# What is the exact relation between a real oscillating body's time period with time?

I took an empty bottle and placed it on the floor, then tilted the bottle to one side such that the the displacement caused a disturbance in its balance but not enough to completely tilt it over. The bottle started oscillating back and forth. But as the time passed and the amplitude of oscillation decreased (as expected for damped oscillation) but its time period also decreased with time, that is it started oscillating more vigorously and the finally the oscillation stopped. Why?

(I'm sorry if I am missing something obvious here)

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No, you're not missing the obvious: this is a good question. The simple harmonic oscillator shoved down your throat in freshman physics and engineering courses is a linear system: any solution scaled by a scale factor is also a solution. So the period cannot depend on amplitude. The basic equation defining this beast is

$$\ddot{x} = - \omega^2 x\tag{1}$$

where $\omega$ is the frequency (in radians per second) and $x$ the displacement. In this system, the restoring force is proportional to the displacement (and directed against the latter).

Given that the bottle is curved, you are almost certainly witnessing the amplitude dependent variation of period that arises in a nonlinear simple harmonic oscillator. You have something akin to a pendulum, where gravity acts normal to the motion at the bottom of the swing, and the component of gravity in the direction of motion is proportional to the sine of the swing angle. Thus we get the equation:

$$\ddot{\theta} = - \omega^2 \sin(\theta)\tag{2}$$

where now $\theta$ is the swing angle. Notice that the restoring force in this model is (realistically) limited: it cannot be greater than $\omega^2$. For small oscillations we again get the same equation as in (1). But watch what happens when I put all this into the trusty Mathematica slave below:

I'm plotting the solution to (2) for $\omega^2=1$ for two beginning speeds: $\dot{x}(0)=1$ and $\dot{x}(0)=1/2$. Witness that the bigger amplitude one has the longer period.

This is exactly what you've just seen with your bottle.

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