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Chapter 7: Problem 869

A steel wire is stretched with a definite load. If the young's modulus of the wire is \(\mathrm{Y}\). For decreasing the value of \(\mathrm{Y}\). (A) Radius is to be decreased (B) Radius is to be increased (C) Length is to be increased (D) None of the above

Short Answer

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(B) Radius is to be increased

Step by step solution

01

Define stress and strain

Stress (\(\sigma\)) is given by the force (\(F\)) applied on an object divided by its cross-sectional area (\(A\)), and strain (\(\epsilon\)) is given by the change in length (\(\Delta L\)) divided by the original length (\(L\)): Stress: \[\sigma = \frac{F}{A}\] Strain: \[\epsilon = \frac{\Delta L}{L}\]
02

Establish Young's Modulus relationship

Young's Modulus (Y) is defined as the ratio of stress to strain: \[Y = \frac{\sigma}{\epsilon}\] Combine the stress and strain definitions from Step 1: \[Y = \frac{\frac{F}{A}}{\frac{\Delta L}{L}}\]
03

Calculate the relationship between radius, length, and stress

The cross-sectional area (A) of the wire is given by the formula: \[A=\pi r^2\] where r is the radius of the wire. Substitute this into the equation for stress: \[\sigma = \frac{F}{\pi r^2}\]
04

Analyze the effects of increasing/decreasing radius and length on stress

Using the equation for stress (\(\sigma = \frac{F}{\pi r^2}\)), we can see that: - If the radius (r) is decreased, the stress on the wire will increase. - If the radius (r) is increased, the stress on the wire will decrease. Since the problem asks for a decrease in the effective Young's Modulus, which should result in less stress on the wire, the correct answer is: (B) Radius is to be increased

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