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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 ignore 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 4.78×10^22 kg k g and 3130 km k m , respectively.

Required:
If the submarine intends to submerge to a depth of 150 m , what pressure must it be designed to withstand?

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Answer:

[tex]193743.49\ \text{Pa}[/tex]

Explanation:

G = Gravitational constant = [tex]6.674\times 10^{-11}\ \text{Nm}^2/\text{kg}^2[/tex]

M = Mass of Europa = [tex]4.74\times 10^{22}\ \text{kg}[/tex]

R = Radius of Europa = [tex]\dfrac{3130}{2}=1565\ \text{km}[/tex]

h = Depth = 150 m

[tex]\rho[/tex] = Density of water = [tex]1000\ \text{kg/m}^3[/tex]

Acceleration due to gravity is given by

[tex]g=\dfrac{GM}{R^2}[/tex]

Pressure is given by

[tex]P=\rho gh\\\Rightarrow P=\rho \dfrac{GM}{R^2}h\\\Rightarrow P=1000\times \dfrac{6.674\times 10^{-11}\times 4.74\times 10^{22}}{1565000^2}\times 150\\\Rightarrow P=193743.49\ \text{Pa}[/tex]

The submarine should be designed to withstand a pressure of [tex]193743.49\ \text{Pa}[/tex].

The pressure due to column of liquid increases with the gravitational

attraction, density of the liquid and height of the liquid column.

  • The submarine should be able to withstand a pressure of approximately 194863.65 Pa.

Reasons:

Mass of Europa, M = 4.78 × 10²² kg

Diameter of Europa, r = 3,130 km

Required:

The pressure a submarine must be designed to withstand when submerged to a depth of 150 meters.

Solution:

Gravitational acceleration on Europa is given by the formula;

[tex]\displaystyle g = \mathbf{\frac{G \cdot M}{r^2}}[/tex]

G = Universal gravitational constant = 6.67408 × 10⁻¹¹ m³/(kg·s²)

[tex]Radius \ of \ the \ satellite\ Europa = \dfrac{3,130 \ km}{2} = \mathbf{1,565 \ km}[/tex]

[tex]\displaystyle g = \frac{6.67408 \times 10^{-11} \times 4.78 \times ^{22}}{1565000^2} \approx 1.303[/tex]

The gravitational acceleration on Europa, g ≈ 1.303 m/s²

Therefore;

Pressure due to a liquid's weight, P = ρ·g·h

The pressure at 150 m, P₁₅₀ = 150 × 1.303 × 997 = 194863.65

The pressure the submarine must be designed to withstand is, P₁₅₀ ≈ 194863.65 Pa.

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