Question

A thick uniform rubber rope of density $1.5gc{m}^{-3}$ and Young's modulus $5\times {10}^{6}N{m}^{-2}$ has a length of 8 m. When hung from the ceiling of a room, the increase in length of the rope due to its own weight will be equal to

• A. $9.6\times {10}^{-2}m$
• B. $19.2\times {10}^{-3}m$
• C. $9.6\times {10}^{-3}m$
• D. $9.6m$

Answer 1

The correct option is A

$9.6\times {10}^{-2}m$

The weight of the rope can be assumed to act at its mid-point. Now, the extension $l$ is proportional to original length $L$. If the weight of the rope acts at its mid-point, the extension will be that produced by half the rope. So, replacing $L$ by $\frac{L}{2}$ in the expression for Young's modulus, we have
$Y=\frac{\frac{FL}{2}}{Al}=\frac{FL}{2Al}$
$\Rightarrow l=\frac{FL}{2AY}=\frac{\rho ALg\times L}{2AY}=\frac{g{L}^2\rho }{2Y}$
Now, $\rho =1.5gc{m}^{-3}=1500kg{m}^{-3}$,
$\therefore l=\frac{10\times (8{)}^2\times 1500}{2\times 5\times {10}^6}=9.6\times {10}^{-2}m$
Hence, the correct choice is (a).