Q125 / 180 · COMEDK 2015 · Physics

A crane with a steel cable of length $11 \mathrm{~m}$ and radius $2.0 \mathrm{~cm}$ is employed to lift a block of concrete of mass 40 tons in a building site. If the Young's modulus of steel is $2.0 \times 10^{11} \mathrm{~Pa}$, what will be roughly the increase in the length of the cable while lifting the block? (Take, $g=10 \mathrm{~ms}^{-2}$ )

  • A
    $0.75 \mathrm{~cm}$
  • B
    $1.25 \mathrm{~cm}$
  • Correct answer
    C
    $1.75 \mathrm{~cm}$
  • D
    $2.50 \mathrm{~cm}$

Solution

Given, length, $l=11 \mathrm{~m}$ Radius, $\quad r=2 \mathrm{~cm}$ Radius, $\quad=2 \times 10^{-2} \mathrm{~m}$ Mass, $m=40$ tons $$ =4 \times 10^{4} \mathrm{~kg} $$ Increase in length, $\Delta l=$ ? We know that, Young's modulus, $$ \begin{aligned} Y &=\frac{m g l}{\pi r^{2}(\Delta l)} \\ \Rightarrow \quad \Delta l &=\frac{m g l}{\pi r^{2} Y} \\ &=\frac{4 \times 10^{4} \times 10 \times 11}{3.14 \times\left(2 \times 10^{-2}\right)^{2} \times 2 \times 10^{11}} \\ &=1.75 \times 10^{-2} \mathrm{~m} \\ &=1.75 \mathrm{~cm} \end{aligned} $$

Hence, the correct answer is option: 'C'.

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