A research submarine has a 30-cm-diameter window that is 8.6cm thick. The manufacturer says the window can withstand forces up to 1.2*10^6 N. What is the submarine's maximum safe depth in salt water?

The pressure inside the submarine is maintained at 1.0 atm.

What is missing is the density of salt water. Let's denote it ρ.

The pressure at depth d then is:

p = p0 + ρ g d

, where p0 is the atmospheric pressure at the surface of the sea. We assume it is equal to the 1 atm inside the sub.

Hence the pressure difference between inside and outside of the window is:

Δp = ρ g d

The force on the window is this pressure difference, multiplied by the area of the circular window of diameter D. The latter equals

A = ¼ π D².

So

F = p A = ¼ π D ²ρ g d.

This should be less than Fmax.

Hence d < Fmax / (¼ π D ²ρ g).