Q. 14.19 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.
Let the height of each mercury column be h.
The total length of mercury in both the columns = 2h.
Let the cross-sectional area of the mercury column be A.
Let the density of mercury be
When either of the mercury columns dips by a distance x, the total difference between the two columns becomes 2x.
Weight of this difference is
This weight drives the rest of the entire column to the original mean position.
Let the acceleration of the column be a Since the force is restoring
which is the equation of a body executing S.H.M
The time period of the oscillation would be