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# Properties of Matter Study Guide (page 3)

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Updated on Sep 27, 2011

#### Example

Lake Baikal in Russia has a depth of 5,371 ft. Find the pressure exerted on the bottom of the lake by the water layer.

#### Solution

Convert first into 51 units and then solve for the unknown: pressure.

h = 5,371 ft

p = 1,000 kg/m3

p = ?

h = 5,371 ft = 5,371 ft · 0.3048 m/1 ft = 1,637 m

p = p · h · g = 1,000 kg/m3 · 1,637 m · 9.8 m/s2 = 16,040 N/m2 = 1.604 · 104N/m2

p = 1.604 · 107N/m2

## Archimedes's Principle

Recall that in mechanics, we defined the condition of equilibrium as referring to the net force on the object being zero. For an object to be at rest on a table, gravity has to be counteracted by another force, an equal and opposite force, and we call that reaction force. In a similar way, you can relate to the new concept we will discuss here: buoyancy. When an object is immersed in a fluid and floats at some level, the weight will act as if weighted down and there is no other force to be noted, or is there? Well, a piece of wood will not sink. And that is because the wood is pushed up by a force. We call this phenomena buoyancy. How large is the force of buoyancy? This is the subject of Archimedes's principle.

The force of buoyancy is usually symbolized by B and according to the principle, the expression is:

B = pfluid · Vfluid · g

Because in some cases, the object floats inside the liquid, the volume displaced is equal to the object's volume: Vliquid = Vobject. In other cases, the object floats at the surface of the liquid, and only part of the volume of the object will have to be taken into account.

### Archimedes's Principle

The upward force exerted by a fluid on an object immersed in the fluid is equal to the weight of the displaced fluid.

#### Example

A totally submerged object floats in water as shown in Figure 10.4. Find the ratio of the object's density to the density of water.

#### Solution

First, let's show the forces acting on this object. The horizontal forces determined by the surrounding pressure will cancel each other out because at equal levels, the pressure is the same, and therefore, the forces will be the same. The object floats inside the liquid, so the weight and the buoyancy force are also equal. Using Archimedes's principle, we can write the equation for the vertical forces.

W – B=0

W – B = m · gpliquid · Vliquid · g = 0

m = pobject · Vobject

0 = pobject · Vobject gpliquid · Vliquid · g

Vobject = Vliquid

0 = (pobjectpliquid) · Vliquid · g

0 = (pobjectpliquid)

## Pascal's Principle

As we mentioned at the beginning of this lesson, pressure is exerted in all directions in the same manner. Therefore, when the pressure changes, it will change in all directions and is also transmitted through the liquid. The manner of transmission constitutes the subject of Pascal's principle.

This phenomenon is very important in the way hydraulic presses function: A liquid transmits pressure between two mobile pistons reducing, the forces necessary to counteract a weight (see the following example).

### Pascal's Principle

Any pressure applied to a static fluid is transmitted equally in all directions through the liquid.

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