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I just encountered a problem involving lift and oscillations where I found the following differential equation: $$\ddot y = -\frac{ \rho gA}{m}y = -\omega^2 y$$

What's the relation between $\ddot y$, $y$ and $\omega$? Does $\ddot y = -\omega^2y $ or $\ddot y =\omega^2y$ apply in all situations? If yes, what's the reasoning behind it? I know it makes sense in terms of the units but what's the physical reason behind it?

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The equation for a simple harmonic oscilator is $\ddot{x}=-\omega^2x$, so by comparison in your equation you can state that in your system there is oscilatory motion with $\omega^2=\rho g A /m$.

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Oscillations such as simple harmonic motion works well (the equation you wrote in the question) for small amplitudes. Consider a simple pendulum oscillating about a fixed point joined by a string. If you gently push the bob sideways, you would generate oscillations. The restoring force acting here is the component of the force acting along the trajectory:

$$F=-mgsin(\theta)$$But for the motion to be simple harmonic motion requires $F$ to be direcly proportional to $-\theta$ here. That works only if $sin(\theta)$ is approximately $\theta$ (which is true for very small angles of $\theta$ like 15 degrees or smaller.

For larger angles, the motion is more complicated and your expression will not be accurate.

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