Harmonic oscillation x=sin?nt?

Question

Hey guys,
Hope someone can give me a hand on this one?

A particle is subjected to undamped harmonic oscillation described by the equation:
x=sinωnt
Prove that the linear acceleration of the particle is given by a=-ω2.x where ω is the angular velocity and x is the linear displacement from the centre of oscillation.

Answer 1

What is n?

 

The acceleration is the second derivative of the position, which in this case is the same as the displacement from the center of oscillation.

 

By the chain rule, the first derivative is dx/dt = ωn cos ωnt.

 

Now take the second derivative:  d²x/dt² = ωn * (-ωn sin ωnt) = -(ωn)² sin ωnt.

 

But sin ωnt is the same as x, so this can be rewritten d²x/dt² = -(ωn)² x.

 

That's very close to what you want, but if you can clarify where n fits in, you should be able to get to the answer.