First slide

Periodic Motion

Question

A particle moves in the x–y plane according to the equation; $\overrightarrow r = \left( {\widehat i + \widehat j} \right)\left( {A\sin \omega t + B\cos \omega t} \right)$ . Motion of the particle is:

Moderate

A

periodic
B

SHM
C

along a straight line
D

all of the above

Solution

$\overrightarrow r = \left( {A\sin \omega t + B\cos \omega t} \right)\widehat i + \left( {A\sin \omega t + B\cos \omega t} \right)\widehat j$
$\therefore x = A\sin \omega t + B\cos \omega t$
$\Rightarrow x = \sqrt {{A^2} + {B^2}} \sin \left( {\omega t + \phi } \right)$ and $y = \sqrt {{A^2} + {B^2}} \sin \left( {\omega t + \phi } \right)$
where $\phi = {\tan ^{ - 1}}\left( {\frac{B}{A}} \right)$
$\therefore r = \sqrt {2\left( {{A^2} + {b^2}} \right)} \sin \left( {\omega t + \phi } \right)$
Also, y/x = 1 ⇒ y = x
So the particle is executing SHM with amplitude $\sqrt {2\left( {{A^2} + {B^2}} \right)} .$

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