Page:Elementary Text-book of Physics (Anthony, 1897).djvu/62

wheel. From these facts the relation of the applied force to the force obtained in any combination of pulleys is evident.

(3) The Inclined Plane is any frictionless surface, making an angle with the line of direction of the force applied at a point upon it. Eesolving the force $P$ (Fig. 17), making an angle $\phi$ with the normal to the plane, into its components $P\cos {\phi }$ and $P\sin {\phi }$ perpendicular to and parallel with the plane, $P\sin {\phi }$ is alone effective to produce motion. Consequently, a force $P\sin {\phi }$ acting parallel to the surface will balance a force $P$, making an angle $\phi$ with the normal to the surface.

If the plane be taken as the hypothenuse of a right-angled triangle $ABC$, of which the base $AB$ perpendicular to the line of direction of the force, then, by similarity of triangles, the angle $BAG$ equals $\phi$: whence the force obtained parallel to $AC$ is equal to the force applied multiplied by the sine of the angle of inclination of the plane. If the components of the force applied be taken, thfe one, as before, perpendicular to the plane $AC$, and the other parallel to the base $AB$, the force obtained parallel to $AB$ is equal to the force applied multiplied by the tangent of the angle of inclination of the plane.

(4) The Wheel and Axle is essentially a continuously acting lever.

(5) The Wedge is made up of two similar inclined planes set together, base to base.

(6) The Screw is a combination of the lever and the inclined plane.

The special formulas expressing the relations of the force applied to the force obtained by the use of these combinations are deduced from those for the more elementary mechanical powers.

Any arrangement of the mechanical powers, designed to do work, is called a machine. The more nearly the value of the work done approaches that of the energy expended, the more closely the machine approaches perfection. The elasticity of the materials we are compelled to employ, friction, and other causes which mod-