Question

A boy has a catapult made of a rubber cord of length $42cm$ and diameter $6.0mm$. The boy stretches the cord by $20cm$ to catapult a stone of mass $20g$. The stone flies off with a speed of $20m{s}^{-1}$. Find young's modulus for rubber. Ignore the change in the cross section of the cord in stretching.

Answer 1

Given : Diameter of cord $D=6mm=0.006m$ $L=42cm=0.42m$ $\Delta L=20cm=0.2m$

Mass of the stone $m=20g=0.02kg$

Velocity of the stone $v=20m/s$

Cross-section area of catapult $A=\frac{\pi D^2}{4}=3.14\times \frac{(0.006{)}^2}{4}=2.826\times {10}^{-5}{m}^2$

Energy stored in the catapult $E=\frac{1}{2}\times Stress\times Strain\times volume$

$\therefore$ $E=\frac{1}{2}\times Y\times (Strain{)}^2\times volume$ $(Y=Stress/Strain)$

$\therefore$ $E=\frac{1}{2}Y\times (\frac{\Delta L}{L}{)}^2\times AL=\frac{1}{2}YA\frac{(\Delta L{)}^2}{L}$

Energy stored in the catapult gives the kinetic energy to the stone.

$\therefore$ $\frac{1}{2}YA\frac{(\Delta L{)}^2}{L}=\frac{1}{2}m{v}^2$

$⟹$ $Y=\frac{m{v}^{2}L}{A(\Delta L{)}^2}=\frac{0.02(20{)}^2\times 0.42}{2.826\times {10}^{-5}\times (0.2{)}^2}=2.97\times {10}^{6}N/m^2$