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Young's Modulus of Elasticity

A rubber rope...

Question

A rubber rope of length $8 m$ is hung from the ceiling of a room. What is the increase in length of rope due to its own weight? (Given : Young's modulus of elasticity of rubber $= 5 \times 106 N / m$ and density of rubber $= 1.5 \times 10^{3} k g / m^{3}$ . Take $g = 10 m / s^{2})$ .

1.5 m m

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6 m m

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24 m m

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96 m m

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Solution

The correct option is D $96 m m$
$Y = \frac{M g}{A} \times \frac{L / 2}{△ L}$
(Length is taken as $L / 2$ because weight acts at C.G.)
Now, $M = A L ρ$ .
(For the purpose of calculation of mass, the whole of geometrical length $L$ is to be considered)
$∴ Y = \frac{A L ρ g L}{2 A △ L}$
or $△ L = \frac{ρ g L^{2}}{2 Y} = \frac{1.5 \times 10^{3} \times 10 \times 8 \times 8}{2 \times 5 \times 10^{6}} m$
$= 9.6 \times 10^{- 2} m = 9.6 \times 10^{- 2} \times 10^{3} m m$
$= 96 m m$ .

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