Page:Elementary Text-book of Physics (Anthony, 1897).djvu/63

 the conditions required by theory, make the attainment of such perfection impossible.

The ratio of the useful work done to the energy expended is called the efficiency of the machine. Since in every actual machine there is a loss of energy in the transmission, the efficiency is always a proper fraction.

46. Application of the Principle of Work to the Mechanical Powers. (1) The Lever.—If the lever be displaced through a small angle $$\phi$$ about its fixed point or fulcrum, the distances traversed by the points of application of the two forces which act on its ends, these forces being supposed parallel, are equal to $$l \phi$$ and $$l' \phi$$. The lever is in equilibrium, by the principle of work, when the work done by the force $$P$$ during this rotation is equal and of opposite sign to that done by the force $$P'$$, or when $$Pl - P'l' = 0.$$ Hence equilibrium obtains, as already shown, when $$Pl = P'l'.$$.

(2) The Pulley.—In any combination of pulleys the forces which are equivalent to the reaction of the parts of the system always occur in pairs, and the work done by the two members of each of these pairs is equal in magnitude but opposite in sign, so that all such forces may be neglected in the enumeration of those forces which are concerned in doing work in the system, and which must be considered in applying the principle of work. The only forces which need be considered are the external force $$W$$, the weight, and the external force $$P$$, the power, which is applied to equilibrate the weight. If $$w$$ and $$p$$ represent the distances passed over by $$W$$ and $$P$$ respectively, when the cord is drawn through the pulleys, equilibrium will obtain when $$Ww = Pp.$$ The dependence of the ratio $$\frac{W}{P}$$ upon the number of pulleys and their arrangement may always be deduced from this equation.

(3) The Inclined Plane.—If the weight W which rests upon the plane be acted upon by the force $$P$$ parallel with the surface of the plane, equilibrium will obtain when the work done by each of these forces, during any small displacement of the weight up or down