# Compared to a low-density spherical particle, a high-density spherical particle of the same size will sink through water at what rate?

Jan 1, 2017

The high-density particle will sink at a faster rate.

#### Explanation:

Stokes' Law

Stokes' law states that the settling velocity $v$ of small spheres in a fluid is given by

v = (gd^2(ρ_p - ρ_f))/(18eta)

where

$g$ = the gravitational acceleration ($\text{m·s"^"-2}$)
$d$ = the diameter of the particles ($\text{m}$)
ρ_p = the mass density of the particles ($\text{kg·m"^"-3}$)
ρ_f = the mass density of the fluid ($\text{kg·m"^"-3}$)
$\eta$ = the dynamic viscosity of the fluid ($\text{kg·m"^"-1""s"^"-1}$)

Thus, for two particles with the same diameter but different densities sinking in the same fluid, the relative sinking rates are

v_2/v_1 = ((color(red)(cancel(color(black)(g)))d_2^2color(red)(cancel(color(black)((ρ_p - ρ_f)))))/(color(red)(cancel(color(black)(18eta)))))/((color(red)(cancel(color(black)(g)))d_1^2color(red)(cancel(color(black)((ρ_p - ρ_f)))))/(color(red)(cancel(color(black)(18eta))))) = d_2^2/d_1^2 = (d_2/d_1)^2

That is, the ratio of their sinking rates equals the square of the ratio of their densities.

The consequence to phytoplankton

Sinking due to gravity is a significant source of mortality for phytoplankton because most cells are slightly denser than water.

Cells with the same size but twice the density as other cells will sink four times as fast.

Thus, the denser cells will need a higher degree of turbulence to persist in the water column.