Problem 50 - University Physics with Modern Physics

Question

Submarines on Europa. Some scientists are eager to send a remote-controlled submarine to Jupiter's moon Europa to search for life in its oceans below an icy crust. Europa's mass has been measured to be \(4.8 \times 10^{22}\) kg, its diameter is \(3138 \mathrm{km},\) and it has no appreciable atmosphere. Assume that the layer of ice at the surface is not thick enough to exert substantial force on the water. If the windows of the submarine you are designing are 25.0 \(\mathrm{cm}\) square and can stand a maximum inward force of 9750 N per window, what is the greatest depth to which this submarine can safely dive?

Step-by-Step Solution

Answer

The submarine can dive safely to a depth of about 118.18 km in Europa's ocean.

Step 1: Calculate the radius of Europa

First, we find the radius of Europa using its diameter. The formula for radius is \( r = \frac{d}{2} \), where \( d \) is the diameter. Diameter of Europa, \( d = 3138 \text{ km} = 3138 \times 10^3 \text{ m} \). Thus, the radius \( r = \frac{3138 \times 10^3}{2} = 1569 \times 10^3 \text{ m} \).

Step 2: Calculate the gravitational acceleration on Europa

The gravitational acceleration \( g \) on the surface of a celestial body is given by the formula: \[ g = \frac{G \, M}{r^2} \] where \( G = 6.674 \times 10^{-11} \text{ m}^3\text{/kg/s}^2 \) is the gravitational constant, \( M = 4.8 \times 10^{22} \text{ kg} \) is the mass of Europa, and \( r = 1569 \times 10^3 \text{ m} \) is the radius of Europa. Substituting these values, we get: \[ g = \frac{6.674 \times 10^{-11} \times 4.8 \times 10^{22}}{(1569 \times 10^3)^2} \approx 1.32 \text{ m/s}^2 \].

Step 3: Determine the maximum pressure the window can withstand

The pressure \( P \) exerted by a force on an area is given by the formula: \[ P = \frac{F}{A} \] where \( F = 9750 \text{ N} \) is the maximum force the window can withstand, and \( A = 0.25 \times 0.25 \text{ m}^2 = 0.0625 \text{ m}^2 \) is the area of the window (since it is 25 cm square).So, the maximum pressure is: \[ P = \frac{9750}{0.0625} = 156000 \text{ Pa} \] (Pascals).

Step 4: Calculate the depth to which pressure equals maximum pressure

The pressure under a fluid is given by the equation: \[ P = \rho g h \] where \( \rho \) is the density of the fluid, \( g \) is the gravitational acceleration, and \( h \) is the depth. Rearrange to solve for \( h \): \[ h = \frac{P}{\rho g} \]Assuming the density of Europa's ocean is similar to water, \( \rho = 1000 \text{ kg/m}^3 \). Using \( P = 156000 \text{ Pa} \) and \( g = 1.32 \text{ m/s}^2 \), we find: \[ h = \frac{156000}{1000 \times 1.32} \approx 118181.82 \text{ meters} \].

Step 5: Final Calculation and Result

Rewriting the depth \( h \) in kilometers (since it's a very deep depth), we have:\[ h \approx 118.18 \text{ km} \]This is the maximum depth the submarine can safely dive without breaking its windows.

Key Concepts

Europa's gravitational acceleration

When exploring celestial bodies, it's crucial to understand the local gravity, which affects everything from vehicle design to surface operations. Europa, one of Jupiter's moons, has a distinct gravitational feature due to its mass and size. The gravitational acceleration on Europa can be calculated using Newton's law of universal gravitation.By knowing Europa's mass, which is approximately \(4.8 \times 10^{22}\) kg, and its radius (calculated from its diameter of \(3138\) km), we find its gravitational acceleration using the equation: \[ g = \frac{G \cdot M}{r^2} \]Here, \(G\) is the gravitational constant \(6.674 \times 10^{-11}\) \(\text{m}^3/\text{kg/s}^2\). This equation helps us determine that Europa's surface gravity is about \(1.32 \text{ m/s}^2\), significantly lower than Earth's gravity. This relatively weak gravitational pull impacts how objects move and interact on Europa's icy surface.

Pressure Calculation

In designing submarines for extraterrestrial environments like those of Europa, understanding pressure is essential. Pressure is the force exerted per unit area, and it increases with depth in a fluid, owing to the weight of the fluid above pushing down.For a submarine's window, the maximum pressure it can withstand without breaking can be calculated by dividing the maximum force by the window's area. Using the formula:\[ P = \frac{F}{A} \]For our scenario, where \(F = 9750\) N and the window area \(A = 0.0625\) \(\text{m}^2\), the maximum pressure is \(156000\) Pascals. This number tells us how deep the submarine can safely go without the windows being compromised by water pressure.Understanding this helps engineers develop safer and more efficient designs, crucial for exploring the vast and treacherous environments beneath extraterrestrial ice sheets.

Fluid Mechanics in Space

Fluid mechanics is a cornerstone of engineering practices, especially crucial in unusual extraterrestrial conditions, such as Europa's under-ice oceans. In these situations, the principles are similar to those on Earth but must account for unique variables like lower gravitational forces.Using the formula for pressure due to a fluid's depth:\[ P = \rho g h \]We can determine how deep a vehicle can submerge. Here, \(\rho\) is the fluid density (assumed to be similar to water on Earth at \(1000 \text{ kg/m}^3\)), \(g\) is the gravitational acceleration we calculated earlier, and \(h\) is the sought depth.By rearranging the formula to solve for depth (\(h\)), it becomes:\[ h = \frac{P}{\rho g} \]This tells us that for a submarine on Europa, supported by \(1.32 \text{ m/s}^2\) gravity, reaching depths of approximately \(118.18\) kilometers is feasible without structural failure, considering the limitations on pressure. This comprehensive understanding is vital for ensuring the reliability of scientific operations in such extreme environments.