First slide

Elastic Modulie

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

Moderate

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

Hysteresis loss corresponding to elasticity per unit volume of a substance is given by the area of hysteresis loop, i.e., stress-strain curve coresponding to one complete loading and deloading.
Area of an ellipse = $π$ x semi-majoraxis x 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 by hysteresis loop is A = $A_{1} + A_{2} - A_{3}$
$A = 2 [\frac{π}{4} \times 8 \times 4 \times 10^{2}] - [8 \times 4 \times 10^{2}]$
= work done per cycle
= energy lost per cycle per unit volume

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