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Equation of a transverse wave propagating in a string is $y = 2 \sin (50 πt - 10 πx) mm$ where x is in metre. If at t = 0, elastic potential energy stored in the string at $x = \frac{10}{4} cm$ is 2 J/m, what is the elastic potential energy stored in the string at $x = \frac{10}{3} cm$ ?

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1 J/m

6 J/m

8 J/m

4 J/m

answer is D.

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Detailed Solution

$y = 2 \sin (50 πt - 10 πx) mm$

$∴ \frac{\partial y}{\partial x} = - 20 π . \cos (50 πt - 10 πx)$

At t = 0, $\frac{\partial y}{\partial x} = - 20 π \cos (10 πx)$

Now elastic potential energy stored per unit length is proportional to ${(\frac{\partial y}{\partial x})}^{2}$

$∴ \frac{u_{2}}{u_{1}} = \frac{\cos^{2} (10 {πx}_{2})}{\cos^{2} (10 {πx}_{1})} = \frac{\cos^{2} (10 π \times \frac{10}{4 \times 100})}{\cos^{2} (10 π \times \frac{10}{3 \times 100})}$

$\Rightarrow \frac{u_{2}}{2} = \frac{\cos^{2} (\frac{π}{4})}{\cos^{2} (\frac{π}{3})} \Rightarrow u_{2} = 4 J / m$

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