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# Archimedes' Law

## The force exerted by a fluid on a floating or submerged object is equal to the weight of the water displaced.

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Archimedes' Principle and Buoyancy

Credit: Flickr:JasonParis
Source: http://www.flickr.com/photos/jasonparis/5495994580/

This cargo ship displaces an amount of water such that the weight of the displaced water is exactly equal to the weight of the ship and its cargo. The displacement of water is what produces the buoyancy to float this ship. When this photo was taken, the ship was empty so it did not sink very deep in the water to displace the necessary weight. When this ship is fully loaded with cargo, the water line will be where the black paint meets the red paint.

### Archimedes’ Principle and Buoyancy

If an object is submerged in a liquid, the object displaces a volume of the liquid equal to the volume of the submerged object. Legend has it that this observation was made by Archimedes when he sat in a bath tub that was filled to the top of the tub. The volume of water that overflowed was equal to his own volume. The forces exerted by the fluid on the sides of the submerged object are balanced. However, the forces exerted by the fluid on the top and bottom of the object are not equal. The force exerted by the liquid below the object is greater than the force exerted by the liquid above it; the liquid exerts a net upward force on the submerged or floating object. This force is called buoyancy, and its magnitude is equal to the weight of the displaced waterArchimedes' Principle states that the buoyant force is equal to the weight of the displaced liquid.

Example Problem: The density of steel is 9000. kg/m3 and the density of water is 1000. kg/m3. If a cube of steel that is 0.100 m on each side is placed in a tank of water and weighed while under water, what is the apparent weight of the cube?

Solution: The volume of the cube is 0.00100 m3.

The mass of the cube is 9.00 kg.

The weight of the cube when not submerged in water = (9.00 kg)(9.80 m/s2) = 88.2 N

The mass of water displaced by the cube = 1.00 kg

The weight of the water displaced by the cube = 9.80 N

The buoyant force on the steel cube = 9.80 N

Apparent weight of cube under water = 88.2 N - 9.80 N = 78.4 N

Example Problem: A hollow metal cube 1.00 m on each side has a mass of  600. kg.  How deep will this cube sink when placed in a vat of water?

Solution: Since the weight of the cube is 5880 N, it will need to displace 5880 N of water in order to float.

Volume of submerged portion of cube =(1.00 m)(1.00 m)(x m)=x m3\begin{align*}= (1.00 \ \text{m})(1.00 \ \text{m})(x \ \text{m}) = x \ \text{m}^3 \end{align*}

Mass of water displaced =1000x kg\begin{align*}=1000x\ \text{kg}\end{align*}

Weight of water displaced =9800x N\begin{align*}=9800 x \ \text{N}\end{align*}

9800x=5880\begin{align*}9800x=5880\end{align*}

x=0.600 m\begin{align*}x = 0.600 \ \text{m}\end{align*}

The cube will sink such that 0.60 m are underwater and 0.40 m are above water.

#### Sink or Swim

MIT students explain buoyancy in the video ''Sink or Swim,'' at http://youtu.be/tWNmQyHDbrw.

#### Summary

• If an object is submerged in a liquid, the object will displace a volume of the liquid equal to the volume of the submerged object.
• The forces exerted by the fluid on the sides of the submerged object are balanced, but the forces exerted by the fluid on the top and bottom of the object are not equal.
• The liquid exerts a net upward force on the submerged or floating object, called buoancy.
• The magnitude of buoancy is equal to the weight of the displaced water.
• Archimedes' Principle states that the buoyant force is equal to the weight of the displaced liquid.

#### Practice

Questions

The following video is on buoyancy. Use this resource to answer the two questions that follow.

1. Why were they unable to use the overturned boat as a submarine?
2. Why does the bubble in the tube shrink as it is taken lower in the pool?

1. If a ship has a mass of 200,000,000 kg, how much water would it need to displace in order to float?
2. A 40.0 kg cylinder that is 1.2 m tall and has a 0.20 m diameter falls off the pier and gets completely submerged when it hits the water.  What is the buoyant force on the cylinder when it is under water?  Will it sink or float?   The formula for finding the volume of a cylinder is πr2h\begin{align*}\pi r^2h\end{align*}.
3. The formula for the volume of a sphere is 4πr3\begin{align*}4 \pi r^3\end{align*}.  A 1.7 × 10-6 kg air bubble has a diameter of 1.00 cm and is submerged in water.  What is the buoyant force on the bubble?

#### Review

Questions

1. A cylinder with a radius of 11 cm and a height of 3.4 cm has a mass of 10.0 kg.
1. What is the weight of this cylinder?
2. What is the weight of this cylinder when it is submerged in water?
2. A wooden raft is 2.00 m wide, 3.00 m long, and 0.200 m deep.  The raft and its occupants have a mass of 700. kg.  How deep will the raft sink below the water when floating?
3. For the raft in problem #2, how many 50. kg people can be added to the raft before it sinks completely under water?
4. The density of gold is 19,320 kg/m3 and the density of mercury is 13,500 kg/m3.  If a cube of gold that is 0.100 m on each side is placed in a tank of mercury and weighed while under the surface, what is the apparent weight of the cube?