Oneend of a U-tube containing mercury is connected to a suction pump andthe other end to atmosphere. A small pressure difference ismaintained between the two columns. Show that, when the suction pumpis removed, the column of mercury in the U-tube executes simpleharmonic motion.
Oneend of a U-tube containing mercury is connected to a suction pump andthe other end to atmosphere. A small pressure difference ismaintained between the two columns. Show that, when the suction pumpis removed, the column of mercury in the U-tube executes simpleharmonic motion.
Area of cross-sectionof the U-tube = A
Density of the mercurycolumn = ρ
Acceleration due togravity = 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 heightof the mercury column in the two arms
k is a constant,given by 
Time period, 
Where,
m is the mass ofthe mercury column
Let l be thelength of the total mercury in the U-tube.
Mass of mercury, m= Volume of mercury × Density of mercury
= Alρ
∴
Hence, the mercurycolumn executes simple harmonic motion with time period
.
Area of cross-sectionof the U-tube = A
Density of the mercurycolumn = ρ
Acceleration due togravity = 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 heightof the mercury column in the two arms
k is a constant,given by
Time period,
Where,
m is the mass ofthe mercury column
Let l be thelength of the total mercury in the U-tube.
Mass of mercury, m= Volume of mercury × Density of mercury
= Alρ
∴
Hence, the mercurycolumn executes simple harmonic motion with time period.
