Acceleration

From v = v_o cos ω t = x₀ ω cos ω t

where x_o is the maximum displacement

differentiating we get

a

= d v d t = − ω2 ( x 0 s i n ω t ) = − ω2 x

Variation with time of acceleration

In terms of x:

Therefore, a = - xo ω² sin ω t
                   = - ω² (x_o sin ω t)
which is the defining equation for S.H.M. !
               a     = - ω² x

Variation with displacement of acceleration

since

        a = – a₀sin ω t

where a_o is the maximum acceleration
where by a₀ = ω² (x_o)