greater than any of the detached bodies which are found upon its
surface. If one of these bodies suspended at any proposed height
above the surface of the earth be disengaged, it will be observed to
descend perpendicularly to the earth, that is, in the direction of the
earth’s centre. The force with which it descends will also be found
to be in proportion to the mass, without any regard to the species of
the body. These circumstances are consistent with the account which
we have given of gravitation. But by that account we should expect,
that as the falling body is attracted with a certain force towards the
earth, the earth itself should be attracted towards it by the same
force; and instead of the falling body moving towards the earth, which
is the phenomenon observed, the earth and it should move towards each
other, and meet at some intermediate point. This, in fact, is the case,
although it is impossible to render the motion of the earth observable,
for reasons which will easily be understood.
Since all the bodies around us participate in this motion, it would
not be directly observable, even though its quantity were sufficiently
great to be perceived under other circumstances. But setting aside
this consideration, the space through which the earth moves in such a
case is too minute to be the subject of sensible observation. It has
been stated (107), that when two bodies attract each other, the space
through which the greater approaches the lesser, bears to that through
which the lesser approaches the greater, the same proportion as the
mass of the lesser bears to the mass of the greater. Now the mass of
the earth is more than 1000,000,000,000,000 times the mass of any body
which is observed to fall on its surface; and therefore if even the
largest body which can come under observation were to fall through an
height of 500 feet, the corresponding motion of the earth would be
through a space less than the 1000,000,000,000,000th part of 500 feet,
which is less than the 100,000,000,000th part of an inch.
The attraction between the earth and detached bodies on its surface is
not only exhibited by the descent of these bodies when unsupported,
but by their pressure when supported. This pressure is what is called
_weight_. The phenomena of weight, and the descent of heavy bodies,
will be fully investigated in the next chapter.
(111.) It is not alone by the direct fall of bodies that the
gravitation of the earth is manifested. The curvilinear motion of
bodies projected in directions different from the perpendicular, is
a combination of the effects of the uniform velocity which has been
given to the projectile by the impulse which it has received, and the
accelerated velocity which it receives from the earth’s attraction.
Suppose a body placed at any point P, _fig. 21._, above the
surface of the earth, and let P C be the direction of the earth’s
centre. If the body were allowed to move without receiving any impulse,
it would descend to the earth in the direction P A, with an
accelerated motion. But suppose that at the moment of its departure
from P, it receives an impulse in the direction P B, which would
carry it to B in the time the body would fall from P to A, then, by
the composition of motion, the body must at the end of that time be
found in the line B D, parallel to P A. If the motion in
the direction of P A were uniform, the body P would in this case
move in the straight line from P to D. But this is not the case. The
velocity of the body in the direction P A is at first so small as
to produce very little deflection of its motion from the line P B.
As the velocity, however, increases, this deflection increases, so that
it moves from P to D in a curve, which is convex, towards P B.
The greater the velocity of the projectile in the direction P A,
the greater sweep the curve will take. Thus it will successively take
the forms P D, P E, P F, &c., and that velocity can be
computed, which (setting aside the resistance of the air) would cause
the projectile to go completely round the earth, and return to the
point P from which it departed. In this case, the body P would continue
to revolve round the earth like the moon. Hence it is obvious, that the
phenomenon of the revolution of the moon round the earth, is nothing
more than the combined effects of the earth’s attraction, and the
impulse which it received when launched into space by the hand of its
Creator.
(112.) This is a great step in the analysis of the phenomenon of
gravitation. We have thus reduced to the same class two effects
apparently very dissimilar, the rectilinear descent of a heavy body,
and the nearly circular revolution of the moon round the earth. Hence
we are conducted to a generalisation still more extensive.
As the moon’s revolution round the earth, in an orbit nearly circular,
is caused by the combination of the earth’s attraction, and an original
projectile impulse, so also the singular phenomena of the planets’
revolution round the sun in orbits nearly circular, must be considered
an effect of the same class, as well as the revolution of the
satellites of those planets which are attended by such bodies. Although
the orbits in which the comets move deviate very much from circles, yet
this does not hinder the application of the same principle to them,
their deviation from circles not depending on the sun’s attraction, but
only on the direction and force of the original impulse which put them
in motion.
(113.) We therefore conclude that gravitation is the principle
which, as it were, animates the universe. All the great changes and
revolutions of the bodies which compose our system, can be traced
to or derived from this principle. It still remains to show how
that remarkable law, by which this force is declared to increase or
decrease in the same proportion as the square of the distance from
the attracting body is decreased or increased, may be verified and
established.
