The Laws of Falling Bodies.

65

They also admit of illustration by producing a number of small iangles in a large right-angled triangle, as in fig. 10. These lesser triangles are alternately inverted, and, for the sake of distinction, alternately shaded and white. The perpendicular side of

Seconds

the large triangle is divided into any number of equal parts-1, 2, 3, 4-which may represent equal portions of time (seconds). Lines are drawn from these points so as to be parallel to the three sides of the triangle. The smaller triangles or spaces are thereby produced. It will be perceived, by reference to fig. 10, that in the first second, a falling body falls through one space or triangle. The space passed through in the second period will be represented by three triangles, two shaded and one white, which, added to the

Fig. 10.

first, will make four spaces. So with the third period, five spaces, which, added to the preceding, make nine, and with the fourth period seven spaces, which, added to those already passed, make sixteen-1, 4, 9, 16.

140. Knowing this rate of progression, we may easily compute the velocity acquired by a falling body and the distance through which it falls in any time; and the height of a precipice, or of a bridge, or the depth of a well, may be ascertained by marking the time required for a body to drop through the space.

141. Such being the law of the velocities produced by gravity, we have at once the means of knowing the amount of force or momentum expressed from gravity in any time or during any length of fall.

The moving force possessed by a ton hammer falling for half a second would be expressed as a mass of one ton moving 16'1 feet a second, or 16'1 tons moving at a rate of one foot a second; and if it fall for a length of time twice, thrice, or four times as long, its moving force would be just twice, thrice, or four times increased.

Thus, should we wish to double the blow which a pile-engine head gives, we should have to double the duration of its fall; but that implies that we should have to do more than double the extent of its fall; we should (see Table, Art. 139) have to quad

56

Momentum a true Measure of Force.

ruple its former extent of fall. So that, if it fell before through ten feet, in order to give double the blow it would have to be dropped through forty feet.

It is interesting to reflect that a railway carriage going fifty miles an hour has the same amount of moving force as it would have by falling through a height of thirty-six feet. When we think of the force which a stone of a hundredweight would acquire in falling through such a height, we need not wonder at the fearful results of railway collisions.

142. We have assumed that the momentum, or velocity and mass together, are a proper estimate of any force. This can be directly shown with Attwood's machine.

We add any mass, W, to one of the balanced masses, A and B, to act as the moving power. Now, as the force which the earth's attraction for w generates in any time must be the same whatever masses A and B have, we find that this force produces in the same time a double or triple velocity when the mass is reduced to one-half or a third: and it produces a half or a third of the velocity when the mass B is doubled or tripled.

A

If A and B be each 151⁄2 ounces, and w one ounce, we find A move down with a velocity at the end of one second only of that acquired by falling freely; that is to say, of about one foot per second.

Fig. 11. Again, if A and B be each 311⁄2 ounces, while w remains one ounce, then the velocity produced in the whole mass at the end of one second would be only one-half foot per second. But the whole amount of gravity must be the same in both cases, and the same as would be on the weight, w, falling alone; when it has to move more than its own mass the force is correspondingly diffused and the velocity correspondingly lessened.

143. In the practical application of these facts, allowance must of course be made for the interference of the air; this increases with the increase of velocity in the falling body, and becomes ultimately so great as just to counterbalance any increase of velocity from gravity; so that if the height from which a body falls exceed a few hundred feet probably, the motion would ultimately become uniform. This is similar to the balance that sets in between the accelerating force of a steam-engine, and the friction or loss of apparent force between the wheels and rails.

Retardation of Force.

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144. Attwood's machine gives us the means also of showing what is meant by, and of measuring, the velocity at any instant in the case of a constantly varying motion. This is effected simply by putting the additional weight in the shape of a bar on the top of the weight, A (fig. 11), and fixing a ring on a stand, so that while A moves down vertically it passes through the ring and leaves the bar behind, and goes on at a uniform rate afterwards in virtue of the velocity it had acquired at the instant that the bar was removed. With the means of adjusting the ring to any height we please, it is obvious that we can ascertain the velocity at any instant. Thus we actually realize the definition of a varying velocity, viz. that it is the space which the body would describe in the unit of time if at any instant its velocity were suddenly to become uniform.

"Retarded force?

145. The transference or transformation of force between masses is in no instance absolutely instantaneous.

We have seen that force is in no case produced instantaneously, but always with more or less of acceleration. So we may say, in like manner, that force can in no case be made to disappear instantaneously, but only more or less gradually.

