Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/167

Rh energy, which is given by the expression $$\frac{m\ v^2}{2}$$ foot poundals per second.

Case 2.—If, on the other hand, the particles and the surface of the earth be perfectly elastic, the former will rebound with a velocity equal to that with which they strike, and the system as a whole will not lose energy. If the body be arranged to deal continually with the same set of particles, none being allowed to escape, then it may be supported without any continued expenditure of energy—that is to say, without any work being done. Such a case is exemplified in the dynamical theory of heat when a loaded piston is supported by gaseous pressure in a closed cylinder. We could also suppose it to be effected by imbuing the supported body with sufficient intelligence and skill so to direct the particles that they would always rebound within its reach.

We have already seen (§ 4) that in Case 1 the weight supported is equal in absolute units to $$m\ v .$$ But in Case 2 the particles impinging on the body impart as much momentum as they do in leaving it; hence the supporting force $$= 2\ m\ v .$$

In both cases it will be observed that the projected particles act as carriers of momentum between the earth's surface and the dynamically supported body, the weight of which is eventually carried down and distributed on the surface beneath; and, moreover, we are unable to conceive of any arrangement of material particles used for dynamic support, however complex, that will not eventually transmit the stress produced by the weight of the body down to the surface of the earth. (Compare § 6.)

§ 112. Aerodynamic Support.—We may now examine and discuss the behaviour of an incompressible and frictionless (inviscid) atmosphere with respect to an aerofoil traversing it.

When a loaded aerofoil is dynamically supported by a fluid, we know that its weight is eventually sustained by the surface Rh