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Mechanical Properties of Solids — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsMechanical Properties of Solids5 Marks
Question
  1. Prove that the work done in stretching a wire per unit volume is $\frac{1}{2}\times\text{tension}\times\text{extension.}$
  2. Prove that the work done per unit volume in stretching a wire for every type of strain $=\frac{1}{2}\times\text{stress}\times\text{strain}.$
✓

Answer

  1. $\frac{\text{Work done}}{\text{Volume}}=\frac{1}{2}\frac{\text{F}}{\text{L}}.\frac{\Delta\text{l}}{\text{l}}$
$\therefore\text{Work done}=\frac{1}{2}\text{F}\times\text{A}\frac{\Delta\text{l}}{\text{l}}$
$=\frac{1}{2}\text{Tension}\times\text{extension}$
  1. The wire has length l, area of cross-section A made of material constant Y. Let a force F be applied and at any instance, x be the extension associated (x < L), where L is the maximum extension. At this instant,
$\text{F}=\frac{\text{AY}.\text{x}}{\text{l}}$
Since force is a variable with x,
Work done to stretch is,
$\text{W}=\int\limits^\text{L}_{0}\text{Fdx}$
$\text{W}=\frac{1}{2}\frac{\text{AY}}{\text{l}}.\text{L}^2$
$\text{W}=\frac{1}{2}(\text{Al})\Big(\frac{\text{Y.L}}{\text{l}}\Big)\Big(\frac{\text{L}}{\text{l}}\Big)$
$=\frac{1}{2}\times\text{Volume}\times\text{Stress}\times\text{Strain}$

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