One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

Explanation

When suction pump connected through one end of U-tube, the mercury depressed by h in one limb and in other arm mercury raised by 2h.
The difference in height of mercury in both the limbs = 2y and pressure of `P = 2hpg` in this arms. Here p= density of mercury and g is the acceleration due to gravity. If the cross sectional area of arms is A, then force produced due to height of 2h,
`F= PA`
`=2hpg xxA = (2pgA)h`
`therefore F propto h " "therefore F = kh`
Where `k= 2pgA` is constant.
This force is opposite to displacement, `F= -kh`
Hence, restoring force acts on oscillation of mercury column is directly proportional to displacement and opposite to displacement. So the motion of mercury in U-tube is simple harmonic motion.