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Help me solve this - Properties of Solids and Liquids - JEE Main

A boy's catapult is made of rubber cord which is 42\; cm long, with 6\; mm diameter of cross-section and of negligible mass. The boy keeps a stone weighing 0.02\; kg on it and stretches the cord by 20\; cm by applying a constant force. When released, the stone flies off with a velocity of 20\; 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 :
 

  • Option 1)

    10^{8}\; N\! m^{-2}

  • Option 2)

    10^{3}\; N\! m^{-2}

  • Option 3)

    10^{6}\; N\! m^{-2}

     

  • Option 4)

    10^{4}\; N\! m^{-2}

Answers (1)
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For string

\\l=42\; cm\\d=6\; mm\Rightarrow r=3\; mm\\A=\pi\; r^{2}\\\Delta l=20\; cm

for stone

\\m=0.02\; kg\\V=20\; m/s

So apply energy conservation.

P.E. stored in string = K.E of stone

\Rightarrow \frac{1}{2}\times Y\times \left ( \frac{\Delta L}{L} \right )^{2}\times A\times L=\frac{1}{2}\; mv^{2}

\Rightarrow Y = \frac{mv^{2}\times L}{(\Delta L)^{2}\times A}

            = \frac{2\times 10^{-2}\times 20\times 20\times 42\times 10^{-2}}{20\times 20\times 10^{-4}\times 3.14 \times 9\times 10^{-6} }

      Y\approx 10^{6}\; N/m^{2}


Option 1)

10^{8}\; N\! m^{-2}

Option 2)

10^{3}\; N\! m^{-2}

Option 3)

10^{6}\; N\! m^{-2}

 

Option 4)

10^{4}\; N\! m^{-2}

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