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14. waves and sound — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysics14. waves and sound1 Mark
MCQ
A wave is represented by the equation $y = 10 sin \,\,2\pi \,\,(100t-0.02x) + 10 \,\,sin \,\,2\pi\,\, (100t+0.02x)$. The maximum amplitude and loop length are respectively
  • A
    $20$ units and $30$ units
  • ✓
    $20$ units and $25$ units
  • C
    $30$ units and $20$ units
  • D
    $25$ units and $20$ units
✓

Answer

Correct option: B.
$20$ units and $25$ units
b
$x=10 \sin 2 \pi(100 t-0.02 x)+10 \sin 2 \pi(100 t+0.02 x)$

$\Rightarrow 10[\sin A+\sin B]$

where

$B=2 \pi(100 t-0.02 x)$ and

$A=2 \pi(100 t+0.02 x)$

Thus,

$\Rightarrow 10\left[2 \sin \frac{A+B}{2} \sin \frac{A-B}{2}\right]$

$\Rightarrow 20 \sin (2 \pi 100 t) \sin (2 \pi 0.02 x)$

Comparing the above equation with standard standing wave equation, we get amplitude $=20$ and wave vector

$k=\frac{2 \pi}{\lambda}=2 \pi \times 0.02 \Rightarrow \lambda=50$

Therefore, the loop length $=\frac{\lambda}{2}=25$

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