Q.19:- 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.
Area of cross-section of the U-tube = A
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)
2h is the height of the mercury column in the two arms
k is a constant, given by k = –F/h = 2Aρg
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
Hence, the mercury column executes simple harmonic motion with time period 2π √l/2g .