MHT CET · Physics · Oscillations
Equation of simple harmonic progressive wave is given by \(y=\frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \cos \omega t\) then the resultant amplitude of the wave is \(\left(\cos 90^{\circ}=0\right)\)
- A \(\frac{a \pm b}{a b}\)
- B \(\frac{\sqrt{a} \pm \sqrt{b}}{a b}\)
- C \(\frac{\sqrt{a} \pm \sqrt{b}}{\sqrt{a b}}\)
- D \(\sqrt{\frac{a+b}{a b}}\)
Answer & Solution
Correct Answer
(D) \(\sqrt{\frac{a+b}{a b}}\)
Step-by-step Solution
Detailed explanation
\(y=\frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \sin \left(\omega t+\frac{\pi}{2}\right)\)
Here phase difference \(=\frac{\pi}{2}\)
The resultant amplitude
\(=\sqrt{\left(\frac{1}{\sqrt{a}}\right)^2+\left(\frac{1}{\sqrt{b}}\right)^2}=\sqrt{\frac{1}{a}+\frac{1}{b}}=\sqrt{\frac{a+b}{a b}}\)
Here phase difference \(=\frac{\pi}{2}\)
The resultant amplitude
\(=\sqrt{\left(\frac{1}{\sqrt{a}}\right)^2+\left(\frac{1}{\sqrt{b}}\right)^2}=\sqrt{\frac{1}{a}+\frac{1}{b}}=\sqrt{\frac{a+b}{a b}}\)
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