# Water level in rotating glass [closed]

I have a glas with water in a train and the train is driving a curve (v and R given). How does the level of the water change? I know that I have the gravity and centripetal force, but how do I calculate the level change?

The following derivation holds if the radius $r$ of the turn is large compared to the glass itself. Consider the following diagram:

Due to the turn at speed $v$ the glass and the water experience a centripetal acceleration $a_c$, pointing to the centre of the turn. Assuming the glass is stationary then consider an infinitesimal element of water $dm$. It it subject to weight $dW=gdm$ and an inertial force $dI=a_cdm$.

The resultant force $dR$ is the vector sum of $dI$ and $dW$ and is perpendicular to the surface of the water.

Basic trigonometry shows that:

$$\tan\theta=\frac{dI}{dW}=\frac{a_c}{g},$$

with $a_c=\frac{v^2}{r}$, we get:

$$\large{\tan\theta=\frac{v^2}{gr}}$$

Where $\theta$ is the angle of inclination of the water's surface due to centripetal acceleration.