+---------------+---+---+---+---+----+----+----+----+----+----+

|Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

+---------------+---+---+---+---+----+----+----+----+----+----+

|Final velocity | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

+---------------+---+---+---+---+----+----+----+----+----+----+

The numbers in the second row express the spaces through which a body

with the final velocity would move in one second, the unit being, as

usual, the space through which a body falls freely in one second.

(125.) Having thus developed theoretically the laws which characterise

the descent of bodies, falling freely by the force of gravity, or by

any other uniform force of the same kind, it is necessary that we

should show how these laws can be exhibited by actual experiment.

There are some circumstances attending the fall of heavy bodies which

would render it difficult, if not impossible, to illustrate, by the

direct observation of this phenomenon, the properties which have

been explained in this chapter. A body falling freely by the force

of gravity, as we shall hereafter prove, descends in one second of

time through a height of about 16 feet[1]; in two seconds, it would,

therefore, fall through four times that space, or 64 feet; in three

seconds, through 9 times the height, or 144 feet; and in four seconds,

through 256 feet. In order, therefore, to be enabled to observe the

phenomena for only four seconds, we should command an height of at

least 256 feet. But further; the velocity at the end of the first

second would be at the rate of 32 feet per second; at the end of the

second second, it would be 64 feet per second; and towards the end of

the fall it would be about 120 feet per second. It is evident that this

great degree of rapidity would be a serious impediment to accurate

observation, even though we should be able to command the requisite

height. It appears therefore that the number expressed by _g_ in the

preceding formulæ is 16·083.

[1] More exactly through 16-1/12 feet, or 193 inches.

It occurred to Mr. George Attwood, a mathematician and natural

philosopher of the last century, that all the phenomena of falling

bodies might be experimentally exhibited and accurately observed, if

a force of the same kind as gravity, viz. an uniformly accelerating

force, be used, but of a much less intensity; so that while the motion

continues to be governed by the same laws, its quantity may be so much

diminished, that the final velocity, even after a descent of many

seconds, shall be so moderated as to admit of most deliberate and exact

observation. This being once accomplished, nothing more would remain

but to find the height through which a body would fall in one second,

or, what is the same, the proportion of the force of gravity to the

mitigated but uniform accelerating force thus substituted for it.

(126.) To realise this notion, Attwood constructed a wheel turning on

its axle with very little friction, and having a groove on its edge

to receive a string. Over this wheel, and in the groove, he placed a

fine silken cord, to the ends of which were attached equal cylindrical

weights. Thus placed, the weights perfectly balance each other, and no

motion ensues. To one of the weights he then added a small quantity, so

as to give it a slight preponderance. The loaded weight now began to

descend, drawing up on the other side the unloaded weight. The descent

of the loaded weight, under these circumstances, is a motion exactly

of the _same kind_ as the descent of a heavy body falling freely by

the force of gravity; that is, it increases according to the same

laws, though at a very diminished rate. To explain this, suppose that

the loaded weight descends from a state of rest through one inch in

a second, it will descend through 4 inches in two seconds, through 9

in three, through 16 in four, and so on. Thus in 20 seconds, it would

descend through 400 inches, or 33 feet 4 inches, a height which, if it

were necessary, could easily be commanded.

It might, perhaps, be thought, that since the weights suspended at the

ends of the thread are in equilibrium, and therefore have no tendency

either to move or to resist motion, the additional weight placed upon

one of them ought to descend as rapidly as it would if it were allowed

to fall freely and unconnected with them. It is very true that this

weight will receive from the attraction of the earth the same force

when placed upon one of the suspended weights, as it would if it were

disengaged from them; but in the consequences which ensue, there is

this difference. If it were unconnected with the suspended weights,

the whole force impressed upon it would be expended in accelerating

its descent; but being connected with the equal weights which sustain

each other in equilibrium, by the silken cord passing over the wheel,

the force which is impressed upon the added weight is expended, not

as before, in giving velocity to the added weight alone, but to it

together with the two equal weights appended to the string, one of

which descends with the added weight, and the other rises on the

opposite side of the wheel. Hence, setting aside any effect which the

wheel itself produces, the velocity of the descent must be lessened

just in proportion as the mass among which the impressed force is to

be distributed is increased; and therefore the _rate_ of the fall

bears to that of a body falling freely the same proportion as the added

weight bears to the sum of the masses of the equal suspended weights

and the added weight. Thus the smaller the added weight is, and the

greater the equal suspended weights are, the slower will the rate of

descent be.

To render the circumstances of the fall conveniently observable, a

vertical shaft (see _fig. 24._) is usually provided, which is

placed behind the descending weight. This pillar is divided to inches

and halves, and of course may be still more minutely graduated, if

necessary. A stage to receive the falling weight is moveable on this

pillar, and capable of being fixed in any proposed position by an

adjusting screw. A pendulum vibrating seconds, the beat of which ought

to be very audible, is placed near the observer. The loaded weight

being thus allowed to descend for any proposed time, or from any

required height, all the circumstances of the descent may be accurately

observed, and the several laws already explained in this chapter may be

experimentally verified.

