Problem 71 - College Physics

Question

Exploring Europa's oceans. 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 \(4.78 \times 10^{22}\) kg and 3130 \(\mathrm{km}\) , respectively. (a) If the submarine intends to submerge to a depth of \(100 \mathrm{m},\) what pressure must it be designed to withstand? (b) 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 100 \(\mathrm{m}\) on Europa?

Step-by-Step Solution

Answer

The submarine must withstand 131500 Pa on Europa; test it at 3.08m depth on Earth.

Step 1: Calculate Europa's Gravitational Acceleration

Europa's gravitational acceleration can be calculated using the formula: \[ g = \frac{G \, M}{r^2} \]where \(G\) is the gravitational constant \(6.674 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2\), \(M\) is the mass of Europa \(4.78 \times 10^{22} \, \text{kg}\), and \(r\) is the radius of Europa \(\frac{3130 \, \text{km}}{2} = 1565 \, \text{km} = 1.565 \times 10^6 \, \text{m}\).

Step 2: Compute Gravitational Acceleration on Europa

Using the formula from Step 1:\[ g = \frac{6.674 \times 10^{-11} \, \times 4.78 \times 10^{22}}{(1.565 \times 10^6)^2} \approx 1.315 \, \text{m/s}^2 \]This is the gravitational acceleration on the surface of Europa.

Step 3: Calculate the Pressure at 100m Depth on Europa

The pressure at a depth \(h\) in a fluid of density \(\rho\) is given by:\[ P = P_0 + \rho \cdot g \cdot h \]where \(P_0\) is the surface pressure (0 Pa in Europa since there is no atmosphere), \(\rho\) is the density of water \(1000 \, \text{kg/m}^3\), \(g\) is the gravitational acceleration on Europa \(1.315 \, \text{m/s}^2\), and \(h\) is 100 m. Thus:\[ P = 1000 \, \times 1.315 \, \times 100 = 131500 \, \text{Pa} \].

Step 4: Find Equivalent Depth in Earth's Ocean

On Earth, the pressure at depth \(h\) is also given by: \[ P = P_0 + \rho \cdot g \cdot h \]where \(P_0\) is the atmospheric pressure at sea level (approximately 101325 Pa), \(\rho\) is the same density of water \(1000 \, \text{kg/m}^3\), \(g\) is Earth's gravitational acceleration \(9.81 \, \text{m/s}^2\), and we need to find \(h\) such that:\[ 131500 = 101325 + 1000 \, \times 9.81 \, \times h \].

Step 5: Solve for Earth's Ocean Depth

Rearranging the equation from Step 4:\[ 131500 - 101325 = 1000 \times 9.81 \times h \]\[ 30175 = 9810 \times h \]\[ h = \frac{30175}{9810} \approx 3.08 \, \text{m} \].

Key Concepts

Gravitational Acceleration

Gravitational acceleration is a key concept in planetary physics, helping us understand how different bodies in space exert forces on objects on their surface. For Europa, determining this value allows us to predict how an object, like a submarine, will behave under the moon's gravitational pull. Europa's gravitational pull is calculated using Newton's law of universal gravitation:

• The formula is: \( g = \frac{G \times M}{r^2} \\), where \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2) \), \( M \) is the mass of Europa \( (4.78 \times 10^{22} \, \text{kg}) \), and \( r \) is the radius of the moon.
• In our specific case for Europa, the radius is half of the moon's diameter, giving \( r = 1.565 \times 10^6 \, \text{m}\).

The computed gravitational acceleration \( g \) of 1.315 \( \text{m/s}^2 \) shows that Europa's gravity is much weaker than Earth's, explaining why objects are less heavily pulled down than on our planet.

Pressure Calculation

Pressure calculation is essential when designing equipment like submarines, ensuring they can withstand extreme conditions underwater. The pressure at a certain depth inside Europa's fluid oceans is found using the relationship:

• Pressure, \( P = P_0 + \rho \cdot g \cdot h \\), involves adding the atmospheric pressure \( P_0 \) to the hydrostatic pressure due to the fluid's weight above.
• For Europa, \( P_0 = 0 \) Pa because there is no atmosphere. With a water density \( \rho = 1000 \, \text{kg/m}^3\), gravitational acceleration \( g = 1.315 \, \text{m/s}^2\) and submersion depth \( h = 100 \, \text{m} \), the pressure is calculated to be 131500 Pa.

This non-trivial computation ensures that technology meant for space exploration is built to survive under unknown iterations of familiar environmental factors.

Physics Problem Solving

Physics problem solving requires a structured approach, especially in multi-step calculations like those involving gravitational forces and pressures. A proper problem-solving strategy often includes:

• Identifying all known values and what we’re required to find. This includes everything from gravitational constants to local gravitational acceleration, which may differ on each celestial body.
• Using correct formulas for each part of the problem, like those for gravitational acceleration and fluid pressure.
• Carefully substituting values with proper units into equations to avoid common mistakes.
• Breaking down complex problems into manageable steps, as done here with calculating pressures both on Europa and equivalent conditions on Earth.

This helps in streamlining the problem-solving process, minimizing errors, and enhancing comprehension, essential skills for any physics student or enthusiast.

Fluid Mechanics

Fluid mechanics explains how fluids behave both on Earth and in space, crucial for understanding underwater or space conditions. In our scenario:

• We calculated the pressure at a certain depth using fluid mechanics principles. This pressure determines whether a submarine can withstand the force exerted by the fluid around it.
• The study of fluid mechanics in space-like environments, as on Europa, involves understanding both hydrostatic (fluid at rest) and hydrodynamic (fluid in motion) pressures.
• Given that Europa lacks an atmosphere, the understanding of hydrostatic pressure becomes even more vital to ensure equipment safety.

Fluid mechanics principles help address these space-specific challenges, equipping us to design technology for harsh environments. This knowledge of pressure calculations and behavior in fluids ensures that exploration missions are prepared for the unique conditions they'll meet.