First slide

Periodic Motion

Question

The displacement of a particle from its mean position (in meter) is given by $\large Y\, = \,0.2\,\sin \left( {10\pi t + 1.5\pi } \right)\cos \left( {10\pi t + 1.5\pi } \right)$ . The motion of the particle is

Easy

A

Periodic but not SHM
B

Non - Periodic
C

SHM with period 0.1sec
D

SHM with period 0.2 sec

Solution

$\large Y\, = \,0.2\,\sin \left( {10\pi t + 1.5\pi } \right)\cos \left( {\pi t + 1.5\pi } \right)$
Multiply and divide with 2,
$\large y\, = \,\frac{{0.2}}{2} \times 2\sin \left( {10\pi t + 1.5\pi } \right)\cos \left( {\pi t + 1.5\pi } \right)$
$\large y = \,0.1\sin \left( {20\pi t + 3\pi } \right)$
$\large \sin ce\,\,\sin 2\theta \, = \,2\sin \theta \cos \theta$
$\large \omega \, = \,20\pi \, \Rightarrow \,\frac{{2\pi }}{T}\, = \,20\pi \, \Rightarrow \,T\, = \,0.1\sec$

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