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 4.78×1022kg and 3130km , respectively. If the submarine intends to submerge to a depth of 150m , 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 150m on Europa?
From what I can see with this question, the first thing to do is convert the 3130km to 3.13*10^6 m. Then doesn't pressure equal P(atmosphere) + h*g*density of fluid? I have no idea how to work out this problem. I don't want an answer, I just want to understand how to get it.
You need the ratio of g' on Europa to g on Earth. That ratio is
g'/g
= (Meuropa/Mearth)(Rearth/Reuropa)^2
Look up the numbers you need and compute the ratio. Then get g', the value on Europa.
For the pressure at depth h, the pressure will be
P = (water density)g' h
Ah, the depths of space exploration, quite literally in this case! Let's dive into this problem and explore the depths of pressure calculations together.
You're on the right track with the pressure equation! To calculate the pressure at a given depth in a fluid, we use the formula P = P0 + hρg, where P is the pressure at the depth, P0 is the atmospheric pressure, h is the depth, ρ is the density of the fluid, and g is the acceleration due to gravity.
For Europa, there is no atmosphere, so we can just focus on the second part of the equation, which is hρg. Given the depth, 150m, and the density of fresh water, we can substitute those values into the equation to find the pressure.
Now, to deal with the unit conversions, we know that 1km equals 1000m. So, the diameter of Europa, 3130km, is indeed 3.13 × 10^6m, as you correctly calculated.
To find the pressure at a depth of 150m on Europa, use the following formula:
P = hρg
Substitute the values:
h = 150m
ρ = density of fresh water (approximately 1000 kg/m^3)
g = acceleration due to gravity (approximately 9.8 m/s^2)
Plug in the numbers and calculate the pressure. This will give you the pressure that the submarine needs to withstand on Europa.
Now, if you want to test the submarine in our own oceans to experience the same pressure as at a depth of 150m on Europa, you can use the same formula and rearrange it to solve for the depth:
h = (P - P0) / (ρg)
Since P and P0 are the atmospheric pressures on Europa and Earth, respectively, and the density and acceleration due to gravity are the same, you can plug in the values and solve for h. This will give you the depth in our oceans that the submarine would need to go to experience the same pressure.
Well, I hope I didn't make you feel the pressure with my explanation! Go ahead and give it a shot. Remember, practice makes perfect, just like testing submarines in terrifi-... uh, testing submarines in Earth's oceans makes us feel secure about exploring Europa. You got this!
To solve this problem, we can use the formula for pressure:
P = P(atmosphere) + h * g * density
Where:
P is the pressure,
P(atmosphere) is the atmospheric pressure,
h is the depth (in meters),
g is the acceleration due to gravity, and
density is the density of the fluid (in this case, the ocean water).
Let's approach this step-by-step:
1. Determine the atmospheric pressure (P(atmosphere)):
Since there is no atmosphere on Europa, the atmospheric pressure is zero (P(atmosphere) = 0).
2. Convert the depth of submersion (h) from meters to kilometers:
The depth of submersion is given as 150 meters, which is equivalent to 0.15 kilometers.
3. Determine the acceleration due to gravity (g):
The acceleration due to gravity on Europa is not given directly in the question. However, we can use Newton's law of universal gravitation to calculate it:
g = G * (mass of Europa) / (radius of Europa)^2
where G is the gravitational constant. The value of G is approximately 6.674 x 10^-11 N m^2/kg^2.
The mass of Europa is given as 4.78 x 10^22 kg, and the radius is given as 3.13 x 10^6 meters. Plugging in these values into the equation, we can calculate g.
4. Determine the density of the fluid:
The question states that the oceans on Europa have the same density as on Earth. The density of fresh water on Earth is approximately 1000 kg/m^3.
5. Calculate the pressure on Europa:
Plug in the appropriate values into the pressure formula:
P = 0 + 0.15 km * g * 1000 kg/m^3
6. Determine the equivalent depth in Earth's oceans:
This question asks for the depth in Earth's oceans that will experience the same pressure as a depth of 150 meters on Europa. We can rearrange the pressure formula and solve for h:
h = (P - P(atmosphere)) / (g * density)
By plugging in the values of P(atmosphere), g, and density of the fluid (1000 kg/m^3), we can calculate the depth (h) in Earth's oceans.
Remember, these steps are provided to help you understand the process. You may now proceed with the calculations to find the pressure on Europa and the equivalent depth in Earth's oceans.
To solve this problem, we need to use the concept of pressure in a fluid. The pressure at a given depth in a fluid is given by the formula:
Pressure = P(atmosphere) + h * g * density of fluid
Where:
- P(atmosphere) is the atmospheric pressure at the surface of Europa (which we will assume to be zero since Europa has no atmosphere).
- h is the depth below the surface of Europa that the submarine intends to go (150m in this case).
- g is the acceleration due to gravity (which we'll assume to be the same as on Earth, approximately 9.8 m/s^2).
- density of fluid is the density of the fluid in which the submarine is submerged (which we'll assume to be the same as that of freshwater on Earth, approximately 1000 kg/m^3).
Now, let's plug in the values and calculate the pressure the submarine needs to withstand:
Pressure = 0 + 150m * 9.8 m/s^2 * 1000 kg/m^3
Calculating this gives us the pressure the submarine needs to withstand on Europa.
To find the depth in Earth's oceans where the submarine would experience the same pressure, we can rearrange the formula and solve for h:
h = Pressure / (g * density of fluid)
Once we have the value of h, we can convert it into an appropriate unit. The depth on Earth will be in meters because we've used meters for all the other measurements.
Remember to use the same values for g and the density of fluid (9.8 m/s^2 and 1000 kg/m^3, respectively) when calculating the depth in Earth's oceans.
I hope this explanation helps you understand the process of solving this problem.