A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit?

#### Solution 1

The given system of water, mercury, and methylated spirit is shown as follows:

Height of the spirit column, *h*_{1} = 12.5 cm = 0.125 m

Height of the water column, *h*_{2} = 10 cm = 0.1 m

*P*_{0} = Atmospheric pressure

*ρ*_{1} = Density of spirit

*ρ*_{2} = Density of water

Pressure at point B = `P_0 + h_1rho_1g`

Pressure at point D = `P_0 + h_2rho_2g`

Pressure at points B and D is the same.

`P_0 + h_1rho_1g =h_2rho_2g`

`rho_1/rho_2 = h_2/h_1`

= 10/12.5 = 0.8

Therefore, the specific gravity of spirit is 0.8.

#### Solution 2

For water column in one arm of U tube, _{ }h_{1} = 10.0 cm; ρ_{1} (density) = 1 g cm^{-3}

For spirit column in other arm of U tube, _{ }h_{2} = 12.5 cm; ρ_{2} =?

As the mercury columns in the two arms of U tube are in level, therefore pressure exerted by each is equal.

Hence _{ }h_{1}ρ_{1}g = _{ }h_{2}ρ_{2}g or ρ_{2} = h_{1}ρ_{1}/h_{2} =10 x 1/12.5 = 0.8 g cm^{-3}

Therefore, relative density of spirit = ρ_{2}/ρ_{1} = 0.8/1 = 0.8