Question

Acceleration of a body with x =A sin(kt) where A and k are constant

Answer 1

x = Asin( Kt)

differentiate wrt t
dx/dt = AKcos( Kt)

again differentiate wrt t
d²x/dt² =- AK²sin( Kt)

you know ,
d²x/dt² = acceleration of any body which moves in trajectory x = Asin(Kt)

so, acceleration = -AK²sin(Kt)
put sin( Kt) = x/A

then, acceleration = -K²( x)
acceleration = K²( -x)

Answer 2

x = Asin( Kt)
dx/dt = AKcos( Kt)
d²x/dt² =- AK²sin( Kt)

We know that,
d²x/dt² = acceleration of any body which moves in trajectory x = Asin(Kt)
sin(Kt)=x/A
Acceleration = -AK²sin(Kt) [Put,sin( Kt) = x/A]
acceleration = -K²(x)
acceleration = K²(-x)

Hope it helps