Question

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 4.78×1022 kg and 3130 km , respectively

If the submarine intends to submerge to a depth of 70.0 m , 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 gauge pressure on Earth as the pressure at a depth of 70.0 m on Europa?

Answer 1

Acceleration due to gravity on surface of a planet is given by:

a=Gm/r², where a is acceleration due to gravity, G is gravitational constant=6.67*10^-11 N-m²/kg² ,m is mass of the planet, r is radius of the planet.

Also, absolute pressure at a depth h of a liquid is given by: P_abs=P_A+dah, where P_abs is absolute pressure, P_A is atmospheric pressure, d is density of liquid, a is gravitational acceleration,h is depth.

So,P_abs=P_A+d(Gm/r² )h, with variables as defined above.

Gauge pressure at a depth h is given by:P_g=dah, where P_g is gauge pressure, d is density of liquid, a is gravitational acceleration, h is depth.

Given mass of planet=4.78*10²² kg, Radius=diameter/2=3130/2 km= 1565 km= 1565000 m,Also,P_atm=0 as there is no atmosphere.Depth=70 m,density of water=1000kg/m³

So,required absolute)pressure that the submarine must withstand (using formula for absolute pressure)

=0+1000*[6.67*10^-11*4.78*10²²/(1565000*1565000)]*70=91121.97 Pa

On earth,a=9.8 m/s², required gauge pressure=91121.97 Pa. Let the required depth be H meters.

So,using formula for gauge pressure,91121.97=1000*9.8*H=>H=91121.97/(1000*9.8)=9.3 meters.