A fluid is a state of matter in which the particles are free to move around one another. No definite shape exists. The term “fluid” encompasses liquids and gases.

Densities of gases are very subject to changes in temperature and pressure while the densities of liquids and solids are much less affected by those changes.

Mass density ρ is the mass m divided by the volume v.

ρ = m/v kg/m3

The buoyant force is an upward force exerted by all fluids on objects submerged in them. This buoyant force is equal to the weight of the displaced fluid. This is Archimedes’ principle: Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces:

FB = Wfluid

For a floating object, the buoyant force equals the object’s weight.

The density of an object determines the depth of submersion.

The king of Syracuse believes his 7.84 N crown is not pure gold. Archimedes weighs the crown when submerged in water; the scale reads 6.86 N. Is the crown pure gold?

The pressure P exerted by a fluid is the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts.

P = F/A

The unit is the N/m2, a pascal (Pa). 105 Pa is one bar of pressure, also called one atmosphere. Pressure is not a vector quantity. F refers only to the magnitude of the force. This force is always perpendicular to the surface.

Air is a fluid that exerts inward pressure on our bodies. Atmospheric pressure at sea level is 1.013 x 105 Pa = 1 atmosphere (1 bar). 14.70 lb/in2.

If P1 is the pressure at the top of a column of fluid, P2 is the pressure at the bottom of a fluid, and h is the height of the column, then:

P2 = P1 + ρgh

This works very well for liquids as they are basically incompressible; however, for gases it only works if h is small enough so that any variation in ρ is negligible.

P2 = P1 + ρgh

If we know P1, we can find P2 by adding ρgh. While ρgh is affected by h, it is not affected by any horizontal distance.

Ex. 4 - Point A and point B are both located a distance of h = 5.50 m below the surface of the water. Find the pressure at each point.

A completely enclosed fluid may be subjected to an additional pressure by the application of an external force. As this external force changes, the pressure at any other point within the confined liquid changes correspondingly. This is Pascal’s principle.

Pascal’s principle - Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and the enclosing walls.

As long as the level at each end is at the same height, ρgh is zero. P2 = P1 + ρgh becomes P2 = P1.

If P2 = P1, then F2/A2 = F1/A1, and

F2 = F1•(A2/A1).

If A2 is larger than A1, a large force F2 can be produced with a small F1. This is used in a hydraulic car lift.

The small piston of a hydraulic car lift has an area of 0.20 m2. A car weighing 1.2 X 104 N sits on a rack mounted on the large piston. The large piston has an area of 0.90 m2. How large a force must be applied to the small piston to support the car?

Ex. 8 - A solid, square, pinewood raft measures 4.0 m on a side and is 0.30 m thick. (a) Determine whether the raft floats in water, and (b) if so, how much of the raft is beneath the surface. ρpine is 550 kg/m3.