Question

A simple accelerometer (an instrument fro measuring acceleration) can be made in the from of a tube
filled with a liquid and bent as shown In Fig During motion, the level of the liquid in the left arm will
be at $a$ height $h_1$, and in the right ar, at a height $h_2$
Determine the acceleration $a$ of a carriage in which the accelerometer is installed, assuming that
the diameter of the tube is much smaller than $h_1$ and $h_2$

Answer 1

In a vessel with a liquid moving horizontally with an acceleration $a$, the surface of the liquid becomes an inclined plane. Its slope $\phi$ is determined from the condition that the sum of the force of pressure $F$ and the force of gravity $mg$ acting on an area element of the surface is equal to $ma$, and the force of pressure is normal to the surface. Hence
$\tan \phi =\frac{a}{g}$.
According to the law of communicating vessels, the surface of the liquid in the arms of the manometer balong top the above-mentioned inclined plane Fig. It follows from geometrical considerations that
$\tan \phi =\frac{h_2-h_1}{h_1+h_1}$
whence
$a=\frac{g(h_2-h_1)}{h_2+h_1}$