It has been shown, that the curvilinear path of a projectile
depends on, and can be derived, by mathematical reasoning, from the
consideration of the intensity of the earth’s attraction, and the
force of the original impulse, or the velocity of projection. In the
same manner, by a reverse process, when we know the curve in which a
projectile moves, we can infer the amount of the attracting force which
gives the curvature to its path. In this way, from our knowledge of the
curvature of the moon’s orbit, and the velocity with which she moves,
the intensity of the attraction which the earth exerts upon her can be
exactly ascertained. Upon comparing this with the force of gravitation
at the earth’s surface, it is found that the latter is as many times
greater than the former, as the square of the moon’s distance is
greater than the square of the distance of a body on the surface of the
earth from its centre.
(114.) If this were the only fact which could be brought to establish
the law of gravitation, it might be thought to be an accidental
relation, not necessarily characterising the attraction of gravitation.
Upon examining the orbits and velocities of the several planets, the
same result is, however, obtained. It is found that the forces with
which they are severally attracted by the sun are great, in exactly the
same proportion as the squares of the several numbers expressing their
distances are small. The mutual gravitation of bodies on the surface of
the earth towards each other is lost in the predominating force exerted
by the earth upon all of them. Nevertheless, in some cases, this effect
has not only been observed, but actually measured.
A plumb-line, under ordinary circumstances, hangs in a direction truly
vertical; but if it be near a large mass of matter, as a mountain,
it has been observed to be deflected from the true vertical, towards
the mountain. This effect was observed by Dr. Maskeline near the
mountain called Skehallien, in Scotland, and by French astronomers near
Chimboraco. For particulars of these observations, see our treatise on
GEODÆSY.
Cavendish succeeded in exhibiting the effects of the mutual gravitation
of metallic spheres. Two globes of lead A, B, each about a foot in
diameter, were placed at a certain distance asunder. A light rod,
to the ends of which were attached small metallic balls C, D, was
suspended at its centre E from a fine wire, and the rod was placed
as in _fig. 22._, so that the attractions of each of the leaden
globes had a tendency to turn the rod round the centre E in the same
direction. A manifest effect was produced upon the balls C, D, by the
gravitation of the spheres. In this experiment, care must be taken that
no magnetic substance is intermixed with the materials of the balls.
Having so far stated the principles on which the law of gravitation is
established, we shall dismiss this subject without further details,
since it more properly belongs to the subject of PHYSICAL ASTRONOMY; to
which we refer the reader for a complete demonstration of the law, and
for the detailed development of its various and important consequences.
CHAP. VII.
TERRESTRIAL GRAVITY.
(115.) GRAVITATION is the general name given to this attraction, by
whatever masses of matter it may be manifested. As exhibited in the
effects produced by the earth upon surrounding bodies, it is called
“terrestrial gravity.”
As the attraction of the earth is directed towards its centre, it might
be expected that two plumb-lines should appear not to be parallel, but
so inclined to each other as to converge to a point under the surface
of the earth. Thus, if A B and C D, _fig. 23._, be two
plumb-lines, each will be directed to the centre O, where, if their
directions were continued, they would meet. In like manner, if two
bodies were allowed to fall from A and C, they would descend in the
directions A B and C D, which converge to O. Observation,
on the contrary, shows, that plumb-lines suspended in places not far
distant from each other are truly parallel; and that bodies allowed
to fall descend in parallel lines. This apparent parallelism of the
direction of terrestrial gravity is accounted for by the enormous
proportion which the magnitude of the earth bears to the distance
between the two plumb-lines or the two falling bodies which are
compared. If the distance between the places B, D, were 1200 feet, the
inclination of the lines A B and C D would not amount to a
quarter of a minute, or the 240th part of a degree. But the distance,
in cases where the parallelism is assumed, is never greater than, and
seldom so great as, a few yards; and hence the inclination of the
directions A B and C D is too small to be appreciated by any
practical measure. In the investigation of the phenomena of falling
bodies, we shall, therefore, assume, that all the particles of the
same body are attracted in parallel directions, perpendicular to an
horizontal plane.
(116.) Since the intensity of terrestrial gravity increases as the
square of the distance decreases, it might be expected that, as a
falling body approaches the earth, the force which accelerates it
should be continually increasing, and, strictly speaking, it is so. But
any height through which we observe falling bodies to descend bears so
very small a proportion to the whole distance from the centre, that
the change of intensity of the force of gravity is quite beyond any
practical means of estimating it. The radius, or the distance from
the surface of the earth to its centre, is 4000 miles. Now, suppose
a body descended through the height of half a mile, a distance very
much beyond those used in experimental enquiries, the distances from
the centre, at the beginning and end of the fall, are then in the
proportion of 8000 to 8001, and therefore the proportion of the force
of attraction at the commencement to the force at the end, being that
of the squares of these numbers, is 64,000,000 to 64,016,001, which, in
the whole descent, is an increase of about one part in 4000; a quantity
practically insignificant. We shall, therefore, in explaining the laws
of falling bodies, assume that, in the entire descent, the body is
urged by a force of uniform intensity.
