Floating and Sinking

Two forces decide the initial motion

An object in a fluid experiences weight downwards and upthrust upwards. Weight is the gravitational force on the object. Upthrust is the resultant upward force exerted by the surrounding liquid or gas.

The directions and sizes of these forces determine the resultant force:

• if upthrust is greater than weight, the resultant force is upwards and the object accelerates upwards;
• if weight is greater than upthrust, the resultant force is downwards and the object accelerates downwards;
• if upthrust equals weight, the resultant force is zero, so an object already at rest can remain at rest.

An upward resultant force does not mean that upthrust will remain greater than weight indefinitely. The forces can change as the object moves and a new equilibrium may be reached.

How a floating object finds its level

Upthrust equals the weight of the fluid displaced by the immersed part of an object. It may be represented by

$U=\rho_{\text{fluid}}V_{\text{displaced}}g$

where $U$ is upthrust, $\rho_{\text{fluid}}$ is fluid density, $V_{\text{displaced}}$ is displaced volume and $g$ is gravitational field strength.

Suppose an object is placed gently on a liquid. As it moves farther down, more of it becomes submerged, so it displaces a greater volume of liquid. Its upthrust therefore increases. If it can displace fluid whose weight equals its own weight before it becomes completely submerged, it settles partially submerged.

A floating object at rest is in equilibrium:

$U=W$

Its upthrust and weight are equal and opposite, so the resultant force is zero. A heavier floating object must displace a greater weight of the same fluid, so it settles lower until the increased upthrust balances its weight.

Why density predicts floating or sinking

Density is mass per unit volume. For a completely submerged object, compare its average density with the density of the surrounding fluid.

If the object's average density is greater than the fluid's density, its weight is greater than the greatest upthrust available when its whole volume is submerged. It therefore sinks.

If the object's average density is lower than the fluid's density, it can displace its own weight of fluid before becoming completely submerged. It can therefore float with only part of its volume below the surface.

If the object and fluid have equal densities, a fully immersed object can have upthrust equal to weight. With no other forces acting, it can remain suspended within the fluid rather than rising or sinking.

This comparison can also be seen algebraically. For an object of volume $V$ that is fully immersed,

$W=\rho_{\text{object}}Vg$

and

$U=\rho_{\text{fluid}}Vg$

The volume and gravitational field strength are common to both expressions, so the comparison between $U$ and $W$ depends on the two densities.

The effect of fluid density

For the same displaced volume, a denser fluid has a greater mass and weight. It therefore produces a greater upthrust. An object may sink in a less dense fluid but float in a denser one.

A floating object also needs to displace less volume of a denser liquid before the displaced liquid's weight equals its own. It consequently floats higher in the denser liquid. In a less dense liquid, it must become more deeply submerged to obtain the same upthrust.

The relevant comparison is always between the object's average density and the density of the fluid it is actually in—not simply whether the object is less dense than water.

Hollow objects and ships

An object made from a dense material can still float if its shape gives it a sufficiently low overall average density. A steel ship encloses a large volume of air. Its total mass divided by its entire external volume is therefore lower than the density of water, even though solid steel itself is denser than water.

The hull also allows the ship to displace a large volume of water. It settles until the weight of displaced water—and therefore the upthrust—equals the ship's weight. Adding cargo increases the ship's weight, so it must sit lower and displace more water to reach equilibrium again.

The same principles apply in gases. For example, an object in air experiences upthrust equal to the weight of the air it displaces; it rises only if this upthrust exceeds its total weight.

Force test

• $U>W$: upward resultant force; the object accelerates upwards.
• $U<W$: downward resultant force; the object accelerates downwards.
• $U=W$: zero resultant force; an object at rest is in equilibrium.

A floating object at rest has upthrust equal to weight.

Displacement and fluid density

$U=\rho_{\text{fluid}}V_{\text{displaced}}g$

For the same displaced volume, increasing fluid density increases upthrust. A floating object sits higher in a denser fluid because it needs to displace less volume to balance its weight.

Density test

Compare average object density with fluid density:

• $\rho_{\text{object}}<\rho_{\text{fluid}}$: the object can float partially submerged;
• $\rho_{\text{object}}>\rho_{\text{fluid}}$: the object sinks;
• $\rho_{\text{object}}=\rho_{\text{fluid}}$: a fully immersed object can remain suspended.

Hollow objects may float because their overall average density includes their air-filled volume.