First slide

Equation of wave

Question

Equation of a transeverse 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 = 10/4 cm is 2 J/m, what is the elastic potential energy stored in the string at x = 10/3 cm?

Difficult

A

1 J/m
B

6 J/m
C

8 J/m
D

4 J/m

Solution

$y = 2 \sin (50 πt - 10 πx) mm$
$∴ \frac{\partial y}{\partial x} = - 20 . \cos (50 πt - 10 πx)$
At t = 0, $\frac{\partial y}{\partial x} = - 20 πcos 10 πx$
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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