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) Where, 2h is the height of the mercury column in the two arms k is a constant, given by k=−F/h=2Aρg Time Period, T=2πkm=2π2Aρgm 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=2π2AρgAlρ=2π2gl
