What will be the equation of displacement in the following different conditions?

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Solution

The general equations of a simple harmonic motion,
$x = A s i n (ω t + ϕ)$
1. In this situation as particle is at mean position and just started its oscillation so, for this $ϕ = 0$
$x = A s i n (ω t)$

2. As particle is returning from extreme position first time, so,
equation will become $x = A s i n (ω t + π)$

3. As particle has covered the distance $3 A$ , that means
So, $x = A s i n (ω t + \frac{3 π}{2})$
You can refer phasor diagram for more clearity.

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