# 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

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