the volume of air bubble is doubled as it rises from the bottom of lake to its surface. the atmospheric presure is 75 cm of the mercury and the ratio of the density of mercury to that of lake water is 40/3. what is the depth of of the lake ?
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Depth of the lake = 79.92 m
Radius of the bubble at the bottom of lake = R.
Pressure of atmosphere = P = 75 cm of Hg.
Relative density of Hg to lake water = d = 40/3
Density of lake water = 13.6 gm/ml ÷ 40/3 = 40.8 /40 gm/ml
=> d = 1.02 gm/ml or 1020 kg/m³
Surface Tension of lake water = S. P2 = Air Pressure Inside the bubble.
P1 = Air Pressure outside the bubble.
We know that: P2 - P1 = 4S/R
However the excess pressure in a soap bubble due to Surface tension is very small. It is about 20 or 25 Pa. Where atmospheric pressure is of the order of 10⁵ Pa. So we neglect Surface Tension factor here.
As the radius of the air bubble doubles at the top, its volume becomes 8 times. We assume that the Lake temperature is the same at the bottom and at the top. Then as per Boyle's law, Pressure at the top of lake is 1/8 of that at the bottom.
Pressure at the bottom = ρ g h = 8 * P_atm
1020 kg/m³ * 9.81 m/sec² * h = 8 * 75/76 * 1.013 * 10⁵ Pa
h = 79.92 meters approx
Radius of the bubble at the bottom of lake = R.
Pressure of atmosphere = P = 75 cm of Hg.
Relative density of Hg to lake water = d = 40/3
Density of lake water = 13.6 gm/ml ÷ 40/3 = 40.8 /40 gm/ml
=> d = 1.02 gm/ml or 1020 kg/m³
Surface Tension of lake water = S. P2 = Air Pressure Inside the bubble.
P1 = Air Pressure outside the bubble.
We know that: P2 - P1 = 4S/R
However the excess pressure in a soap bubble due to Surface tension is very small. It is about 20 or 25 Pa. Where atmospheric pressure is of the order of 10⁵ Pa. So we neglect Surface Tension factor here.
As the radius of the air bubble doubles at the top, its volume becomes 8 times. We assume that the Lake temperature is the same at the bottom and at the top. Then as per Boyle's law, Pressure at the top of lake is 1/8 of that at the bottom.
Pressure at the bottom = ρ g h = 8 * P_atm
1020 kg/m³ * 9.81 m/sec² * h = 8 * 75/76 * 1.013 * 10⁵ Pa
h = 79.92 meters approx
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