First slide

Simple hormonic motion

Question

Out of the following functions representing motion of a particle which represents SHM :
(i) $y = \sin ωt - \cos ωt$
(ii) $y = \sin^{3} ωt$
(iii) $y = 5 \cos (\frac{3 π}{4} - 3 ωt)$
(iv) $y = 1 + ωt + ω^{2} t^{2}$

Moderate

A

only (i)
B

only (iv) does not represent SHM
C

only (i) and (iii)
D

only (i) and (ii)

Solution

$For (i), y = \sin ωt - \cos ωt = \sqrt{2} [\frac{1}{\sqrt{2}} \sin ωt - \frac{1}{\sqrt{2}} \cos ωt] = \sqrt{2} [\sin ωt \cos \frac{π}{4} - \cos ωt \sin \frac{π}{4}] = \sqrt{2} \sin (ωt - \frac{π}{4}) it is SHM For (ii), y = \sin^{3} ωt (∴ \sin 3 A = 3 \sin A - 4 \sin^{3} A) y = \frac{1}{4} (3 \sin ωt - \sin 3 ωt) = \frac{3}{4} \sin ωt - \frac{1}{4} \sin 3 ωt$
It is superposition of two SHMs of different frequencies. So it is not SHM.
$For (iii), y = 5 \cos (\frac{3 π}{4} - 3 ωt)$

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