Chap. XI.

MECHANICS.

925

Quantity of Work on an inclined Plane _The inclined plane is a machine of the greatest

utility in the arts, and facilitates the removal of the heaviest burthens to a considerableheight. If, for example, it be required to build a tower of stone, a number of menwould be necessary to raise the materials vertically ; but by means of slopes or inclined planes,men, horses, or carriages could readily ascend it, and the power required to draw theweights would be much reduced.

To move a stone along an inclined plane A B, the workmen are accustomed to turnit on its edges perpendicular to the inclinations, or if it be cylindrical, they roll it along,so that in its different operations the resistance of

the friction is avoided ; the effort, F, which is hereparallel to the length of the plane, A B, has thenonly to overcome the resistance of the weight, P,of the body ; and as we have shown, we shallhave

Fig. 1236.

If the inclination of the plane be such that its length, A B, is 100 times its correspond-ing height, B C, the effort required to be exercised against the body is in the proportionof 1 to 100, and consequently the labour is the same whether the body be raiseddirectly or along an inclined plane, this latter having no other use than that of facilitatingthe elevation of the burthen. When a body descends along an inclined plane, its relativegravity, or the force which accelerates its motion, is to the absolute force of gravity, as theheight of the inclined plane is to its length ; or, the force urging the body to descend inthe direction of the plane, is to the whole force of gravity, as the height of the plane is toits length.

An inclined plane is any plane surfacewhich forms an angle with the horizon,

and is measured by ABC. The line A Bis the length of the plane B C, the lengthof its base, C A, its height.

There is, then, in an inclined plane anequilibrium when the power is to theweight, as the height of the inclined plane,

is to the length of its base multiplied by

the cosine of the angle which the direction l23, ‘

of the power makes with the inclined plane.

When a body W is placed upon a horizontal plane, it is obvious that it cannot be in equili-brio, unless a perpendicular let fall from its centre of gravity cuts the plane within the baseof its body : when the plane becomes inclined to the horizon, the same condition isnecessary, and another force, P, is requisite to prevent it sliding down the plane in virtueof its gravity. In order that an equilibrium may take place, the resultant of the two forcesP and W must be in a line perpendicular to the plane, and their direction must lie in thesame plane, and as the direction of W is in a vertical plane passing through the centre ofgravity of the body, the direction of P must be in the same plane.

The weight, power, and pressure on the plane are then respectively as radius, the sine.

and cosine of the plane’s elevation : for if a cylinder be sustained upon an inclined plane by

a power holding one end of a cord parallel to the plane, whilst the other end is fixed, thepower is to the weight of the cylinder, as half the height is to the length of the plane: thisis obvious, for half the relative weight of the cylinder is sustained by the end of the ropewhich is fixed.

The Wedge, is a solid with three planes inclined to eachother, and two triangular sides: to free it from friction ¥5%%%,

the faces should be polished, and the force which acts upon /

it is directed perpendicularly. That the three forces act- /,

ing on the three planes may be in equilibrio, each of / /

them must be in proportion to the length of the plane / /

on which it acts, and they must be applied at such parts / /

that their directions meet in one part; otherwise they will / /

be in opposition, and produce a rotary motion. /c./

The shore C, for instance, when placed against a smooth //

vertical surface, as the wall of a building, or employed to / /

raise a weight by means of any force applied at the foot by / /

drawing it along the surface, AB; the horizontal force //

applied will be to the weight, as the distance, A B, is //

to the perpendicular height of the wall: and on this A / / _,_L.J a

principle the equilibrium of arches, domes and roofs is

__ i 1 Fiff. 12M.

estimated.