4.1.3 Law of Floatation

If an object is floating in a fluid, its weight W_o must be exactly balanced by upthrust U. Since upthrust is equal to the weight of the displaced fluid W_f,

$\displaystyle \displaystyle {{W}_{\text{o}}}={{W}_{f}}\text{ }$

This is called the law of floatation. Obviously, this law applies only when an object is floating. It does not apply when an object is sunk. It also does not apply when forces other than W_o and U are involved. (e.g. an object is tied to a string and lowered into the fluid)

We can write W_o and W_f as $\displaystyle \displaystyle {{\rho }_{o}}{{V}_{o}}g$ and $\displaystyle \displaystyle {{\rho }_{f}}{{V}_{f}}g$ respectively, where V_o is the total volume of the object, and V_f is the volume of the displaced fluid.

Object is less dense than fluid

If $\displaystyle {{\rho }_{o}}<{{\rho }_{f}}$ , then the object will float with $\displaystyle \displaystyle \frac{{{{\rho }_{o}}}}{{{{\rho }_{f}}}}$ of its volume submerged.

$\displaystyle \displaystyle \begin{aligned}{{W}_{o}}&={{W}_{f}}\\{{\rho }_{o}}{{V}_{o}}g&={{\rho }_{f}}{{V}_{f}}g\\\frac{{{{V}_{f}}}}{{{{V}_{o}}}}&=\frac{{{{\rho }_{o}}}}{{{{\rho }_{f}}}}\end{aligned}$

Object is denser than fluid

If $\displaystyle {{\rho }_{o}}>{{\rho }_{f}}$ , then the object will sink. This is because even when the object is already fully submerged, the volume of fluid it can displace is only V_o. The upthrust it can achieve is only $\displaystyle {{U}_{{\max }}}={{\rho }_{f}}{{V}_{o}}g$ , which is obviously smaller than ${{m}_{o}}g={{\rho }_{o}}{{V}_{o}}g$ , which is the weight of the object W.

Object has the same density as the fluid

If $\displaystyle {{\rho }_{o}}={{\rho }_{f}}$ , then object can hover at equilibrium at any depth in the fluid. As long as the object is fully submerged, the weight of the displaced fluid is exactly equal to the weight of the object.