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Applications of Levers.

27

As levers of the second class may be enumerated the oars of a boat. The resistance of, the water to the motion of the feather of the oar represents the ful

crum, the hand of the oars

man is the power, and the

boat, or rather the water it displaces, is the resistance. The knife fixed at one end

Fig. 21.

and used in slicing roots, or cutting bread, is a lever of the second kind. Nutcrackers (fig. 21) afford a third illustration, as also does the common wheelbarrow.

The third kind of lever is less frequently met with. The pedals used in pianos and in grindstones are instances. In the latter case the pedal consists of a wooden board AC (fig. 22) forming a lever.

The fulcrum is at C on a bolt fixed to the frame; the power is the foot of the man turning, and the resistance, which is the motion to be transmitted to the wheel, is applied at A by means of a rod joined to a crank in the centre of the mill.

In the common fire-tongs each leg is a lever of the third kind.

The hand of a man pushing open a gate while standing near the hinges moves through much less space than the end of the gate, and must act, therefore, with greater force.

The most beautiful and numerous instances are met with in the muscular system of men and animals, almost all motions of which are effected by this mechanism.

CHAPTER IV.

35. Universal attraction.— It is stated that Newton, sitting one day in his garden saw an apple fall from a tree, was led by this circumstance to reflect upon the cause why bodies fell to the ground, and ultimately to the discovery of the important laws which govern the motion of the earth and of the stars.

They may be thus stated:

1. All bodies in nature exert a mutual attraction upon each other at all distances, in virtue of which they are continually tending towards each other.

2. For the same distance the attractions between bodies are proportional to their masses.

3. The masses being equal the attraction varies with the distance, being inversely proportional to the square of the distances asunder. To illustrate this, we may take the case of two spheres which, owing to their symmetry, attract each other just as if their masses were concentrated in their centres. If without other alteration the mass of one sphere were doubled, trebled, etc., the attraction between them would be doubled, trebled, etc. If, however, the mass of one sphere being doubled, that of the other were increased three times, the distance between their centres remaining the same, the attraction would be increased six times. Lastly, if, without altering their masses, the distance between their centres were increased from 1 to 2, 3, 4, .. units, the attraction would be diminished the 4th, 9th, 16th. . . part of its former intensity.

36. Gravitation.- The term gravitation is applied more especially to the attraction exerted between the heavenly bodies. The sun, being that member of our planetary system which has the largest mass, exerts also the greatest attraction, from which it might seem that the earth and the other planets ought to fall into the sun in virtue of this attraction. This would indeed be the case,

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Gravity.

29

if they were only acted upon by the force of gravitation; but owing to their inertia, the original impulse which they once received, constantly tends to carry them away from the sun in a straight line. This acquired velocity, combined with gravitation, makes the planets describe curves about the sun which are almost circular and are called their orbits.

37. Gravity.—This is the force in virtue of which bodies fall when they are no longer supported, that is, tend towards the centre of the earth. It is a particular case of universal attraction; and is due to the reciprocal attraction exerted between the earth and bodies placed on its surface it acts equally upon all bodies, whether they are at rest or in motion; whether they are solids, liquids, or gases. Some bodies, such as clouds and smoke, appear not to be influenced by this force, for they rise in the atmosphere instead of sinking; yet this, as will afterwards be seen, is no exception to the action of gravity.

Gravity, being a particular case of universal attraction, acts upon bodies proportionally to their mass and inversely as the square of their distance; that is, a body which contains twice or thrice as much matter as another, is attracted by the earth with a twofold or threefold force; or, in other words, weighs twice or thrice as much. In like manner one and the same body could be moved to twice or thrice its present distance from the centre of the earth, it would have one-fourth or one-ninth of its present weight; we say the centre and not the surface of the earth, for it is demonstrated in treatises on mechanics that the attractive force of the earth which causes bodies to fall must be calculated from its centre.'

From the magnitude of the earth's radius, which is about 4,000 miles, all bodies on its surface may be considered to be virtually at the same distance from the centre, and we may therefore conclude that their difference in weight is merely due to their difference in mass.

38. The weight of a body increases from the equator to the poles.-The force which makes bodies fall is not exactly the same at all points of the earth's surface. Two causes make it increase from the equator to the poles: the daily rotation of the earth about its axis, and the flattening at the poles. For the rotation of the earth gives rise to a centrifugal force acting from the centre to the surface, that is, in the opposite direction to the force of gravity. Hence bodies are continually acted upon by two forces in opposite directions; the force of gravity which draws them towards the centre, and the centrifugal force which tends to drive them away

from it. So that it is really the excess of the second force over the first which makes bodies fall. But as the centrifugal force decreases from the equator to the poles (30), the excess of gravity over this force becomes greater, and thus the weights of bodies increase as they come nearer the poles.

The flattening of the earth concurs in producing the same effect ; for, in consequence of it, bodies placed on the surface of the earth are nearer the centre at the poles, than they are at the equator, and are therefore more attracted. It must be added, that the increase in weight due to these two causes is very small, and is inappreciable by ordinary balances.

39. Vertical and horizontal lines.-At any point of the earth's surface, the direction of gravity, that is, the line which a falling body describes, is called the vertical line. The vertical lines drawn at different points of the earth's surface converge very nearly to the earth's centre. Hence, owing to the great distance from the surface of the earth to its centre, for points on the surface a and

b (fig. 23), not far apart, these verticals may assume to be parallel; but they are less parallel the further apart the points, as shown by the verticals a and d. For points situated on the same meridian the angle contained between the vertical lines equals the difference between the latitudes of those points.

At each point on the surface of the earth a man standing upright is in the direction of the vertical. But, as we have just seen, this direction changes from one place to another, and the same is the case with the position of the inhabitants of the various countries on the earth. As the earth is spherical, it follows that at two points, exactly opposite, two men will be in inverted positions in reference

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Weight of a Body.

31

to each other; from which is derived the term antipodes (opposite as regards the feet), given to the inhabitants of two diametrically opposite places.

A plane or a line is said to be horizontal when it is perpendicular to the direction of the vertical. The surface of water in a state of equilibrium is always horizontal. In speaking of the level we shall learn how the horizontality of any surface or line is determined.

40. Plumb-line. The vertical line at any point of the globe is generally determined by the plumb-line (fig. 24), which consists of a cylindrical weight attached to the end of a string. In obedience to the action of gravity this weight draws the string in the direction of this force, and when it is at rest the string is in the vertical direction. To ascertain by the aid of the plumb-line whether a given surface, a wall for example, is vertical, a small metal plate is used,

the side of which is equal to the diameter of the weight. In the centre of this plate is a small hole, through which passes the string: holding in one hand the plate, and in the other the string, the edge of the plate is placed against the wall (fig. 24); 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 outwards; it is inclined inwards if the weight touches the wall when the plate is a little removed from it.

41. Weight of a body.-The weight of a body is the sum of