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

ρ = m/v kg/m3

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.

Ex. 1 - The body of a man whose weight is about 690 N contains about 5.2 x 10-3 m3 of blood. (a) Find the blood’s weight and (b) express it as a percentage of the body weight. Blood’s ρ = 1060 kg/m3.

Specific gravity is the density of a substance divided by the density of water at 4° C.

Density of substanceSpecific gravity = -----------------------------

1.000 x 103 kg/m3

Specific gravity has no units.

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. Pressure is not a vector quantity. F refers only to the magnitude of the force. This force is always perpendicular to the surface.

Ex. 2 - Suppose the pressure acting on the back of a swimmer’s hand is 1.2 x 105 Pa. The surface area of the back of the hand is 8.4 x 10-3 m2. (a) Find the magnitude of the force that acts on it. (b) discuss the direction of the force.

Air is a fluid that exerts inward pressure on our bodies. Atmospheric pressure at sea level is 1.013 x 105 Pa = 1 atmosphere. 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 + rgh 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.

Ex. 5 - Blood in the arteries is flowing, but the effects of this flow can be ignored an the blood can be treated as a static fluid. Estimate the amount by which the blood pressure P2 in the anterior tibial artery at the foot exceeds the blood pressure P1 in the aorta at the heart when the body is (a) reclining horizontally and (b) standing (h = 1.35 m).

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, andF2 = 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.

Ex. 7 - In a hydraulic car lift, the input piston has a radius of r1 = 0.0120 m. The output plunger has a radius of r2 = 0.150 m. The combined weight of the car and plunger is F2 = 20 500 N. The lift uses hydraulic oil that has a density of 8.00 x 102 kg/m3. What input force is needed to support the car and the output plunger when the bottom surfaces of the piston and plunger are at (a) the same level and (b) with h = 1.10 m and the output plunger lower than the input?

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

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. rpine is 550 kg/m3.

Ex. 10 - Normally, a Goodyear airship contains about 5.40 x 103 m3 of helium whose density is 0.179 kg/m3. Find the weight of the load WL that the airship can carry in equilibrium at an altitude where the density of air is 1.20 kg/m3.