Question

The Jovian moon Europa may have oceans (covered by ice, which can be ignored). What would the pressure be \(1.00 \mathrm{~km}\) below the surface of a Europan ocean? The surface gravity of Europa is \(13.5 \%\) that of the Earth's.

Short answer

Answer: The approximate pressure 1 km below the surface of a Europan ocean is 1,319,720 Pa.

Step-by-step solution

Calculate the acceleration due to gravity on Europa

To calculate the acceleration due to gravity on Europa, we need to find 13.5% of Earth's gravity (g = 9.81 m/s²). Europa's gravity = 0.135 × 9.81 m/s² Europa's gravity ≈ 1.32 m/s²

Convert depth to meters

As the depth given is in kilometers, we'll need to convert it to meters before using it in our formula: Depth in meters = 1.00 km × 1000 Depth in meters = 1000 m

Calculate the pressure at the given depth

Now, we can use the hydrostatic pressure formula, P = ρgh, and plug in our values: ρ = 1000 kg/m³ (density of water) g ≈ 1.32 m/s² (surface gravity of Europa) h = 1000 m (depth below surface) P = (1000 kg/m³) × (1.32 m/s²) × (1000 m) P ≈ 1319720 Pa So the pressure 1 km below the surface of the Europan ocean is approximately 1,319,720 Pa.

Key concepts

Acceleration Due to Gravity

The acceleration due to gravity is a vital concept in physics describing the rate at which an object will accelerate towards the center of a celestial body, such as a planet or moon, due to its gravitational pull. It is an important factor in calculating hydrostatic pressure in fluids. Earth’s standard acceleration due to gravity is approximately 9.81 m/s², affecting everything from the way we move to how objects fall.

Gravity not only influences the motion of objects but also plays a key role in shaping the structure of the universe, including the formation of galaxies and the behavior of celestial bodies in space. The force of gravity on different planets or moons varies depending on their mass and radius. Understanding this concept helps students relate gravity to the everyday experience, like how much they would weigh on another planet or moon, such as Jupiter's moon Europa.

Europa’s Surface Gravity

Jupiter's moon Europa presents a unique example where its surface gravity, which is roughly 13.5% of Earth's, significantly influences calculations when studying its potential oceans. The calculation of pressure at a depth within these oceans begins by grasping Europa's lesser gravity. This lesser gravity means objects on Europa would weigh less and that liquids in Europa's oceans exert less pressure at depth compared to Earth's oceans. For instance, the pressure at an equivalent depth in the ocean on Earth would be greater due to the stronger gravitational acceleration.

Comparing Gravitational Acceleration: On Earth, the standard acceleration due to gravity is 9.81 m/s², whereas on Europa it's only 1.32 m/s², a value derived from multiplying Earth's gravity by 0.135. Understanding Europa's lower gravitational force is essential for any calculations involving motion or force on its surface or within its speculated subsurface oceans.

Pressure Formula

The hydrostatic pressure formula, symbolically represented as P = ρgh, is a pivotal equation in physics, used to calculate the pressure exerted by a fluid at rest due to the gravitational force. This formula explains that pressure (P) at a certain depth in a fluid is the product of the fluid's density (ρ), the acceleration due to gravity (g), and the height of the fluid column above the point (h). In the context of a liquid like water, with a roughly constant density of 1000 kg/m³, what changes when you move between celestial bodies is the gravity variable.

Using the Formula: Given the lower gravity on Europa, the pressure under its oceans differs from that on Earth. As illustrated in the exercise, when calculating the pressure at 1 kilometer below Europa's ocean surface, the identified factors are applied: a reduced gravity (g) consistent with Europa's gravity, the depth converted to meters (h), and the standard density of water (ρ). By doing so, you can understand how different celestial bodies affect fluid pressure, a concept crucial for applications ranging from submarine design to the study of extraterrestrial oceans.