Question

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 executes simple harmonic motion.

Answer 1

The restoring force,

F = weight of the mercury column of the height 2ξ

F = -(Volume) x density x g

or, F = -(A x 2ξ).ρg

= -2Aρgξ    ....(i)

Where A = Area of cross section of the U-tube

ρ = density of mercury

k = -F/ξ = 2A.ρg

Time period, T = 2π√{m/k} = 2π√{m/2Aρg}

Let L = length of the whole mercury column.

Therefore, mass of mercury,

M = Volume x density

= A.L.ρ.

Where L is the total length of the mercury column.

or, L = 2h, where 'h' is the height of the mercury column in U-tube. It shows that the mercury column executes S.H.M.