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Determination of the Centre of Gravity.

35

string: holding in one hand the plate, and in the other the string, the edge of the plate is placed against the wall (fig. 26); if the weight just touches it the wall is vertical; if the cylinder does not touch the wall, it shows that the wall is inclined inwards; it is inclined outwards if the weight touches the wall when the plate is a little removed from it.

42. Weight of a body.-The weight of a body is the sum of the partial attractions which the earth exerts upon each of its molecules. Hence the weight of a body must increase as its mass does ; that is, if it contains twice or thrice as much matter, its weight must be twice or thrice as great. The weight of a body is not to be confounded with gravity; this is the cause which produces the fall of bodies; the weight is only the effect. We shall presently see how weight is determined by means of the balance; gravity is measured by the aid of the pendulum.

43. Centre of gravity.-We have seen that all the partial attractions which the earth exerts upon each of the molecules of a body are equivalent to a single force, which is the weight of the body. Now, it may be shown in mechanics that, whatever be the shape of any body, there is always a certain point through which this single force, the weight, acts, in whatever position the body be placed in respect to the earth; this point is called the centre of gravity of the body.

To find the centre of gravity of a body is a purely geometrical problem; in many cases, however, it can be at once determined. For instance, the centre of gravity of a right line is the point which bisects its length; in the circle and sphere it coincides with the geometrical centre; in cylindrical bars it is the middle point of the axis; in a square or a parallelogram it is at the point of intersection of the two diagonals. These rules, it must be remembered, presuppose that the several bodies are of uniform density.

44. Experimental determination of the centre of gravity.— The centre of gravity of a body may also be found by experiment. When its weight is not too great, it is suspended by a string in two different positions; the centre of gravity of the body is necessarily below the point of suspension, and therefore in the prolongation of the vertical cord which sustains it. If then, in two different positions, the vertical lines of suspension be prolonged, they cut one another, and the point of intersection is the centre of gravity sought.

In the case of thin flat substances, like a piece of cardboard or

å sheet of tin plate, the centre of gravity may be found by balancing the body in two different positions on a horizontal edge; for instance, sliding them near the edge of a table until they are ready

lines, that is, must be at the point of their intersection, g: or, more accurately, a little below this point, in the interior of the

body, and at an equal dis

tance from its two faces.

If the body be thicker, three positions of equilibrium must be found; the centre of gravity is then at the point of intersection of the three planes passing vertically through the lines of contact when the body is in equilibrium.

45. Equilibrium ofheavy bodies. As the centre of gravity is the point where the whole action of gravity is concentrated, it follows that whenever this point rests upon any support, the action of gravity is destroyed, and therefore the body remains in equilibrium. There are, however, several cases, according as the body has one or more points of support.

Fig. 28.

Where the body has only one point of support, equilibrium is

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Different States of Equilibrium.

37

only possible when the centre of gravity either coincides with this point or is exactly above or below it in the same vertical line; for then the action of gravity is destroyed by the resistance of the fixed point through which this

force passes.
The plumb-line (fig. 26)
is a case of this kind, the centre of
gravity being below the point of sup-
port. Another example is the case
of a stick balanced on the finger, as
seen in fig. 28, in which the letter g in-
dicates the position of equilibrium ex-
actly over the point of support.

If the body has two points of support, it is not necessary for equilibrium that its centre of gravity coincide with either of these points, or be exactly above or below; it is sufficient if it be exactly below or above the right line which joins these two points, for the action of gravity may then be decomposed into two forces applied at the

Fig. 29.

points of support, and destroyed by the resistance of these points. A man on stilts (fig. 29) is an example of this case of equilibrium.

Lastly, if a body rests on the

ground by three or more points of support (fig. 30), equilibrium is produced whenever the centre of gravity is within the base formed by these points of support; that is, whenever the vertical let fall from the centre of gravity to the earth is within the points of support; for gravity cannot then overturn the body beyond its points of support, and its only effect is to settle it more firmly on the ground.

Fig. 30.

46. Different states of equilibrium.-Although a body supported by a fixed point is in equilibrium whenever its centre of gravity is in the vertical line through that point, the fact that the centre of gravity is always tending to occupy the lowest possible

position leads us to distinguish between three states of equilibriumstable, unstable, neutral.

A body is said to be in stable equilibrium if it tends to return to its first position after the equilibrium has been slightly disturbed. Every body is in this state when its position is such that the slightest alteration of the same elevates its centre of gravity; for the centre of gravity will descend again when permitted, and after a few oscillations the body will return to its original position.

The pendulum of a clock continually oscillates about its position of stable equilibrium, and an egg on a level table is in this state when its long axis is horizontal. We have another illustration in the toy represented in fig. 31.

These little figures, which are hollow and light, are loaded at the

base with a small mass of lead, so that the centre of gravity is very low. Hence when the figure is inclined, the centre of gravity is raised, and gravity tending to make it descend, the figure reverts to its original position after a number of oscillations on the right and left of its final position of equilibrium.

A body is said to be in unstable equilibrium when, after the slightest disturbance, it tends to depart still more from its original position. A body is in this state when its centre of gravity is vertically above the point of support, or higher than it would be in any adjacent position of the body. An egg standing on its end, or a stick balanced upright on the finger, is in this state (fig. 28). As soon as the stick is out of the vertical its centre of gravity descends, and, gravity acting with increasing force, the stick falls,

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Examples of Stable Equilibrium.

39

if care be not taken to bring the point of support below the centre of gravity, by which equilibrium is restored.

Neutral equilibrium.-A body is in a state of neutral equilibrium when it remains at rest in any position which may be given to it. This can only be the case when an alteration in the position of the body neither raises nor lowers its centre of gravity. A perfect sphere resting on a horizontal plane is in this state.

Fig. 32 represents three cones A, B, C, placed respectively in stable, unstable, and neutral equilibrium upon a horizontal plane. The letter g in each shows the position of the centre of gravity.

47. Examples of stable equilibrium.—It follows, from what has been said, that the wider the base on which a body rests, the greater is its stability; for then, even with a considerable inclination, its centre of gravity falls within its base.

The well-known leaning towers of Pisa and Bologna are so much out of the vertical that they seem ready to fall at any moment;

and yet they have remained for centuries in their present position, because the perpendiculars let fall from their centres of gravity are within the base. Fig. 33 represents the tower of Bologna, built in the year 1112, and known as the Garisenda. Its height is 165 feet, and it is 7 or 8 feet out of the vertical. The leaning is due to the foundations having given way. The tower on the side is that of Asarelli, the highest in Italy.

In the cases we have hitherto considered, the position of the centre of gravity is fixed; this is not the case with men and animals, whose centre of gravity is continually varying with their attitudes, and with the loads they support.

When a man, not carrying anything, stands upright, his centre of gravity is about the middle of the lower part of the pelvis, that is, between the two thigh-bones. This, however, is not the case with a man carrying a load, for, his own weight being added to that