The rate at which this transference takes place may be various ; the force of a body may be exhausted regularly or irregularly.

As the velocity and momentum of a falling body are gradually increased and at a uniform rate, so in an ascending body they are uniformly diminished.

A bullet, shot directly upwards, loses every instant a part of its velocity and force, till at last it comes to rest in the sky, and there a soaring eagle might see the messenger of death motionless and harmless for a moment, ere it start again on its downward course. 146. The following examples show the gradual nature of all retardation of force :

The shock of two railway carriages meeting, is prevented or lessened by the resisting elasticity of the buffers which gradually overcomes the motion.

It is soft gas expanding that begets gradually the death-carrying force in the cannon-ball; and soft air, or cotton, or wool, resisting in a strong, close tube-if the bullet could be directed exactly into it would again gradually absorb the moving force from the ball.

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Examples of retarded Force.

Were the attempt made, however, to stop the ball suddenly by a block of the hardest granite, the block would be shivered by the blow.

Bales of cotton or thick masses of cork attached round a ship will receive cannon balls, and bring them to rest, without themselves suffering much, while the naked firmer side of the ship would be penetrated. The cotton or cork offers an increasing resistance through considerable space, while the oak opposes its hard front at once and must instantly suffice or be torn.

A hard body, that it may at once destroy such a motion as we are supposing, must be able to oppose as much force in perhaps the hundredth of an inch, that is in the extent to which its elasticity will let it yield without breaking, as the moving cause gave through a much greater space; and, when it cannot do this, it must be penetrated by the moving body.

Could we suspend a vast mass of rock like a pendulum, and cause it to swing down from a considerable elevation, it would arrive at the bottom (or vertical line) with force sufficient to shake a thick wall or rampart to its foundation; but if it were merely allowed to continue its course like a pendulum, and ascend on the other side, the continued action of gravity now opposing its motion would bring the great mass to powerless rest again, just when it had reached an elevation equal to that from which it fell.

A heavy ship moving quickly with the tide or wind could not be stopped instantly by a short rope or chain of any magnitude; if the attempt were made to stop at once so vast a momentum, something would certainly give way. But a rope of very moderate size kept tight between the shore and the ship, and from time to time allowed to slip a little round a wooden block, when the tightness threatened its breaking, would accomplish the end very soon and safely.

A hempen or silken elastic rope supporting the scale of a weighing-beam would, for a similar reason, resist a greater weight falling into the scale than would be resisted by an iron chain, which, however, would be stronger than the rope for bearing a quiescent weight.

On the other hand, iron is stronger than hemp or rope when used as a chain cable for a ship to withstand the sudden force of waves. This will be understood on considering that the heavy chain hangs as a curve in the water, while the rope, being nearly of the density of the water, is supported in it almost as a straight line from the

Newton's Laws of Motion.

69

anchor to the ship: therefore when a great wave dashes against the ship, the bent iron chain will have a sort of elasticity from its great weight, and offer a continued resistance till it is drawn nearly straight; but the straight rope can yield by its elasticity only a very little way, and its weight is of no consequence in the resistance.

"Laws of motion."

147. After these general explanations of force, and the means and mode of its measurement, we are prepared to understand Newton's famous Laws of Motion; and we shall now consider these in detail, as they are much more comprehensive than at first sight appears from the simple statement of them.

The first Law of Motion may be given thus

A body free from the interference of external matter or force will either remain for ever at rest, or will move uniformly in a straight line.

Absolute proof of this, we may remark, it is impossible to find, because we never see matter entirely isolated, free from the action of other matter and of outside force. Still we can be certain of its truth by the same kind of reasoning as we employ in many inductions from experience. Finding that any departure from agreement with this law is distinctly traceable to some external interference, and that the more we can remove all external forces the more nearly we approach to a complete realisation of the principle, we conclude that, were it possible to get rid of every outside influence, we should see the law operating in perfection.

148. The first part of the law no one will deny. A body at rest requires force to set it in motion. Keep this away, and the object will sleep through all time, dead as the everlasting hills.

Stated thus, there is not the least difficulty about it. But when we expand the statement, or view it from the other side, and say that the action of the smallest force on any mass at rest must produce a corresponding amount of motion or departure from this state of rest, we come in contact with a popular prejudice-known under the name of the law of inertia-that conceives matter at rest as offering a positive resistance to be set in motion.

The ideas attached to the use of the words inertia, inertness, deadness, and resistance of matter, being often confused and erroneous, require some elucidation.