A uniform vertical tube of 40cm is sealed at upper end is lowered into Mercury and length of air column become 35cm calculate depth of immersion of tube if atm is 77cm of mercury
16cm
To calculate the depth of immersion of the tube in Mercury, we need to first determine the change in pressure acting on the air column.
Given:
Length of the air column in air (L1) = 40 cm
Length of the air column in Mercury (L2) = 35 cm
Atmospheric pressure (P1) = 77 cm of Mercury
Since the tube is sealed at the upper end, the pressure at the upper end is equal to atmospheric pressure.
Step 1: Calculate the pressure at the lower end of the tube using the equation:
P2 = P1 + ρgh
where:
P2 is the pressure at the lower end of the tube,
P1 is the atmospheric pressure, and
ρ is the density of Mercury (13.6 g/cm³) and g is the acceleration due to gravity (9.8 m/s²).
Converting the units of pressure and density:
P2 = 77 cm * (13.6 g/cm^3) * (1000 kg/m^3 / 13.6 g/cm^3) * (1 cm / 100 m)^2 * 9.8 m/s^2
P2 = 101,320 Pa
Step 2: Calculate the pressure difference:
ΔP = P1 - P2
ΔP = 77 cm * (13.6 g/cm^3) * (1000 kg/m^3 / 13.6 g/cm^3) * (1 cm / 100 m)^2 * 9.8 m/s^2
ΔP = 0 Pa
Step 3: Calculate the depth of immersion:
Depth = L1 - ΔL
Since ΔP = 0, there is no change in pressure and hence no change in the height of the air column.
Depth = 40 cm
Therefore, the depth of immersion of the tube in Mercury is 40 cm.
To calculate the depth of immersion of the tube in the mercury, we need to use the principles of hydrostatics.
We know that the atmospheric pressure is 77 cm of mercury (Hg). This means that the pressure at the surface of the mercury column is equal to the atmospheric pressure. Let's call this pressure "P1".
At the bottom of the mercury column, the pressure is higher due to the weight of the mercury above it. The pressure at the bottom of the mercury column is equal to the pressure at the bottom of the air column plus the pressure due to the height difference between the bottom of the mercury column and the top of the air column. Let's call this pressure "P2".
The pressure at a certain depth in a fluid can be calculated using the equation P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height or depth.
In this case, the density of mercury (ρ) is much higher than the density of air, so we can consider the density of air to be negligible compared to the density of mercury. Therefore, we can assume that the pressure at the top of the air column is equal to the atmospheric pressure.
Using these principles, we can set up the following equation:
P1 + ρ(air)gh = P2 + ρ(Hg)gh
Since P1 = atmospheric pressure = 77 cm Hg and P2 = pressure at the bottom of the mercury column, we can rearrange the equation to solve for h, the depth of immersion of the tube in the mercury:
h = (P1 - P2) / (ρ(Hg) - ρ(air))
Substituting the given values, we have:
h = (77 cm Hg - 35 cm Hg) / (density of mercury - density of air)
Now we just need to calculate the density of mercury and the density of air.
The density of mercury is approximately 13,600 kg/m^3 and the density of air is approximately 1.225 kg/m^3.
Converting the densities to the same units (cm), we have:
density of mercury = 13,600 kg/m^3 * (100 cm/m)^3 ≈ 13,600,000 cm^3
density of air = 1.225 kg/m^3 * (100 cm/m)^3 ≈ 1.225 cm^3
Now we can substitute the values into the equation:
h = (77 cm - 35 cm) / (13,600,000 cm^3 - 1.225 cm^3)
h ≈ 42 cm / 13,598,775 cm^3
h ≈ 0.000003093 cm
Therefore, the depth of immersion of the tube in the mercury is approximately 0.000003093 cm.