We’ll conclude by designing a boat. FIGURE 14.23 is a physicist’s idea of a boat. Four massless but rigid walls are attached to a solid steel plate of mass $m_{\text{o}}$ and area A. As the steel plate settles down into the water, the sides allow the boat to displace a volume of water much larger than that displaced by the steel alone. The boat will float if the weight of the displaced water equals the weight of the boat.

FIGURE 14.23 A physicist’s boat.

In terms of density, the boat will float if ${\rho }_{\text{avg}}<{\rho }_{\text{f}}$. If the sides of the boat are height h, the boat’s volume is $V_{\text{o}}=Ah$ and its average density is ${\rho }_{\text{avg}}=m_{\text{o}}\text{/}{V}_{\text{o}}=m_{\text{o}}\text{/}Ah$. The boat will float if

Thus the minimum height of the sides, a height that would allow the boat to float (in perfectly still water!) with water right up to the rails, is

As a quick example, a $5\text{m}\times 10\text{m}$ steel “barge” with a 2-cm-thick floor has an area of $50{\text{m}}^2$ and a mass of 7900 kg. The minimum height of the massless walls, as given by Equation 14.17, is 16 cm.

Real ships and boats are more complicated, but the same idea holds true. Whether it’s made of concrete, steel, or lead, a boat will float if its geometry allows it to displace enough water to equal the weight of the boat.