homework and exercises - Independence of Period and Amplitude in Simple Harmonic Motion

In Simple Harmonic Motion, the period $T$ of an oscillation, is said to be independent of the amplitude $A$ of an oscillation, but why is that so?

Attempting to derive from the equations of Simple Harmonic Motion, doesn't seem to get me anywhere :

$$x(t) = A\cos{(\omega t)}$$ $$\implies x(t) = A\cos{(\frac{2\pi}{T} t)}$$

But it is unclear to me how show the independance of $T$ from $A$ from the above equation, or even if it can be shown through a derivation here.

Answer

A simple harmonic motion is one where the acceleration (or restoring force) is directly proportional to the displacement and in the opposite direction of the displacement. For a mass $m$ on a spring with spring constant $k$, the differential equation describing the motion becomes:

$m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx$

That equation has as solution:

$x(t) = A\cos\left(\omega t + \varphi\right)$

with $\omega = \sqrt{\frac{k}{m}}$

$A$ and $\varphi$ are determined by the initial conditions only: for example, if the mass m is released from position $x_0$ at t=0, then $A=x_0$ and $\varphi=0$. The frequency is determined by the ratio $k/m$ and is independent of the initial conditions.