Velocity LO (f)

From x = x_o sin ω t

differentiating we get

v

= d x d t = x 0 ω ( s i n ω t ) = v 0 c o s ( ω t )

where v₀ = x₀ ω is the maximum velocity

Variation with time of velocity

In terms of x:

From mathematical identity cos² ωt + sin² ωt = 1,

rearranging

cos² ωt = 1 - sin² ωt

$c$ o s ω t = ±( 1 − s i n 2 ωt )

since

v = x₀ω cos ωt

where x₀ is the maximum displacement

$v$ = ±x 0 ω( 1 − s i n 2 ωt )
$v$ = ±x 0 ω( 1 − ( x x 0 ) 2 )

$v$ = ±ω( x 0 2 − x 2 )

Variation with displacement of velocity