# Buoyancy

What do an ice cube and a block of wood have in common? Throw either material into water, and it will float. Well, mostly; each object will have its bottom part immersed, but the upper part will ride high and dry. People often say that wood and ice float because they are "lighter than water", but this is true only if we compare the masses of equal volumes of the substances. In other words, we need to compare the masses-per-unit-volume, meaning the densities, of each material with that of water. So we would more properly say that objects capable of floating in water must have densities smaller than that of water.

The “true” weight of an object is the downward force exerted by gravity on the object's mass. When the object is immersed in a fluid, this downward force is opposed by a net upward force (the buoyancy) that results from the displacement of this fluid by the object. The magnitude of this upward force is equal to the weight of fluid displaced by the object. So the apparent weight of the object immersed in the fluid is its true weight minus its buoyancy.

The source of the upward “buoyancy force” produced by the displaced fluid is basically the force of gravity acting on the fluid itself. In a glass of water, for example, each tiny layer of the liquid presses down on, and adds it weight to the layers beneath it, creating a hydrostatic pressure gradient that increases with depth. If we now drop a solid object into the water, the object will experience not only the force of gravity acting on its own mass (which we call its “true weight”), but also the hydrostatic forcees due to the fluid pressing against each point on its surface.

Because this hydrostatic pressure (indicated here by the length and weight of the blue arrows) increases with depth, there is more force exerted on the bottom of the object than on its top. The overall effect is to create a net upward “buoyancy force” that opposes the true weight of the object.The magnitude of this force is equal to the mass of liquid displaced by the object.

Example 4

An object weighs 36 g in air and has a volume of 8.0 cm3. What will be its apparent weight when immersed in water?

SOLUTION

When immersed in water, the object is buoyed up by the mass of the water it displaces, which of course is the mass of 8 cm3 of water. Taking the density of water as unity, the upward (buoyancy) force is just 8 g. The apparent weight will be

$(36\; g) – (8\; g) = 28\; g$

Air is of course a fluid, and buoyancy can be a problem when weighing a large object such as an empty flask. The following problem illustrates a more extreme case:

Example 5

A balloon having a volume of 5.000 L is placed on a sensitive balance which registers a weight of 2.833 g. What is the "true mass" of the balloon if the density of the air is 1.294 g L–1?

SOLUTION

The mass of air displaced by the balloon exerts a buoyancy force of

(5.000 L) × (1.294 g L –1) = 6.470 g.

Thus the true weight of the balloon is this much greater than the apparent weight: (2.833 + 6.470) g = 9.303 g.

Example 6

A piece of metal weighs 9.25 g in air, 8.20 g in water, and 8.36 g when immersed in gasoline. a) What is the density of the metal? b) What is the density of the gasoline?

SOLUTION

When immersed in water, the metal object displaces (9.25 – 8.20) g = 1.05 g of water whose volume is (1.05 g) / (1.00 g cm–3) = 1.05 cm3. The density of the metal is thus

(9.25 g) / (1.05 cm3) = 8.81 g cm–3

The metal object displaces (9.25 - 8.36) g = 0.89 g of gasoline, whose density must therefore be (0.89 g) / (1.05 cm3) = 0.85 g cm–3.

### Floating – "the tip of the iceberg"

When an object floats in a liquid, the portion of it that is immersed has a volume that depends on the mass of the same volume of displaced liquid.

Figure: A photomontage of what a whole iceberg might look like. Image used with permission from Wikipedia.

Example 7

A cube of ice that is 10 cm on each side floats in water. How many cm does the top of the cube extend above the water level? (Density of ice = 0.917 g cm–3.)

SOLUTION

The volume of the ice is (10 cm)3 = 1000 cm3 and its mass is (1000 cm3) x (0.917 g cm–3) = 917 g. The ice is supported by an upward force equivalent to this mass of displaced water whose volume is (917 g) / (1.00 g cm–3) = 917 cm3 . Since the cross section of the ice cube is 100 cm2, it must sink by 9.17 cm in order to displace 917 cm3 of water. Thus the height of cube above the water is (10 cm – 9.17 cm) = 0.83 cm.

... hence the expression, “the tip of the iceberg”, implying that 90% of its volume is hidden under the surface of the water.

### Positive, negative, and neutral buoyancy

An object (such as an iceberg) whose density is smaller than that of the water in which it is immersed, will float on the surface. This occurs because the volume V of the more dense fluid required to support the object is smaller than the volume of the object itself. Thus only that portion of the object having this same volume V need be submerged in order to counter the object's weight. This condition is known as positive buoyancy.

If the density of the object exceeds that of the fluid, the weight of fluid it displaces (the buoyant force) will be smaller than the object’s true weight, and thus unable to support it. In this condition of negative buoyancy, the object experiences a net downward force at any depth, so it will sink until it strikes the bottom.

Note

It is interesting to note that it takes much less force to lift a sunken object such as a ship up to just under the water's surface, than it does to lift it completely out of the water, at which point the buoyancy force that reduces its apparent weight is no longer present.

### Neutral buoyancy and its control

If the density of an object is the same as that of the fluid, the upward and downward forces balance out to zero.  In this condition of neutral buoyancy, the object "floats" within the fluid.

• Fish, marine mammals, scuba divers, and submarines must be able alter their average densities by switching between positive-, negative- and neutral buoyancy in order to ascend, descend, or maintain themselves at a constant  depth under water.
• Mammals, including dolphins and humans, are naturally close to having neutral buoyancy, and use their appendages to move up or down.

• Most fish possess an organ known as a swim bladder whose degree of inflation enables them to switch buoyancy modes. These bladders are not connected to the mouth; they are inflated or deflated by oxygen gas that is extracted from (or returned to) the blood.
• Unclothed humans commonly tend to have a slight degree of positive buoyancy, enabling them to easily float on the water surface, especially if they possess generous amounts of body fat. People who are lean and more muscular will more likely be negatively buoyant. Scuba divers are trained to control their buoyancy by adjusting their lung inflation. The use of weight belts or external bladders provides additional flexibility.
• Submarines employ tanks that are filled with compressed air for surface cruising, or with seawater for under-water maneuvering.