# A thick rope of density ρ and length L is hung from a rigid support. The increase in length of the rope…

Q: A thick rope of density ρ and length L is hung from a rigid support. The increase in length of the rope due to its own weight is (Y is the Young’s modulus)

(a) $\displaystyle \frac{1}{4Y}\rho L^2 g$

(b) $\displaystyle \frac{1}{2Y}\rho L^2 g$

(c) $\displaystyle \frac{\rho L^2 g}{Y}$

(d) $\displaystyle \frac{\rho L g}{Y}$

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Ans: (b)
Sol: Tension at mid is

T = Mg/2

Young’s modulus of Elasticity is

$\displaystyle Y = \frac{T L}{A \Delta L}$

$\displaystyle \Delta L = \frac{T L}{A Y}$

$\displaystyle \Delta L = \frac{M g L}{2A Y}$

$\displaystyle \Delta L = \frac{(A L \rho ) g L}{2 A Y}$

$\displaystyle \Delta L = \frac{\rho g L^2}{2 Y}$