Calculating buoyant force: dependency on shape

To find the buoyant force, we must find the weight of water displaced. We can do this by using the densities of water and steel given in Table 11.1. We note that, since the steel is completely submerged, its volume and the water’s volume are the same. Once we know the volume of water, we can find its mass and weight.

First, we use the definition of density $\rho =\frac{m}{V}$ to find the steel’s volume, and then we substitute values for mass and density. This gives

Because the steel is completely submerged, this is also the volume of water displaced, $V_{\text{w}}$. We can now find the mass of water displaced from the relationship between its volume and density, both of which are known. This gives

By Archimedes’ principle, the weight of water displaced is $m_{\text{w}}g$, so the buoyant force is

The steel’s weight is $m_{\text{w}}g=9\text{.}\text{80}\times {\text{10}}^7\text{N}$, which is much greater than the buoyant force, so the steel will remain submerged. Note that the buoyant force is rounded to two digits because the density of steel is given to only two digits.

Here we are given the maximum volume of water the steel boat can displace. The buoyant force is the weight of this volume of water.

The mass of water displaced is found from its relationship to density and volume, both of which are known. That is,

The maximum buoyant force is the weight of this much water, or

The maximum buoyant force is ten times the weight of the steel, meaning the ship can carry a load nine times its own weight without sinking.