922

THEORY AND PRACTICE OF ENGINEERING.

Book II.

If two weights on any machine are in e<]uilibrio, and moved by any means, the centre otgravity of the weight and the power will always be in the same horizontal right line: inthe lever the centre of gravity is at the fulcrum; and therefore neither ascends nor descends.

In the wheel and axis, and in the pulley, the weight and power approach or recede fromeach other by spaces which are reciprocally as the bodies, and their centre of gravity istherefore at rest.

Upon the inclined plane, the perpendicular velocities of the power and weights arereciprocally as their quantities, and the distances of the centre of gravity from each beingin the same ratio, is also at rest.

In the combination of any of these powers, or of any machine whatever, where theequilibrium continues, the ascent and descent of the power and weight being reciprocallyas their quantities, the centre of gravity neither ascends nor descends.

To show the value of ascertaining the centre of gravity in a body in the arts of con-struction, we will suppose a piece of timber supported at the two ends B and C, as in thefollowing diagram. If G be the centre of gravity of the whole weight sustained, and theline FGII be drawn perpendicular to the horizon, and CF and BH drawn, then theweight of the whole body is asFII; the pressure at the topC, as B II; thrust of the pres-sure at the base B, as F B.

The pressure and the thrustare also in the direction of therespective lines which representthem. If the beam support anyweight, the beam and weightmust be considered as one body,whose centre of gravity is G.

Then the end C is supported bythe plane BCE, and the otherend B may be supposed to besustained by a plane perpen-dicular to B F ; therefore theweight and forces at C and Bare respectively as FH, B H,and F B. The angles B C F andB D F are right angles, and a circle described upon the diameter B F will passthrough (.' I)

If B C be any beam bearing any weight, G, the centre of gravity of the whole, and if itlean against a perpendicular wall C A, and be supported in that position, draw BA, CF,parallel, FGD perpendicular to the horizon, and draw FB.

Then the whole weight is represented by the line - FD

The pressure at the top, C, by - - - - B I)

Thrust at the bottom, B, by - - - FB

and the same by the several directions.

If the point which constitutes the centre of gravity be supported by a force equal to theweight of the body, it will remain in equilibrio in every possible position; and if the centreof gravitv be either vertically above or vertically below the point of support, the body willbe in equilibrio, but will not, as before, be inditferent to any position. When the centre Isvertically above a point of support, the smallest inclination will altogether destroy theequilibrium ; when it is vertically below' a point of suspension, if the equilibrium be dis-turbed, it will after some oscillations be restored.

If a body rest upon a horizontal plane, and the vertical line passing through the centreof gravity be within the base, as we have already seen, the body will stand, if without itwill fall, and when that line is within the base, the farther the point in which it meets it isfrom the perimeter on any side, the firmer will be the stability of the body, for the weightmay be considered as acting entirely on the centre of gravity ; consequently, the more dis-tant any point in the perimeter of the base is from the vertical of the centre, the longer isthe vertical lever by which the weight acts, and the greater the momentum to resist thatof a given force similarly applied, tending to produce a rotation about an axis passingthrough that point. When the centre of gravity is high, the body is more easily overturnedthan it would bc.were the centre lower : a carriage loaded will overturn more readilyon a declivity, and when the weight is piled up very high, the chance of accident is stillfurther increased, because its centre of gravity is more easily thrown outside its base, orbeyond the wheels or points of support. In every body possessing weight, there is a pointthrough which the resultant of the gravitation of its individual particles w’ill pass,whatever be the position of any given points in it with respect to the constant verticaldirection of this force, that point is the centre of gravity.

Fig. 1230.

F E

Fig. 1229.