(127.) The laws which govern the descent of bodies by gravity, being

reversed, will be applicable to the ascent of bodies projected upwards.

If a body be projected directly upwards with any given velocity, it

will rise to the height from which it should have fallen to acquire

that velocity. The earth’s attraction will, in this case, gradually

deprive the body of the velocity which is communicated to it at the

moment at which it is projected. Consequently, the phenomenon will be

that of _retarded motion_. At each part of its ascent it will have the

same velocity which it would have if it descended to the same place

from the highest point to which it rises. Hence it is clear, that all

the particulars relative to the ascent of bodies may be immediately

inferred from those of their descent, and therefore this subject

demands no further notice.

To complete the investigation of the phenomena of falling bodies, it

would now only remain to explain the method of ascertaining the exact

height through which a body would descend in one second, if unresisted

by the atmosphere, or any other disturbing cause. As the solution

of this problem, however, requires the aid of principles not yet

explained, it must for the present be postponed.

CHAP. VIII.

OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES.

(128.) In the last chapter, we investigated the phenomena of bodies

descending freely in the vertical direction, and determined the laws

which govern, not their motion alone, but that of bodies urged by any

uniformly accelerating force whatever. We shall now consider some of

the most ordinary cases in which the free descent of bodies is impeded,

and the effects of their gravitation modified.

(129.) If a body, urged by any forces whatever, be placed upon a

hard unyielding surface, it will evidently remain at rest, if the

resultant (76) of all the forces which are applied to it be directed

perpendicularly against the surface. In this case, the effect produced

is pressure, but no motion ensues. If only one force act upon the

body, it will remain at rest, provided the direction of that force be

perpendicular to the surface.

But the effect will be different, if the resultant of the forces which

are applied to the body be oblique to the surface. In that case this

resultant, which, for simplicity, may be taken as a single force, may

be considered as mechanically equivalent to two forces (76), one in the

direction of the surface, and the other perpendicular to it. The latter

element will be resisted, and will produce a pressure; the former will

cause the body to move. This will perhaps be more clearly apprehended

by the aid of a diagram.

Let A B, _fig. 25._, be the surface, and let P be a particle

of matter placed upon it, and urged by a force in the direction

P D, perpendicular to A B. It is manifest, that this force

can only press the particle P against A B, but cannot give it any

motion.

But let us suppose, that the force which urges P is in a direction

P F, oblique to A B. Taking P F as the diagonal of a

parallelogram, whose sides are P D and P C (74), the force

P F is mechanically equivalent to two forces, expressed by the

lines P D and P C. But P D, being perpendicular to

A B, produces pressure without motion, and P C, being in

the direction of A B, produces motion without pressure. Thus

the effect of the force P F is distributed between motion and

pressure in a certain proportion, which depends on the obliquity of

its direction to that of the surface. The two extreme cases are, 1.

When it is in the direction of the surface; it then produces motion

without pressure: and, 2. When it is perpendicular to the surface; it

then produces pressure without motion. In all intermediate directions,

however, it will produce both these effects.

(130.) It will be very apparent, that the more oblique the direction

of the force P F is to A B, the greater will be that part

of it which produces motion, and the less will be that which produces

pressure. This will be evident by inspecting _fig. 26._ In this

figure the line P F, which represents the force, is equal to

P F in _fig. 25._ But P D, which expresses the pressure,

is less in _fig. 26._ than in _fig. 25._, while P C,

which expresses the motion, is greater. So long, then, as the obliquity

of the directions of the surface and the force remain unchanged, so

long will the distribution of the force between motion and pressure

remain the same; and therefore, if the force itself remain the same,

the parts of it which produce motion and pressure will be respectively

equal.

(131.) These general principles being understood, no difficulty can

arise in applying them to the motion of bodies urged on inclined planes

or curves by the force of gravity. If a body be placed on an unyielding

horizontal plane, it will remain at rest, producing a pressure on the

plane equal to the total amount of its weight. For in this case the

force which urges the body, being that of terrestrial gravity, its

direction is vertical, and therefore perpendicular to the horizontal

plane.

But if the body P, _fig. 25._, be placed upon a plane A B,

oblique to the direction of the force of gravity, then, according to

what has been proved (129), the weight of the body will be distributed

into two parts, P C and P D; one, P D, producing a

pressure on the plane A B, and the other, P C, producing

motion down the plane. Since the obliquity of the perpendicular

direction P F of the weight to that of the plane A B must be

the same on whatever part of the plane the weight may be placed, it

follows (130), that the proportion P C of the weight which urges

the body down the plane must be the same throughout its whole descent.

(132.) Hence it may easily be inferred, that the force down the

plane is uniform; for since the weight of the body P is always the

same, and since its proportion to that part which urges it down the

plane is the same, it follows that the quantity of this part cannot

vary. The motion of a heavy body down an inclined plane is therefore

an uniformly-accelerated motion, and is characterised by all the

properties of uniformly-accelerated motion, explained in the last

chapter.

