A (radius = 2r): \[ \text{Area of A} = \pi (2r)^2 = 4 \pi r^2 \] For wire B (radius = r): \[ \text{Area of B} = \pi r^2 \]
04

Calculate the stress on A and B using the given formula

We know that both wires have the same load (Force). Let's denote the Force as F. Stress on A: \[ \text{Stress on A} = \frac{\text{Force}}{\text{Area of A}} = \frac{F}{4 \pi r^2} \] Stress on B: \[ \text{Stress on B} = \frac{\text{Force}}{\text{Area of B}} = \frac{F}{\pi r^2} \]
05

Find the relationship between stress on A and stress on B

Divide the stress on B by the stress on A: \[ \frac{\text{Stress on B}}{\text{Stress on A}} = \frac{\frac{F}{\pi r^2}}{\frac{F}{4 \pi r^2}} = \frac{1}{\frac{1}{4}} = 4 \] Thus, the stress on B is 4 times the stress on A. The correct answer is (B) Four times that on A.

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