Question

derive an expression for displacement of a particle executing SHM

Answer 1

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Displacement equation can be obtained uniform circular motion,where SHM can be represented by projection of uniform circular motion along diameter .

Consider a particle P executing uniform circular motion in a circle of radius A .

Let 'w' be angular velocity of particle. The particle P moves from x to point P making an angle Θ in times 't'.

'P'→ X to P→ 't'→Θ

$ᴀɴɢᴜʟᴀʀ \: \: ᴠᴇʟᴏᴄɪᴛʏ \: \: ᴏғ \: \: ᴀ \\ ᴘᴀʀᴛɪᴄʟᴇ \: \: ɪន \: \: ᗯ = \frac{Θ}{t}$

Now the displacement y of projection P on diameter YOY' is given by y=ON.

From OPN

$\sinΘ = \frac{ᴏɴ}{ᴏᴘ} = \frac{y}{ᴀ} \\ y = ᴀ \sinΘ \\ y = ᴀ \sinᴡt$

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