### 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* × 2*h* *×* *ρ** *×* *g) = –2*A**ρ*g*h* = –*k* × Displacement in one of the arms (*h*)

Where,

2*h* is the height of the mercury column in the two arms

*k* is a constant, given by k = –*F*/*h* = 2*A**ρ*g

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 = *Al**ρ*

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