# About the water surface in a accelerated cylinder

After I woke up this morning while sitting at our table I looked at a plastic bottle of cola lying on the floor. Please, don't think it's a mess out here. It just lay there. I put it nicely back on the table but not before I gave the bottle a push.
The water (which is what the zero-energy-cola is in essence made up of) in the bottle started to oscillate, giving the bottle (a kind of cylinder) a strange way of rolling. It was going fast, made a sudden stop, going fast again, came to a stop, going fast, etc., until it finally stopped due to all kinds of friction. That's not so difficult to explain. But now think about a closed cylinder with length $$l$$ and radius $$R$$. The water inside, when the cylinder is at rest, has a surface which reaches a height $$h$$ in the $$z$$-direction, which is one component of the associated cartesian coordinate system), measured from the point of contact with the plane it's lying on.
The bottle starts to accelerate in a horizontal direction (the $$x$$-direction) with an associated value $$a$$. I gave the push in the non-idealized setting of the room here, so let us do this thought experiment (which you can approach in reality but I don't know how to let that happen here; I merely gave the bottle a push), perpendicular to the $$y$$-axis (how else could it be?).
One more thing. The cylinder is rolling without slipping.

I gave sufficient information (at least, I think) to answer the following question for which I don't have the mental energy yet to answer, so I'm calling out:

What will be the angle the flat water surface in the cylinder makes with the $$x$$-axis when the cylinder is accelerated in the $$x$$-direction and magnitude $$a$$?

• The cylinder is also rolling, yes? "Accelerated" could just mean sliding... Sep 2, 2020 at 8:10
• @Philip Yes. I forgot to mention. Sep 2, 2020 at 9:39

## 1 Answer

you may go through the attached answer before the content review team removes it. Sorry I don't have enough time to answer this but I feel like I should atleast give an intuition. You will get better answers soon

$$\theta = tan^{-1}\left(\frac{a}{g}\right)$$

The wobbly motion of the bottle can be explained using the fact that after we push and the bottle gets a momentum, the system is acted upon by no other external forces (neglect friction please). So the Centre of mass of the bottle should continue moving straight with a fixed velocity. So If the water was to move forward inside the bottle, the bottle would slow down and when the water. moves back, the bottle again speeds up... always keeping the centre of mass at a constant speed.

• You've understood exactly what I meant. Sep 2, 2020 at 9:41