Before you read this, I suggest you read posts 17.6 and 17.7

A cruise ship has a mass of about 5 ×10⁷ kg. (This is usually called the tonnage of the ship and is usually measured in US tons or metric tonnes). How can this be measured?

First notice that part of a floating ship is beneath the surface of the water. We know that most of an iceberg is under water. And part of any floating object must be submerged, for reasons that we shall soon find out about. But for an object like a cork, the submerged part is very small.

In post 17.6 we have that when an object displaces a volume V of a liquid of density ρ_o, it experiences an upward, buoyancy (post 17.7) force of – Vρ_og; the minus sign is to show that this force acts in the opposite direction to gravity and g represents the modulus of the gravitational field (posts 16.16 and 16.17). A floating ship doesn’t move upwards or downwards; so the total force acting in the up-and-down direction must be zero (see post 16.13). This means that the sum of the buoyancy force and the gravitation force must be zero. If m represents the mass of the ship, this can be expressed as

mg – Vρ_og = 0.

So the mass of the ship is given by

m = Vρ_o.

We can easily measure the density of seawater, so all we need to know if the volume of the ship that is submerged.

To calculate the submerged volume, we need to know the total depth of the ship’s hull; we then subtract the height (in blue) that we can see above water from the total depth to calculate the submerged (in red) depth. We also need the engineering drawings, made when the ship was designed, that show the shape and dimensions of the hull.

We can see that the (red) submerged depth is not the same throughout the length of the ship. To calculate the volume, we need to describe the submerged length of the ship as a series of thin slices. The thickness of each slice, multiplied by its cross-sectional area, is a good approximation to the volume of the slice. So, if we add the volumes of all the slices, we have a good approximation to the submerged volume.

How can we calculate the cross-sectional area of a slice?

We need to know dimensions of the hull, so that we can divide each submerged cross-sectional slice into a series of strips, like the one shown in the picture above. If each strip is sufficiently thin that the curved end (at the bottom) is almost the same as a straight line, the area of the strip is its width, w, multiplied by its average height, (h₁ + h₂)/2. If we add the areas of all the strips together, we have the cross-sectional area of the slice.

Now we have all the information we need to work out the mass of the ship!

Related posts.

17.7 Solid gold
17.6 Floating
16.44 Density
16.17 Weight
16.16 Gravity