II.

Fluid Statics or Hydrostatics

A fundamental characteristic of any fluid at rest is that the force exerted on any particle within the fluid is the same in all directions. If the forces were unequal, the particle would move in the direction of the resultant force. It follows that the force per unit area, or the pressure exerted by the fluid against the walls of an arbitrarily shaped containing vessel, is perpendicular to the interior walls at every point. If the pressure were not perpendicular an unbalanced tangential force component would exist and the fluid would move along the wall.

This concept was first formulated in a slightly extended form by the French mathematician and philosopher Blaise Pascal in 1647. Known as Pascal’s law, it states that the pressure applied to an enclosed fluid is transmitted equally in all directions and to all parts of the enclosing vessel, if pressure changes due to the weight of the fluid can be neglected. This law has extremely important applications in hydraulics.

The top surface of a liquid at rest in an open vessel will always be perpendicular to the resultant forces acting on it. If gravity is the only force, the surface will be horizontal. If other forces in addition to gravity act, then the “free” surface will adjust itself. For instance, if a glass of water is spun rapidly about its vertical axis, both gravity and centrifugal forces will act on the water and the surface will form a parabola that is perpendicular to the resultant force. If gravity is the only force acting on a liquid contained in an open vessel, the pressure at any point within the liquid is directly proportional to the weight of a vertical column of that liquid. This, in turn, is proportional to the depth below the surface and is independent of the size or shape of the container. Thus the pressure at the bottom of a pipe about 2.5 cm (about 1 in) in diameter and about 15 m (about 50 ft) high that is filled with water is the same as the pressure at the bottom of a lake about 15 m (about 50 ft) deep. Similarly, a pipe about 30 m (about 100 ft) long that is filled with water, and slanted so that the top is only about 15 m (about 50 ft) above the bottom vertically, will have the same pressure exerted at the bottom of the pipe even though the distance along the pipe is much longer. The weight of a column of fresh water about 30 cm (about 12 in) high and with a cross section of about 6.5 sq cm (about 1 sq in) is about 0.196 kg (about 0.433 lb) and this will be the pressure exerted at the bottom. A column about 30 cm (about 12 in) high and about 0.093 sq m (about 1 sq ft) in cross section will weigh 144 times as much, but the pressure, which is force per unit area, will remain identical. The pressure at the bottom of a mercury column about 30 cm (about 12 in) high will be 0.196 × 13.6 = 2.07 kg per 6.5 sq cm (1 sq in) as mercury is 13.6 times as heavy as water. See also Atmosphere; Barometer.

The second important principle of fluid statics was discovered by the Greek mathematician and philosopher Archimedes. The so-called Archimedes’ principle states that a submerged body is subject to a buoyancy force that is equal to the weight of the fluid displaced by that body. This explains why a heavily laden ship floats; its total weight equals exactly the weight of the water that it displaces, and this weight exerts the buoyant force supporting the ship.

A point at which all forces producing the buoyant effect may be considered to act is the center of buoyancy and is the center of gravity of the fluid displaced. The center of buoyancy of a floating body is directly above its center of gravity. The greater the distance between these two, the more stable the body. See Stability.

Archimedes’ principle also makes possible the determination of the density of an object that is so irregular in shape that its volume cannot be measured directly. If the object is weighed first in air and then in water, the difference in weights will equal the weight of the volume of the water displaced, which is the same as the volume of the object. Thus the weight density of the object (weight divided by volume) can readily be determined. In very high precision weighing, both in air and in water, the displaced weight of both the air and water has to be accounted for in arriving at the correct volume and density.