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Chapter 8: Problem 103

If \(\mathrm{S}\) is stress and \(Y\) is Young's modulus of material of a wire, the energy stored in the wire per unit volume is \([\mathbf{2 0 0 5}]\) (A) \(2 S^{2} Y\) (B) \(\frac{S^{2}}{2 Y}\) (C) \(\frac{2 Y}{S^{2}}\) (D) \(\frac{S}{2 Y}\)

Short Answer

Expert verified
The energy stored in the wire per unit volume is \(\frac{S^{2}}{2 Y}\).

Step by step solution

01

Convert the formula into stress and Young's modulus terms

We know that the strain in a wire can be given as, \(Strain = \frac{Stress}{Young's\ modulus} = \frac{S}{Y}\) Then, we substitute this equation into the formula for energy stored per unit volume: \(Energy \ stored\ per\ unit\ volume = \frac{1}{2} 脳 Stress 脳 Strain\) \(Energy\ stored\ per\ unit\ volume = \frac{1}{2} 脳 S 脳 (\frac{S}{Y})\)
02

Simplify the expression for energy stored per unit volume

Now we simplify the expression: \(Energy\ stored\ per\ unit\ volume = \frac{1}{2} 脳 S 脳 (\frac{S}{Y})\) \(Energy\ stored\ per\ unit\ volume = \frac{S^{2}}{2 Y}\)
03

Compare answer with the given options

Our simplified expression for the energy stored per unit volume is \(\frac{S^{2}}{2 Y}\). Comparing this with the options provided in the exercise: (A) \(2 S^{2} Y\) (B) \(\frac{S^{2}}{2 Y}\) (C) \(\frac{2 Y}{S^{2}}\) (D) \(\frac{S}{2 Y}\) We find that option (B) matches our expression. So, the energy stored in the wire per unit volume is \(\frac{S^{2}}{2 Y}\).

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Most popular questions from this chapter

A mercury barometer reads \(75 \mathrm{~cm}\). If the tube be inclined by \(60^{\circ}\) from vertical, the length of mercury in the tube will be (A) \(37.5 \mathrm{~cm}\) (B) \(150 \mathrm{~cm}\) (C) \(\frac{75 \sqrt{3}}{2} \mathrm{~cm}\) (D) \(100 \mathrm{~cm}\)

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