Calculate the amount of the water in a bottle While throwing a water bottle in the air I noticed that its rotational speed is not constant, instead, the amount of its deceleration depends on how full the bottle is. I find it logical but I got an idea:
Say someone threw a bottle in the air and flipped it. Then he tells me how big the bottle was and what its rotational velocity was over time (or maybe its angle over time). Is there a way I could calculate how much water there was in the bottle?
 A: In principle you can since Newton's laws are deterministic but, as chaos theory amply points out, any slight uncertainty in the initial conditions of the bottle (initial position, amount of water, volume, velocity, pressure etc.) will yield different results than what you calculate. 
What you can do is make a model for the bottle and the water like a mass (representing the water) inside of another mass (the bottle) and, using Lagrangian dynamics, calculate the frequency modes ("normal modes" in physics language) for a particular mass of water. Then you can compare the modes with what you see in your experiment and estimate how much water is in the bottle.
Beware that the math is far from simple so using dynamics modelling software like Matlab's Simmechanics would be a start.
A: The sloshing about of the liquid water makes analysis of the motion of the bottle far too difficult. The centre of mass moves about, the moment of inertia (how easily the bottle can be rotated) changes. The situation is much easier to analyse if the water is frozen and cannot move about in the bottle.
Even tossing the bottle of ice into the air can result in complicated motion : the axis of rotation can switch as the centre of mass follows a parabola. While the switching can tell us which axis of rotation has the highest moment of inertia, it is difficult to measure mass from this. Instead, it would be better to swing the bottle like a pendulum. The period of oscillations depends on the moment of inertia, which depends on how full the bottle is. The period can be measured easily and accurately - ie the average of a large number of oscillations.
