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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.

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    $\begingroup$ You can change $A$, without affecting $T$, and you can change $T$ without affecting $A$. $\endgroup$ – garyp Jun 4 '16 at 22:46
  • $\begingroup$ In the full derivation of the SHO, $A=x_0$, the initial displacement at $t=0$. It's fully independent of $\omega$. $\endgroup$ – Gert Jun 4 '16 at 23:20
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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.

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I do not believe you can show it starting by the equation, but it becomes clear when you solve the differential equation that the two quantities are independent: $\omega$ is an arbitrary parameter on the equation of motion, and A is an arbitrary constant that appears in the solution.

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