# 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?

Aug 31, 2015

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.

$\textcolor{b l u e}{d = {\rho}_{\text{oil"/rho_"water}}}$

Most of the time, relative density is compared with water's density at ${4}^{\circ} \text{C}$, which can be approximated to be ${\text{1000 kg/m}}^{3}$.

This means that the density of oil will be

${\rho}_{\text{oil" = d * rho_"water}}$

${\rho}_{\text{oil" = 0.8 * "1000 kg/m"""^3 = "800 kg/m}}^{3}$

The relationship between depth and pressure is given by the formula

$\textcolor{b l u e}{P = \rho \cdot g \cdot h} \text{ }$, 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 ${\text{N"/"m}}^{2}$ is equivalent to "kg"/("m" * "s"^2), and don't forget that you have kilonewtons, not Newtons

$h = \frac{P}{\rho \cdot g} = \left(120 \cdot 10 \text{^3color(red)(cancel(color(black)("kg")))/(color(red)(cancel(color(black)("m"))) * color(red)(cancel(color(black)("s"^2)))))/(9.8color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2))) * 800 color(red)(cancel(color(black)("kg")))/"m"^color(red)(cancel(color(black)(3)))) = color(green)("15.3 m}\right)$

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

$h = \frac{P}{\rho \cdot g} = \left(120 \cdot 10 \text{^3color(red)(cancel(color(black)("kg")))/(color(red)(cancel(color(black)("m"))) * color(red)(cancel(color(black)("s"^2)))))/(9.8color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2))) * 1000 color(red)(cancel(color(black)("kg")))/"m"^color(red)(cancel(color(black)(3)))) = color(green)("12.2 m}\right)$