First slide

Periodic Motion

Question

A particle is subjected to two S H Ms x₁ = A₁ sin wt and x₂ = A₂ sin (wt + $\large \frac{\pi }{4}$ ). The resultant S H M will have an amplitude of

Easy

A

$\large \frac{{{A_1} + {A_2}}}{2}$
B

$\large \sqrt {A_1^2 + A_2^2}$
C

$\large \sqrt {A_1^2 + A_2^2 + \sqrt 2 {A_1}{A_2}}$
D

A₁A₂

Solution

$\large {A_{net}}\, = \,\sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \phi }$
$\large \phi \rightarrow$ Phase angle between two S.H.M waves
$\large \phi \, = \,\frac{\pi }{4}$
$\large {A_{net}}\, = \,\sqrt {A_1^2 + A_2^2 + 2.{A_1}{A_2} \times \cos \left( {\frac{\pi }{4}} \right)}$
$\large \, = \,\sqrt {A_1^2 + A_2^2 + 2.{A_1}{A_2} \times \frac{1}{{\sqrt 2 }}}$
$\large \, = \,\sqrt {A_1^2 + A_2^2 + \sqrt 2 .{A_1}{A_2}}$

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