488. The contrast between heavy bodies and light bodies is so marked
that without trial we hardly believe that a heavy body and a light
body will fall from the same height in the same time. That they do
so Galileo proved by dropping a heavy ball and a light ball together
from the top of the Leaning Tower at Pisa. They were found to reach
the ground simultaneously. We shall repeat this experiment on a scale
sufficiently reduced to correspond with the dimensions of the lecture
room.
[Illustration: FIG. 67.]
489. The apparatus used is shown in Fig. 67. It consists of a stout
framework supporting a pulley H at a height of about 20 feet above
the ground. This pulley carries a rope; one end of the rope is
attached to a triangular piece of wood, to which two electro-magnets
G are fastened. The electro-magnet is a piece of iron in the form
of a horse-shoe, around which is coiled a long wire. The horse-shoe
becomes a magnet immediately an electric current passes through the
wire; it remains a magnet as long as the current passes, and returns
to its original condition the moment the current ceases. Hence, if I
have the means of controlling the current, I have complete control of
the magnet; you see this ball of iron remains attached to the magnet
as long as the current passes, but drops the instant I break the
current. The same electric circuit includes both the magnets; each
of them will hold up an iron ball F when the current passes, but the
moment the current is broken both balls will be released. Electricity
travels along a wire with prodigious velocity. It would pass over many
thousands of miles in a second; hence the time that it takes to pass
through the wires we are employing is quite inappreciable. A piece of
thin paper interposed between the magnets and the balls will ensure
that they are dropped simultaneously; when this precaution is not taken
one or both balls may hesitate a little before commencing to descend.
A long pair of wires E, B, must be attached to the magnets, the other
ends of the wires communicating with the battery D; the triangle and
its load is hoisted up by means of the rope and pulley and the magnets
thus carry the balls to a height of 20 feet: the balls we are using
weigh about 0·25 lb. and 1 lb. respectively.
490. We are now ready to perform the experiment. I break the circuit;
the two balls are disengaged simultaneously; they fall side by side the
whole way, and reach the ground together, where it is well to place a
cushion to receive them. Thus you see the heavy ball and the light one
each require the same amount of time to fall from the same height.
491. But these balls are both of iron; let us compare together balls
made of different substances, iron and wood for example. A flat-headed
nail is driven into a wooden ball of about 2"·5 in diameter, and by
means of the iron in the nail I can suspend this ball from one of the
magnets; while either of the iron balls we have already used hangs from
the other. I repeat the experiment in the same manner, and you see
they also fall together. Finally, when an iron ball and a cork ball
are dropped, the latter is within two or three inches of its weighty
companion when the cushion is reached: this small difference is due
to the greater effect of the resistance of the air on the lighter of
the two bodies. There can be no doubt that in a vacuum all bodies of
whatever size or material would fall in precisely the same time.
492. How is the fact that all bodies fall in the same time to be
explained? Let us first consider two iron balls. Take two equal
particles of iron; it is evident that these fall in the same time;
they would do so if they were very close together, even if they were
touching, but then they might as well be in one piece: and thus we
should find that a body consisting of two or more iron particles takes
the same time to fall as one (omitting of course the resistance of the
air). Thus it appears most reasonable that two balls of iron, even
though unequal in size, should fall in the same time.
493. The case of the wooden ball and the iron ball will require
more consideration before we realise thoroughly how much Galileo’s
experiment proves. We must first explain the meaning of the word _mass_
in mechanics.
494. It is not correct to define _mass_ by the introduction of the idea
of weight, because the _mass_ of a body is something independent of the
existence of the earth, whereas weight is produced by the attraction of
the earth. It is true that weight is a convenient means of measuring
_mass_, but this is only a consequence of the property of gravity which
the experiment proves, namely, that the attraction of gravity for a
body is proportional to its _mass_.
495. Let us select as the unit of mass the mass of a piece of platinum
which weighs 1 lb. at London; it is then evident that the mass of any
other piece of platinum should be expressed by the number of pounds it
contains: but how are we to determine the mass of some other substance,
such as iron? A piece of iron is defined to have the same mass as a
piece of platinum, if the same force acting on either of the bodies
for the same time produces the same velocity. This is the proper test
of the equality of masses. The mass of any other piece of iron will be
represented by the number of times it contains a piece equal to that
which we have just compared with the platinum; similarly of course for
other substances.
496. The magnitude of a force acting for the time unit is measured by
the product of the mass set in motion and the velocity which it has
acquired. This is a truth established, like the first law of motion, by
indirect evidence.
497. Let us apply these principles to explain the experiment which
demonstrated that a ball of wood and a ball of iron fall in the
same time. Forces act upon the two bodies for the same time, but
the magnitudes of the forces are proportional to the mass of each
body multiplied into its velocity, and, since the bodies fall
simultaneously, their velocities are equal. The forces acting upon
the bodies are therefore proportional to their masses; but the force
acting on each body is the attraction of the earth, therefore, the
gravitation to the earth of different bodies is proportional to their
masses.
498. We may here note the contrast between the attraction of
gravitation and that of a magnet. A magnet attracts iron powerfully and
wood not at all; but the earth draws all bodies with forces depending
on their masses and their distances, and not on their chemical
composition.
THE SPACE DESCRIBED BY A FALLING BODY IS PROPORTIONAL TO THE SQUARE OF
THE TIME.
499. We have next to discover the law by which we ascertain the
distance a body falling from rest will move in a given time; it is not
possible to experiment directly upon this subject, as in two seconds a
body will drop 64 feet and acquire an inconveniently large velocity;
we can, however, resort to Atwood’s machine (Fig. 66) as a means of
diminishing the motion. For this purpose we require a clock with a
seconds pendulum.
500. On one of the equal cylinders A I place a slight brass rod, whose
weight gives a preponderance to A, which will consequently descend. I
hold the loaded weight in my hand, and release it simultaneously with
the tick of the pendulum. I observe that it descends 5" before the
next tick. Returning the weight to the place from whence it started, I
release it again, and I find that at the second tick of the pendulum it
has travelled 20". Similarly we find that in three seconds it descends
45". It greatly facilitates these experiments to use a little stage
which is capable of being slipped up and down the scale, and which
can be clamped to the scale in any position. By actually placing the
stage at the distance of 5", 20", or 45" below the point from which the
weight starts, the coincidence of the tick of the pendulum with the
tap of the weight on its arrival at the stage is very marked.
501. These three distances are in the proportion of 1, 4, 9; that
is, as the squares of the numbers of seconds 1, 2, 3. Hence we may
infer that _the distance traversed by a body falling from rest is
proportional to the square of the time_.
502. The motion of the bodies in Atwood’s machine is much slower
than the motion of a body falling freely, but the law just stated
is equally true in both cases so that in a free fall the distance
traversed is proportional to the square of the time. Atwood’s machine
cannot directly tell us the distance through which a body falls in one
second. If we can find this by other means, we shall easily be able to
calculate the distance through which a body will fall in any number of
seconds.