The expression for the displacement of particle in simple harmonic motion is given by, x=Asin(ωt+φ) Differentiate the above expression with respect to t
dx
dt
=Aωcos(ωt+φ) =Aω[1−sin2(ωt+φ)]
1
2
=Aω[1−
x2
A2
]
1
2
=ω√A2−x2 It can be written as,
dx
dt
=ω√A2−x2 =ω√(−1)(x2−A2) =ω√−1√(x2−A2) =iω√x2−A2 The above represents simple harmonic motion.