Question

A thick rope of rubber of density $1.5\times {10}^3\text{kg/}{\text{m}}^{\text{3}}$ and Young's modulus $5\times {10}^6\text{N/}{\text{m}}^2$ , $8m$ length is hung from the ceiling of a room , the increases in its length due to its own weight is :

$\left(g=10\text{m/}{\text{s}}^{\text{2}}\right)$

• A. $9.6\times {10}^{\text{- 2}}\text{m}$
• B. $19.2\times {10}^{\text{- 7}}\text{m}$
• C. $9.6\times {10}^{\text{- 7}}\text{m}$
• D. $9.6m$

Answer 1

The correct option is A $9.6\times {10}^{\text{- 2}}\text{m}$

Given that,

Density of rubber$\rho =1.5\times {10}^{3}kg/m^3$

Young's modulus $Y=5\times {10}^{6}N{m}^2$

Length $=8m$

We need to calculate the increase length

Mg acts at centre of gravity which is at $\frac{L}{2}$

so we take length of rope$=\frac{L}{2}$

we know $Y=\frac{stress}{strain}=\frac{F/A}{\Delta l/l}=\frac{Fl}{\Delta l.A}$

$\Rightarrow \Delta l=\frac{Fl}{AY}=\frac{mgL}{2AY}=\frac{Al\rho gl}{2AY}$

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

$=\frac{1.5\times {10}^3\times 10\times 8^2}{2\times 5\times {10}^6}$

$\Rightarrow \Delta l=9.6\times {10}^{-2}m$.

So (A) is correct option.