Question

At what depth below the surface of oil, relative density 0.8, will the oil produce a pressure of 120 kN/m2? What depth of water is this equivalent to?

Answer 1

That pressure will be felt at a depth of 15.3 meters.

Explanation:

You need to know two things in order to solve this problem, what relative density is and what the formula that establishes a relationship between depth and pressure looks like.

Relative density (you'll sometimes see this being referred to as specific gravity) is simply the ratio between the density of a substance, in your case oil, and the density of a reference substance, which in your case I assume it's water, at specified conditions.

`color(blue)(d = rho_"oil"/rho_"water")`

Most of the time, relative density is compared with water's density at `4^@"C"`, which can be approximated to be `"1000 kg/m"""^3`.

This means that the density of oil will be

`rho_"oil" = d * rho_"water"`

`rho_"oil" = 0.8 * "1000 kg/m"""^3 = "800 kg/m"""^3`

The relationship between depth and pressure is given by the formula

`color(blue)(P = rho * g * h)" "`, where

`P` - the pressure produced at the depth `h`;
`g` - the gravitational acceleration;
`rho` - the density of the liquid.

Rearrange to solve for `h` - keep in mind that `"N"/"m"^2` is equivalent to `"kg"/("m" * "s"^2)`, and don't forget that you have kilonewtons, not Newtons

`h = P/(rho * g) = (120 * 10""^3color(red)(cancel(color(black)("kg")))/(color(red)(cancel(color(black)("m"))) * color(red)(cancel(color(black)("s"^2)))))/(9.8color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2))) * 800 color(red)(cancel(color(black)("kg")))/"m"^color(red)(cancel(color(black)(3)))) = color(green)("15.3 m")`

To find the depth at which this pressure would be produced in water, simply replace the density of the oil with that of water

`h = P/(rho * g) = (120 * 10""^3color(red)(cancel(color(black)("kg")))/(color(red)(cancel(color(black)("m"))) * color(red)(cancel(color(black)("s"^2)))))/(9.8color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2))) * 1000 color(red)(cancel(color(black)("kg")))/"m"^color(red)(cancel(color(black)(3)))) = color(green)("12.2 m")`