# Accelerating a glass of a water across a table?

Suppose you have a glass of water on the table (the glass is half-filled, say). You accelerate the glass in a straight line across the table. The effect is that the water inside the glass rises up one side of the glass.

Why does this happen?

I suppose we can ask some other questions, like what is the shape of the water in the glass as a function of the acceleration, and what is the maximum acceleration of the glass such that none of the water spills out.

## 1 Answer

The equivalence principle tells us that gravity and acceleration are locally indistinguishable. So if you are accelerating your glass with some acceleration $\mathbf{a}$ to the right, then this is indistinguishable from a gravitational acceleration acting to the left (the red line):

If we add the effective gravitational force to the gravitational acceleration $\mathbf{g}$ acting downwards we get a resultant acceleration shown as $\mathbf{a}_r$.

So to work out the shape of the water just calculate the magnitude and direction of the total acceleration $\mathbf{a}_r$ and tilt the glass so the vector $\mathbf{a}_r$ is vertical.

• This is almost true, just that in the reference frame of the glass, the acceleration of the glass presents itself as inertial force in the opposite direction (the g is also equivalent to glass accelerating upwards, not downwards). So the picture is incorrect. Water level will be higher on the back side of the glass. Jun 12 '15 at 13:22
• @airguru: oops, yes, doh! I'll redo the diagram. Jun 12 '15 at 15:21