14.19 One end of a U-tube containing mercury is connected to a suction pump and the other end to the 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.


Let the area of a cross-section of the U-tube = A
And the density of the mercury column = ρ
Acceleration due to gravity = g
Restoring force, F = weight of the mercury column of a certain height
F = –(volume×density×g)
F = –(A × 2h × ρ ×g) = –2Aρgh = –k × Displacement in one of the arms (h)
where 2h is the height of the mercury column in the two arms and k is a constant

k=-Fh=2Aρg
Time period,

T=2πmk=2πm2Aρg 

where m is the mass of the mercury column.
Let l be the length of the total mercury in the U-tube.
Mass of mercury, m = Volume of mercury × Density of mercury = Alρ

 T=2πAlρ2Aρg=2πl2g

Hence, the mercury column executes simple harmonic motion with the time period 2πl2g