At the surface of a freshwater lake the air pressure is 1.0 atm. At what depth under water in the lake is the water pressure 3.8 atm?
To determine the depth at which the water pressure is 3.8 atm, we need to use the relationship between pressure and depth in a fluid.
The pressure at a given depth in a fluid is given by the equation:
P = P₀ + ρgh
Where:
P is the pressure at the given depth,
P₀ is the atmospheric pressure (1.0 atm in this case),
ρ is the density of the fluid (which is constant for freshwater),
g is the acceleration due to gravity (9.8 m/s² on Earth),
h is the depth.
We can rearrange the equation to solve for h:
h = (P - P₀) / (ρg)
Substituting the given values:
P = 3.8 atm
P₀ = 1.0 atm
ρ = density of freshwater (approximately 1000 kg/m³)
g = 9.8 m/s²
Plugging in the values:
h = (3.8 atm - 1.0 atm) / (1000 kg/m³ * 9.8 m/s²)
Simplifying:
h = 2.8 atm / (9800 kg/(m·s²))
h ≈ 0.0002857 m
Converting the depth to centimeters (cm):
h ≈ 0.0002857 m * 100 cm/m
h ≈ 0.02857 cm
Therefore, at a depth of approximately 0.02857 cm (or 2.857 mm), the water pressure in the lake is 3.8 atm.
To determine the depth underwater in the lake where the water pressure is 3.8 atm, you can use the concept of Pascal's law. According to Pascal's law, the pressure exerted by a fluid (in this case, water) increases with depth.
We can use the equation:
P = P₀ + ρgh
where:
P is the pressure at a given depth,
P₀ is the initial pressure (at the surface),
ρ is the density of the fluid (in this case, water),
g is the acceleration due to gravity, and
h is the depth.
In this case, we know the initial pressure P₀ is 1.0 atm and the pressure P is 3.8 atm. We also know that the density of fresh water is approximately 1000 kg/m³, and the acceleration due to gravity g is approximately 9.8 m/s².
Substituting the values into the equation, we have:
3.8 atm = 1.0 atm + (1000 kg/m³)(9.8 m/s²)h
Simplifying the equation, we get:
2.8 atm = (1000 kg/m³)(9.8 m/s²)h
Solving for h, the depth, we divide both sides of the equation by [(1000 kg/m³)(9.8 m/s²)]:
h = (2.8 atm) / [(1000 kg/m³)(9.8 m/s²)]
Now, let's calculate the value of h:
h ≈ 0.0285 m
Therefore, the depth underwater in the lake where the water pressure is 3.8 atm is approximately 0.0285 meters.