Although the force which attracts all parts of the same body during
its descent in a given place is the same, yet the force of gravity,
at different parts of the earth’s surface, has different intensities.
The intensity diminishes with the latitude, so that it is greater
towards the poles, and lesser towards the equator. The causes of
this variation, its law, and the experimental proofs of it, will be
explained, when we shall treat of centrifugal force, and the motion of
pendulums. It is sufficient merely to advert to it in this place.
(117.) Since the earth’s attraction acts separately and equally on
every particle of matter, without regard to the nature or species of
the body, it follows that all bodies, of whatever kind, or whatever
be their masses, must be moved with the same velocity. If two equal
particles of matter be placed at a certain distance above the surface
of the earth, they will fall in parallel lines, and with exactly the
same speed, because the earth attracts them equally. In the same
manner, a thousand particles would fall with equal velocities. Now,
these circumstances will in no wise be changed if those 1000 particles,
instead of existing separately, be aggregated into two solid masses,
one consisting of 990 particles, and the other of 10. We shall thus
have a heavy body and a light one, and, according to our reasoning,
they must fall to the earth with the same speed.
Common experience, however, is not always consistent with this
doctrine. What are called light substances, as feathers, gold-leaf,
paper, &c., are observed to fall slowly and irregularly, while
heavier masses, as solid pieces of metal, stones, &c., fall rapidly.
Nay, there are not a few instances in which the earth, instead of
attracting bodies, seems to repel them, as in the case of smoke,
vapours, balloons, and other substances which actually ascend. We are
to consider that the mass of the earth is not the only agent engaged in
these phenomena. The earth is surrounded by an atmosphere composed of
an elastic or aeriform fluid. This atmosphere has certain properties,
which will be explained in our treatise on PNEUMATICS, and which are
the causes of the anomalous circumstances alluded to. Light bodies
rise in the atmosphere, for the same reason that a piece of cork rises
from the bottom of a vessel of water; and other light bodies fall more
slowly than heavy ones, for the same reason that an egg in water falls
to the bottom more slowly than a leaden bullet. This treatise is not
the place to give a direct explanation of these phenomena. It will
be sufficient for our present purpose to show, that if there were no
atmosphere, all bodies, heavy and light, would fall at the same rate.
This may easily be accomplished by the aid of an air-pump. Having
by that instrument abstracted the air from a tall glass vessel, we
are enabled, by means of a wire passing air-tight through a hole in
the top, to let fall several bodies from the top of the vessel to the
bottom. These, whether they be feathers, paper, gold-leaf, pieces of
money, &c. all descend with the same speed, and strike the bottom at
the same moment.
(118.) Every one who has seen a heavy body fall from a height, has
witnessed the fact, that its velocity increases as it approaches
the ground. But if this were not observable by the eye, it would be
betrayed by the effects. It is well known, that the force with which a
body strikes the ground increases with the height from whence it has
fallen. This force, however, is proportional to the velocity which it
has at the moment it meets the ground, and therefore this velocity
increases with the height.
When the observations on attraction in the last chapter are well
understood, it will be evident that the velocity which a body has
acquired in falling from any height, is the accumulated effects of the
attraction of terrestrial gravity during the whole time of the fall.
Each instant of the fall a new impulse is given to the body, from which
it receives additional velocity; and its final velocity is composed
of the aggregation of all the small increments of velocity which are
thus communicated. As we are at present to suppose the intensity of the
attraction invariable, it will follow that the velocity communicated to
the body in each instant of time will be the same, and therefore that
the whole quantity of velocity produced or accumulated at the end of
any time is proportional to the length of that time. Thus, if a certain
velocity be produced in a body having fallen for one second, twice that
velocity will be produced when it has fallen for two seconds, thrice
that velocity in three seconds, and so on. Such is the fundamental
principle or characteristic of _uniformly accelerated motion_.
(119.) In examining the circumstances of the descent of a body, the
time of the fall and the velocity at each instant of that time are not
the only things to be attended to. The spaces through which it falls in
given intervals of time, counted either from the commencement of its
fall, or from any proposed epoch of the descent, are equally important
objects of enquiry. To estimate the space in reference to the time and
the final velocity, we must consider that this space has been moved
through with varying speed. From a state of rest at the beginning of
the fall, the speed gradually increases with the time, and the final
velocity is greater still than that which the body had at any preceding
instant during its descent. We cannot, therefore, _directly_ appreciate
the space moved through in this case by the time and final velocity.
