A U-tube open at both ends contains mercury of density 13\text{}600\text{ kg/m}^3. Water o… — JAMB UTME

A U-tube open at both ends contains mercury of density $13\text{}600\text{ kg/m}^3$ . Water of density $1000\text{ kg/m}^3$ is poured into one arm until the water column reaches a height of $27.2\text{ cm}$ . What is the difference in height, in $\text{cm}$ , between the mercury surfaces in the two arms?

Cevap: 2 cm

Cevap

The difference in height between the mercury surfaces in the two arms is

2.0\text{ cm}

At the boundary level where water meets mercury, the pressure produced by the

27.2\text{ cm}

water column must equal the pressure of the mercury column above that same horizontal level. Using

h_w \rho_w = h_m \rho_m

, we solve for the mercury height difference:

h_m = \frac{27.2 \times 1000}{13600} = 2.0\text{ cm}

Adım Adım Çözüm

Equate the hydrostatic pressure exerted by the water column to the hydrostatic pressure exerted by the balancing mercury column at the interface level.

h_w \rho_w g = h_m \rho_m g

At the same horizontal level within a continuous fluid at rest, the pressures must be equal.

Cancel the acceleration due to gravity (

g

) from both sides of the equation.

h_w \rho_w = h_m \rho_m

Gravity acts equally on both liquid columns.

Substitute the known values (

h_w = 27.2\text{ cm}

\rho_w = 1000\text{ kg/m}^3

\rho_m = 13600\text{ kg/m}^3

) into the pressure relation.

27.2 \times 1000 = h_m \times 13600

Inserting the physical quantities isolates the unknown mercury column height

h_m

Solve for the height difference

h_m

of the mercury levels.

h_m = \frac{27200}{13600} = 2.0\text{ cm}

Dividing the water pressure head product by the density of mercury yields the height of the mercury column.

Anahtar Kavram

Hydrostatic pressure equilibrium in immiscible fluids (U-tube manometer)

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