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32 Buoyant Force

Archimedes’ Principle

Archimedes' Principle states that the buoyant force exerted on an object by a fluid is equal to the weight of the fluid displaced by the object.

Reinforcement Exercises

A weight hangs from a scale that reads 4 Newtons. The weight is placed above a cup of water. The water just reaches a spout on the cup which drains into an empty cup sitting on a digital scale which reads 0 Newtons. The weight is submerged in the water, and now the scale holding the weight reads only 1 Newton. Water from the first cup rose and drained through the spout into the previously empty cup sitting on the digital scale, which now reads 3 Newtons.
Demonstration of Archimedes’ Principle. The buoyant force is equal to the weight of the water displaced, which in this case is 3 N. The buoyant force cancels out 3 N worth of the objects weight, so the scale only pulls up with 1 N to hold the object in static equilibrium. As a result, the scale reads an apparent weight of only 1 N. Image Credit:  “Archimedes-principle” by MikeRun via Wikimedia Commons

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Buoyant Force and Density

A given mass of low density tissue will take up more volume relative to the same mass of high density tissue. Taking up the extra volume means more water is displaced when the body is submerged so the buoyant force will be larger compared to the weight than it would be for a more dense body. In turn, that means that apparent weight is smaller relative to actual weight  for bodies of higher density. By comparing weight and apparent weight, the body density can be determined. We will do that in the next chapter, but first we should become more familiar with the Buoyant force.

Everyday Example

The water displaced by a brick weighs less than the brick so the buoyant force cannot cancel out the weight of the brick and it will tend to sink (left diagram). To hold the brick in place you must provide the remaining upward force to balance the weight and maintain static equilibrium. That force is less than the weight in air so the brick appears to weigh less in the water (right diagram).

The diagram on the left shows a brick with an arrow labeled "weight" pointing downward from its center. A shorter "buoyant force arrow points upward. In the diagram on the right a second arrow points upward so that the total length of the upward arrows equals the length of the downward arrow. The new upward arrow is labeled "force supplied by you = apparent weight"
Free body diagrams for bricks in water. The brick on the left is sinking, the brick on the right is being held in place by you.

If you let go of the brick it will be out of equilibrium and sink to the pool bottom. At that point the pool bottom is providing the extra upward force to balance out the weight, and the brick is once again in static equilibrium.

The diagram shows a brick with an arrow labeled "weight" pointing downward from its center. A shorter "buoyant force arrow points upward. A second arrow points upward so that the total length of the upward arrows equals the length of the downward arrow. The second upward arrow is labeled "force supplied by pool bottom."
Free body diagram of a brick sitting on the bottom of a pool.

The water displaced by an entire beach ball weighs more than a beach ball, so if you hold one under water the buoyant force will be greater than the weight. Your hand is providing the extra downward force to balance out the forces and maintain static equilibrium (left diagram). When you let go, the forces will be unbalanced and the ball will begin moving upward (right diagram).

The diagram on the left shows a long arrow labeled "buoyant force" pointing upward from the center of the ball. An arrow labeled "weight" points downward along with a second downward arrow labeled "force supplied by you." The combined lengths of the downward arrows equals the length of the upward arrow. The arrow representing "force supplied by you" has been removed from the diagram on the right, the length of the upward buoyant force arrow is much longer than the length of the downward weight arrow.
Free body diagrams of a beach ball under water. The ball on the left is held in place by you. The ball on the right will float upwards.

The density of ice is only about 9/10 that of water. The weight of the water displaced by only 9/10 of the iceberg has the same weight as the entire iceberg. Therefore, 1/10 of the iceberg must remain exposed in order for the weight and buoyant forces to be balanced and the iceberg to be in static equilibrium.

An iceberg floating with roughly 9/10 of its volume submerged. Image Credit: “Iceberg” created by Uwe Kils (iceberg) and User:Wiska Bodo (sky) via Wikimedia Commons

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Check out this buoyancy simulation which lets you control how much objects of different masses are submerged and shows you the resulting buoyant force along with forces provided by you and a scale at the bottom of the pool (apparent weight).

Buoyancy

Reinforcement Exercises

Not-So-Everyday Example

Submarines control how much water they displace by pumping water in and out of tanks within the submarine. When water is pumped inside, then that water is not displaced by the sub and it doesn’t count toward increasing the buoyant force. Conversely, when water is pumped out that water is now displaced by the sub and the buoyant force increases, which is the concept behind the maneuver in the following video:

  1. "Archimedes-principle"By MikeRun [CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0)], from Wikimedia Commons ↵
  2. "Iceberg" created by Uwe Kils (iceberg) and User:Wiska Bodo (sky). [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons ↵
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