But as the velocity increases uniformly with the time, we shall obtain
the average speed, by finding that which the body had in the middle of
the interval which elapsed between the beginning and end of the fall,
and thus the space through which the body has actually fallen is that
through which it would move in the same time with this average velocity
uniformly continued.
But since the velocity which the body receives in any time, counted
from the beginning of its descent, is in the proportion of that time,
it follows that the velocity of the body after half the whole time of
descent is half the final velocity. From whence it appears, that the
height from which a body falls in any proposed time is equal to the
space through which a body would move in the same time with half the
final velocity, and it is therefore equal to half the space which would
be moved through in the same time with the final velocity.
(120.) It follows from this reasoning, that between the three
quantities, the height, the time, and the final velocity, which enter
into the investigation of the phenomena of falling bodies, there are
two fixed relations: _First_, the time, counted from the beginning of
the fall and the final velocity, are proportional the one to the other;
so that as one increases, the other increases in the same proportion.
_Secondly_, the height being equal to half the space which would be
moved through in the _time_ of the fall, with the _final velocity_,
must have a fixed proportion to these two quantities, viz. the _time_
and the _final velocity_, or must be proportional to the product of the
two numbers which express them.
But since the time is always proportional to the final velocity, they
may be expressed by equal numbers, and the product of equal numbers
is the square of either of them. Hence, the product of the numbers
expressing the time and final velocity is equivalent to the square
of the number expressing the time, or to the square of the number
expressing the final velocity. Hence we infer, that the height is
always proportional to the square of the time of the fall, or to the
square of the final velocity.
(121.) The use of a few mathematical characters will render these
results more distinct, even to students not conversant with
mathematical science.
Let S = the height from which the body falls, expressed in feet.
V = the velocity at the end of the fall in feet per second.
T = the number of seconds in the time of the fall.
_g_ = the number of feet through which a body would fall in one
second.
It will therefore follow that the velocity acquired in one second will
be 2_g_, and the velocity acquired in T seconds will therefore be 2_g_
× T; so that
V = 2_g_ × T [1]
Since the space which a body falls through in T seconds is found by
multiplying the space it falls through in one second by T^2, we shall
have
S = _g_ × T^2 [2]
from which, combined with [1] we deduce
S = V^2/(4_g_) [3]
S = (1/2)V × T [4]
By these formularies, if the height through which a body falls freely
in one second be known, the height through which it will fall in any
proposed time may be computed. For since the height is proportional
to the square of the time, the height through which it will fall in
_two_ seconds will be _four_ times that which it falls through in
_one_ second. In _three_ seconds it will fall through _nine_ times
that space; in _four_ seconds, _sixteen_ times; in _five_ seconds,
_twenty-five_ times, and so on. The following, therefore, is a general
rule to find the height through which a body will fall in any given
time: “Reduce the given time to seconds, take the square of the number
of seconds in it, and multiply the height through which a body falls in
one second by that number; the result will be the height sought.”
The following table exhibits the heights and corresponding times as far
as 10 seconds:
+-------+---+---+---+----+----+----+----+----+----+-----+
|Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+-------+---+---+---+----+----+----+----+----+----+-----+
|Height | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
+-------+---+---+---+----+----+----+----+----+----+-----+
Each unit in the numbers of the first row expresses a second of time,
and each unit in those of the second row expresses the height through
which a body falls freely in a second.
(122.) If a body fall continually for several successive seconds,
the spaces which it falls through in each succeeding second have a
remarkable relation among each other, which may be easily deduced from
the preceding table. Taking the space moved through in the first second
still as our unit, four times that space will be moved through in the
first two seconds. Subtract from this 1, the space moved through in the
first second, and the remainder 3 is the space through which the body
falls in the _second_ second. In like manner if 4, the height fallen
through in the first two seconds, be subtracted from 9, the height
fallen through in the first three seconds, the remainder 5 will be the
space fallen through in the third second. To find the space fallen
through in the fourth second, subtract 9, the space fallen through in
the first three seconds, from 16, the space fallen through in the first
four seconds, and the result is 7, and so on. It thus appears that if
the space fallen through in the first second be called 1, the spaces
described in the second, third, fourth, fifth, &c. seconds, will be
expressed by the odd numbers respectively, 3, 5, 7, 9, &c. This places
in a striking point of view the accelerated motion of a falling body,
the spaces moved through in each succeeding second being continually
increased.
(123.) If velocity be estimated by the space through which the body
would move uniformly in one second, then the final velocity of a body
falling for one second will be 2; for with that final velocity the body
would in one second move through twice the height through which it has
fallen.
(124.) Since the final velocity increases in the same proportion as
the time, it follows that after two seconds it is twice its amount
after one, and after three seconds thrice that, and so on. Thus, the
following table exhibits the final velocities corresponding to the
times of descent: