First slide

Simple hormonic motion

Question

Two S.H.Ms. $s_{1} = a \sin ωt and s_{2} = b \sin ωt$ are superimposed on a particle. The $s_{1} and s_{2}$ are along the directions which make angle 37° with each other.

Moderate

A

the particle will perform S.H.M.
B

the particle will not perform S.H.M.
C

the particle will perform periodic motion but not S.H.M.
D

the motion will not be oscillatory.

Solution

Resultant displacement will be the vector sum of two displacements:
$S = \sqrt{{S_{1}}^{2} + {S_{2}}^{2} + 2 S_{1} S_{2} \cos 37 °}$
= $\sqrt{a^{2} s i n^{2} ω t + b^{2} s i n^{2} ω t + 2 a b s i n^{2} ω t \times \frac{4}{5}}$
S = $\sqrt{a^{2} + b^{2} + \frac{8 ab}{5}} \sin ωt$
Which shows that the particle will perform SHM.

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