First slide

Elastic behaviour of solids

Question

A rubber of volume 2000 cc is alternately subjected to tension and released. The figure shows the stress-strain curve of rubber. Each curve is a quadrant of an ellipse. The amount of energy lost as heat per cycle per unit volume will be

Difficult

A

$(\frac{π}{2} - 1) \times 16 \times 10^{2} J$
B

$(\frac{π}{4} - 1) \times 8 \times 10^{2} J$
C

$(\frac{π}{4} - 1) \times 32 \times 10^{2} J$
D

$(\frac{π}{2} - 1) \times 32 \times 10^{2} J$

Solution

Hystersis loss corresponding to elasticity per unit volume of a substance is given by the area of hysteresis loop, i.e., stress-strain curve corresponding to one complete loading and deloading.
$Area of an ellipse = π \times semi-major axis \times semi-minor axis$
$A_{1} = \frac{1}{4} (π \times 8 \times 4 \times 10^{2}) and A_{2} = \frac{1}{4} (π \times 8 \times 4 \times 10^{2})$
$Also, A_{3} = 8 \times 4 \times 10^{2}$
$Area of hysteresis loop is A = A_{1} + A_{2} - A_{3}$
$\begin{matrix} A = 2 [\frac{π}{4} \times 8 \times 4 \times 10^{2}] - [8 \times 4 \times 10^{2}] \\ = [\frac{π}{2} - 1] \times 32 \times 10^{2} J \end{matrix}$
$= work done per cycle$
$= energy lost per cycle per unit volume$

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