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} $