32 Under Water Weight

Apparent Weight

When an object is held still under water it appears to weigh less than it does in air because the is helping to hold it up (balance its ).  For this reason, the reduced force you need to apply to hold the object is known as the . When a scale is used to weigh an object submerged in water the scale will read the apparent weight. When performing for body composition measurement the apparent weight is often called the (UWW).

Static Equilibrium

When weighing under water we know the must be equal to the difference between the and because the object remains still, which is a state known as For an object to be in static equilibrium, all of the forces on it must be balanced so that there is no . For the case of under water weighing, the buoyant force plus the force provided by the scale must perfectly balance the weight of the object, as long as the object is holding still. We can use arrows to represent the forces on an object and visualize how they are balanced or unbalanced. This type of diagram is known as a (FBD).  The direction of arrows shows the direction of the forces and the arrow lengths shows the size () of the force. In this case we call the arrows and say the forces they represent are vector quantities. The FBD for a person undergoing would look like this:

Arrows emerge upward and downward from dot. An arrow labeled "weight" points downward. An arrow labeled buoyant force points upward and a smaller arrow labeled "force supplied by scale" is added to the end of the buoyant force arrow. The combined length of the upward arrows is equal to the length of the downward weight arrow.
Free body diagram of an object hanging from a scale, submerged in water. The length of the weight arrow is equal to the combined lengths of the force supplied by the scale and the buoyant force. A scale will read the weight that it must supply, therefore it will read an apparent weight for submerged objects that is less than the actual weight.

We learned in the last chapter that scales measure the force that they are supplying to other objects. The scale must supply less to counteract and maintain when the is also helping, therefore the scale will provide a reading that is less than the actual weight.

Archimedes’ Principle

Measuring the and of a body allows us to calculate its because the  that causes the reduction in apparent weight has a special relation to the amount of water being by the body. states that the buoyant force provided by a fluid is equal to the weight of the fluid displaced.

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 of low tissue will take up relative to the same mass of high density tissue. Taking up the volume means more water is when the body is submerged so the will be larger compared to the 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 .

Everyday Example

The water by a brick weighs less than the brick so the 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 . That force is less than the 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 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 .

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 by an entire beach ball weighs more than a beach ball, so if you hold one under water the will be greater than the . 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 of ice is only about 9/10 that of water. The 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 .

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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Reinforcement Exercises

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

Buoyancy

 

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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