# Upthrust and Viscosity

## Density

• Density, ρ is defined as the mass per unit volume. It is measured in
Kgm-3.
Density: Mass per unit volume

## Upthrust

• A fluid will exert a force upward on a body if it is partly or wholly submerged within it. This is because the deeper into a fluid you go, the greater the weight of it and so the greater the pressure. This difference in pressure between the top and the bottom of the object produces an upward force on it. This is called Upthrust.
• According to Archimedes' Principle, the upthrust on an object in a fluid is equal to the weight of the fluid displaced. So the volume of the object multiplied by the density of the fluid.
Upthrust = Weight of Fluid Displaced

## Viscosity

• In a fluid, each 'layer' experts a force of friction of each other 'layer'. This frictional force is also present when solid object moves through a liquid. This force is termed Viscous Drag. Viscous Drag is greater in Turbulent Flow than Laminar Flow.
• The size of the Viscous Drag in a fluid depends on the (coefficient of) Viscosity of that fluid. Viscosity is given the letter η and is measured in Kgm-2s or Pa s. The greater the Viscosity, the greater the Viscous Drag.
• In most liquids, Viscosity decreases as temperature increases, whereas in most gases, Viscosity increases as temperature increases. It is therefore important to always measure the temperature of a fluid when measuring Viscosity.
• It is possible to calculate the drag force exerted on a spherical object in a fluid using Stoke's Law:
F = 6πηrv
• Stoke's Law assumes Laminar Flow, and so low velocities.
• In this equation, v represents Terminal Velocity. This means that the forces acting on the object are balanced. This means that is it possible to form an equation be equating Weight with Upthrust and Viscous Drag (or, in the case of Upward Motion, Upthrust with Weight and Viscous Drag).

$\frac{4}{3}\pi r^3 \rho _{object}\,g = \frac{4}{3}\pi r^3 \rho _{fluid}\,g + 6\pi\eta rv$