4.3.2 Law of Floatation

• When a foreign object is submerged in a fluid, the surrounding fluid will exert pressure forces perpendicularly into each point on the surface of the foreign object.
• Because pressure increases with depth, the resultant of these pressure forces is an vertically upward force. This force is called the force of upthrust, U.

• With some simple reasoning, we can deduce that the magnitude of the upthrust must be equal to the weight of the displaced fluid. Hence

$U={{m}_{f}}g={{rho }_{f}}{{V}_{f}}g$

• To generate a larger upthrust, a body must displace more fluid.
• Similar to weight, upthrust appears to act at a single point, the centre of gravity of the displaced fluid.

• Compare the upthrust U

$U={{m}_{f}}g={{rho }_{f}}{{V}_{f}}g$

• with the weight of the object W

$W={{m}_{o}}g={{rho }_{o}}{{V}_{o}}g$

• When an object is fully submerged, Vf = Vo. This also represents the maximum upthrust the object can generate.

• If ρo > ρf , then W > Umax.
• The object will sink all the way down.
• If ρo = ρf, then W = Umax.
• The object will neither sink nor float.
• It can hover at equilibrium at any depth in the fluid as long as it is fully submerged.
• If ρo < ρf, then W < Umax.
• The object will float partially submerged.
• It will displace just enough fluid to generate the amount of upthrust exactly equal to its weight.
• The lower the object’s density, the higher it will float.