Answer to Question #93266 in Classical Mechanics for Ayo

Question

Determine the equation of motion of an oscillation in a U tube

Answer 1

Let us consider a liquid with density $\rho$ in the U-tube of cross section area $A$. The total length of liquid is $l$. Let the liquid is displaced from equilibrium on distance $y.$ The restoring force

$$F_y=-\Delta m g=-2yA\rho g$$

Using Newton's second law

$$m\ddot y=F_y$$

we obtain

$$m\ddot y=-2yA\rho g$$

$$lA\rho\ddot y=-2yA\rho g$$

Finally, the equation of motion of an oscillation in a U-tube

$$\ddot y+\frac{2g}{l}y=0$$