Europa, a satellite of Jupiter, appears to have an ocean beneath its icy surface. Proposals have been made to send a robotic submarine to Europa to see if there might be life there. There is no atmosphere on Europa, and we shall assume that the surface ice is thin enough that we can neglect its weight and that the oceans are fresh water having the same density as on the earth. The mass and diameter of Europa have been measured to be 4.78×1022 kg and 3130 m, respectively.
If the submarine intends to submerge to a depth of 110 m, what pressure must it be designed to withstand?
If you wanted to test this submarine before sending it to Europa, how deep would it have to go in our oceans to experience the same pressure as the pressure at a depth of 110 m on Europa?
Tbe weight of a column of water of area A and height H is
weight=densiy*area*h*k were k is the acceleration of gravity
k=GMasseuropa/radius^2
Pressure = weight/Area
To determine the pressure the submarine must withstand on Europa, we can use the equation for pressure:
Pressure = Density × gravity × Depth
Where:
- Density is the density of the liquid (fresh water on Earth)
- Gravity is the acceleration due to gravity (on Europa)
- Depth is the depth to which the submarine intends to submerge
Given that the density of fresh water on Earth is approximately 1000 kg/m^3, and the acceleration due to gravity on Europa is approximately 1.315 m/s^2, we can substitute these values into the equation to find the pressure.
Pressure = 1000 kg/m^3 × 1.315 m/s^2 × 110 m
Calculating this expression, the pressure the submarine must withstand on Europa is approximately 144,650 Pa (pascals).
To determine the equivalent pressure at the same depth in our oceans on Earth, we can use the same equation. However, we need to use the acceleration due to gravity on Earth and the density of seawater, which is approximately 1025 kg/m^3.
Given that the acceleration due to gravity on Earth is approximately 9.8 m/s^2, we can substitute these values into the equation:
Pressure = 1025 kg/m^3 × 9.8 m/s^2 × Depth
We want to find the depth at which this pressure is equivalent to the pressure at a depth of 110 m on Europa, so we can set up the following equation:
144,650 Pa = 1025 kg/m^3 × 9.8 m/s^2 × Depth
Solving for Depth, we can rearrange the equation:
Depth = 144,650 Pa / (1025 kg/m^3 × 9.8 m/s^2)
Calculating this expression, the submarine would need to go to a depth of approximately 14.87 m in our oceans on Earth to experience the same pressure as a depth of 110 m on Europa.
To calculate the pressure the submarine must be designed to withstand on Europa, we can use the hydrostatic pressure formula:
P = ρ*g*h
where:
P is the pressure
ρ is the density of the fluid
g is the acceleration due to gravity
h is the depth
Given that the density of the ocean on Europa is the same as that on Earth, which is about 1000 kg/m^3, the acceleration due to gravity on Europa is approximately 1.314 m/s^2, and the depth is 110 m, we can substitute the values into the formula:
P = (1000 kg/m^3)*(1.314 m/s^2)*(110 m)
P ≈ 144,540 Pa
Therefore, the submarine must be designed to withstand a pressure of approximately 144,540 Pa on Europa.
To determine how deep the submarine would have to go in our ocean to experience the same pressure as at a depth of 110 m on Europa, we can use the same formula and rearrange it to solve for the depth:
h = P/(ρ*g)
Using the values of the pressure on Europa (144,540 Pa), the density of seawater on Earth (approximately 1025 kg/m^3), and the acceleration due to gravity on Earth (9.8 m/s^2), we can calculate the depth:
h = (144540 Pa)/(1025 kg/m^3 * 9.8 m/s^2)
h ≈ 14.7 m
Therefore, the submarine would have to go to a depth of approximately 14.7 m in our oceans to experience the same pressure as at a depth of 110 m on Europa.