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. | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-14 Oscillations



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.

 

 

Answer:-

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)
Where,
2h is the height of the mercury column in the two arms
k is a constant, given by k =  –F/h = 2Aρg



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

Hence, the mercury column executes simple harmonic motion with time period 2π √l/2g .