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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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Most popular questions from this chapter

Read the assertion and reason carefully and mark the correct option given below. (a) If both assertion and reason are true and the reason is the correct explanation of the assertion. (b) If both assertion and reason are true but reason is not the correct explanation of the assertion. (c) If assertion is true but reason is false. (d) If the assertion and reason both are false. Assertion: When height of a tube is less then liquid rise in the capillary tube the liquid does not overflow. Reason: Product of radius of meniscus and height of liquid incapilling tube always remains constant. (A) a (B) \(b\) (C) c (D) d

When there is no external force, the shape of liquid drop is determined by (A) Surface tension of liquid (B) Density of Liquid (C) Viscosity of liquid (D) Temperature of air only

Melting point of ice (A) Increases with increasing pressure (B) Decreases with increasing pressure (C) Is independent of pressure (D) is proportional of pressure

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) \(3 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

The upper end of a wire of radius \(4 \mathrm{~mm}\) and length $100 \mathrm{~cm}$ is clamped and its other end is twisted through an angle of \(30^{\circ}\). Then what is the angle of shear? (A) \(12^{\circ}\) (B) \(0.12^{\circ}\) (C) \(1.2^{\circ}\) (D) \(0.012^{\circ}\)
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