Chap. XI.

MECHANICS.

921

in proportion to one another ; for as one increases, so does the other. The height beingequal to half the space which would be moved through in the time of the fall with thetinal velocity, must have a fixed proportion, and the time and final velocity are proportionalto the product of the two numbers which express them.

The phenomena of falling bodies were experimented upon by Mr. George Atwood, whoconstructed for the purpose a very ingenious machine, which still bears his name.

The gravitating forces of bodies are to each other, first, directly as their masses, and,secondly, inversely as the squares of their distances: for if the mass of one body is doublethat of another, its gravity will be doubled, and with a distance twice as great, its gravitywill be as many times less. Putting M and m for the masses ; D and d for the distances,and G and g for the forces of gravity of two bodies; first we have G : g : : M : m :

Or, G= 9 M and if /fl = ^

d 9 1

Secondly : G : y : : d 2 : D*; or, G =g . ; and if g — 1, and d=- 1, G = - t

From whence it follows that G: g ; : M <f * : m D 8 ;

Or, G =g —; whence, if m and d are = 1, G = a .

m D* 9 D 8

The latter expression is usually taken for the above laws, M representing the attractingmass, and D the distance at which it acts upon some other body, as compared with the

attracting force g of this second body, whose mass is equal to 1, and whose distance is equal

to 1, which are considered the units of measurement.

Terrestrial gravitation is the attraction exercised on all bodies by the earth ; and occasionsthe common musket-ball, which proceeds at the rate of 1280 feet in a second, or the 2d poundcannon-ball, which has double that velocity, to fall at last to a place of rest.

The Centre of Gravity of a Body is a point through which a single force, equivalent to

the gravitation of all its particles, passes, and in whatever situation the body is placed,its centre of gravity tends to descend in a line perpendicular to the horizon, called the line

of direction of the weight. If a line be drawn from the centre of gravity of a body perpen-

dicular to the horizon, and this perpendicular fall within the base upon which the body rests,it will stand, but if it falls without the base the body will fall down.

Hence it follows that if the centre of gravity of a body be supported, the whole body is

supported, and the centre of gravity must be considered as the place of the body : if it besustained by any lever or beam, its place is at the point where the beam is cut by a linedrawn from the centre of gravity perpendicular to the horizon.

All the gravity of a body, or the force with which it endeavours to descend, being collectedinto the centre of that body or its centre of gravity, whatever sustains that centre of gravity

sustains the whole weight, and the descent of a body must be estimated by the descent of itscentre of gravity. Also the larger the base is upon which a body stands, and the fartherthe centre of gravity lies within it, the firmer will it stand, and the more difficult will be its

removal. If, on the contrary, the centre of gravity be moved out of this situation, and madeto approach to the extent of its base, its tendency to fall will be increased : a body laidupon an inclined plane, and having one end gradually elevated, will continue to slide as longas the perpendicular falls within the base; but if the perpendicular fall without it, it willroll over.

The centres of gravity of plane figures and solids are determined—

In the Triangle , by drawing two lines from any two of its angles to the middle of itsopposite sides; the point of their intersection is the centre of gravity ; the distance, there-fore, of the centre of gravity from the vertex is two-thirds of the line bisecting the oppositeside.

In the Trapezium , by dividing it into triangles, then finding the centre of gravity ofeach, and uniting the centres so found; the point of their common junction is the centre ofgravity sought.

For the Arc of a Circle, as ^ arc; sine of half the arc : : radius : to the distance of itscentre of gravity from the centre.

For the Sector of a Circle , as arc : chord : : | radius to the distance of its centre ofgravity from the centre : for the parabolic space the distance of the centre of gravityfrom the vertex is at two-fifths of the axis.

In the Cones and Pyramids, the distance of the centre of gravity from .the vertex is three-fourths of the axis.

its axis.

For the Segment of a Sphere let r be radius, x = to the height of the segment, then the dis-

In a Paraboloid the distance of the centre of gravity from the vertex is at two-thirds of

tances of the centre of gravity from the vertex is

8 r — 3 x12 r — 4 x'