First slide

Simple hormonic motion

Question

When two displacements represented by $\large {Y_1}\, = \,a\sin \left( {\omega t} \right)$ and $\large {Y_2}\, = \,b\cos \left( {\omega t} \right)$ are super- imposed the motion is

Easy

A

not a simple harmonic
B

simple harmonic with amplitude $\large \frac{a}{b}$
C

simple harmonic with amplitude $\large \sqrt {{a^2} + {b^2}}$
D

simple harmonic with amplitude $\large \left( {\frac{{a + b}}{2}} \right)$

Solution

y = y₁+y₂ = $\large a\sin \omega t + b\cos \omega t$
$\large \therefore y\, = \,\sqrt {{a^2} + {b^2}} \sin \left( {\omega t + {{\tan }^{ - 1}}\frac{b}{a}} \right)$
so the resulting motion is SHM with amplitude $\large \sqrt {{a^2} + {b^2}}$

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