A boy's catapult is made of rubber cord which is $42 \mathrm{~cm}$ long, with $6 \mathrm{~mm}$ diameter of cross-section and of negligible mass. The boy keeps a stone weighing $0.02 \mathrm{~kg}$ on it and stretches the cord by $20 \mathrm{~cm}$ by applying a constant force. When released, the stone flies off with a velocity of $20 \mathrm{~ms}^{-1}$. Neglect the change in the area of cross-section of the cord while stretched. The Young's modulus of rubber is closest to :
Correct Option: 1
(1) When a catapult is stretched up to length $\ell$, then the stored energy in it $=\Delta \mathrm{k} . \mathrm{E}$
$\Rightarrow \frac{1}{2} \cdot\left(\frac{Y A}{L}\right)(\Delta I)^{2}=\frac{1}{2} m v^{2}$
$\Rightarrow y=\frac{m v^{2} L}{\Delta(\Delta I)^{2}}$
$\mathrm{m}=0.02 \mathrm{~kg}$
$\mathrm{v}=20 \mathrm{~ms}^{-1}$
$\mathrm{~L}=0.42 \mathrm{~m}$
$\mathrm{~A}=\left(\pi \mathrm{d}^{2}\right) /(4)$
$\mathrm{d}=6 \times 10^{-3} \mathrm{~m}$
$\Delta \ell=0.2 \mathrm{~m}$
$=\frac{0.02 \times 400 \times 0.42 \times 4}{\pi \times 36 \times 10^{-6} \times 0.04}$
$=2.3 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}$
So, order is $10^{6}$.
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