Since P F represents the force of gravity, that is, the force

with which the body would descend freely in the vertical direction,

and P C the force with which it moves down the plane, it follows

that a body would fall freely in the vertical direction from P to F

in the same time as on the plane it would move from P to C. In this

manner, therefore, when the height through which a body would fall

vertically is known, the space through which it would descend in the

same time down any given inclined plane may be immediately determined.

For let A B, _fig. 25._, be the given inclined plane, and let

P F be the space through which the body would fall in one second.

From F draw F C perpendicular to the plane, and the space P C

is that through which the body P will fall in one second on the plane.

(133.) As the angle B A H, which measures the elevation

of the plane, is increased, the obliquity of the vertical direction

P F with the plane is also increased. Consequently, according to

what has been proved (130), it follows, that as the elevation of the

plane is increased, the force which urges the body down the plane is

also increased, and as the elevation is diminished, the force suffers a

corresponding diminution. The two extreme cases are, 1. When the plane

is raised until it becomes perpendicular, in which case the weight is

permitted to fall freely, without exerting any pressure upon the plane;

and, 2. When the plane is depressed until it becomes horizontal, in

which case the whole weight is supported, and there is no motion.

From these circumstances it follows, that by means of an inclined plane

we can obtain an uniformly-accelerating force of any magnitude less

than that of gravity.

We have here omitted, and shall for the present in every instance

omit, the effects of _friction_, by which the motion down the plane

is retarded. Having first investigated the mechanical properties of

bodies supposed to be free from friction, we shall consider friction

separately, and show how the present results are modified by it.

(134.) The accelerating forces on different inclined planes may be

compared by the principle explained in (131). Let _figs. 25._

and _26._ be two inclined planes, and take the lines P F in each

figure equal, both expressing the force of gravity, then P C will

be the force which in each case urges the body down the plane.

As the force down an inclined plane is less than that which urges a

body falling freely in the vertical direction, the space through which

the body must fall to attain a certain final velocity must be just so

much greater as the accelerating force is less. On this principle we

shall be able to determine the final velocity in descending through any

space on a plane, compared with the final velocity attained in falling

freely in the vertical direction. Suppose the body P, _fig. 27._,

placed at the top of the plane, and from H draw the perpendicular

H C. If B H represent the force of gravity, B C will

represent the force down the plane (131). In order that the body

moving down the plane shall have a final velocity equal to that of

one which has fallen freely from B to H, it will be necessary that it

should move from B down the plane, through a space which bears the same

proportion to B H as B H does to B C. But since the

triangle A B H is in all respects similar to H B C,

only made upon a larger scale, the line A B bears the same

proportion to B H as B H bears to B C. Hence, in falling

on the inclined plane from B to A, the final velocity is the same as in

falling freely from B to H.

It is evident that the same will be true at whatever level an

horizontal line be drawn. Thus, if I K be horizontal, the final

velocity in falling on the plane from B to I will be the same as the

final velocity in falling freely from B to K.

(135.) The motion of a heavy body down a curve differs in an important

respect from the motion down an inclined plane. Every part of the

plane being equally inclined to the vertical direction, the effect of

gravity in the direction of the plane is uniform; and, consequently,

the phenomena obey all the established laws of uniformly-accelerated

motion. If, however, we suppose the line B A, on which the

body P descends, to be curved as in _fig. 28._, the obliquity

of its direction at different parts, to the direction P F of

gravity, will evidently vary. In the present instance, this obliquity

is greater towards B and less towards A, and hence the part of the

force of gravity which gives motion to the body is greater towards

B than towards A (130). The force, therefore, which urges the body,

instead of being uniform as in the inclined plane, is here gradually

diminished. The rate of this diminution depends entirely on the nature

of the curve, and can be deduced from the properties of the curve by

mathematical reasoning. The details of such an investigation are not,

however, of a sufficiently elementary character to allow of being

introduced with advantage into this treatise. We must therefore limit

ourselves to explain such of the results as may be necessary for the

development of the other parts of the science.

(136.) When a heavy body is moved down an inclined plane by the force

of gravity, the plane has been proved to sustain a pressure, arising

from a certain part of the weight P D, _fig. 25._, which

acts perpendicularly to the plane. This is also the case in moving

down a curve such as B A, _fig. 28._ In this case, also, the

whole weight is distributed between that part which is directed down

the curve, and that which, being perpendicular to the curve, produces

a pressure upon it. There is, however, another cause which produces

pressure upon the curve, and which has no operation in the case of

the inclined plane. By the property of inertia, when a body is put in

motion in any direction, it must persevere in that direction, unless

it be deflected from it by an efficient force. In the motion down an

inclined plane the direction is never changed, and therefore by its

inertia the falling body retains all the motion impressed upon it

continually in the same direction; but when it descends upon a curve,

its direction is constantly varying, and the resistance of the curve

being the deflecting cause, the curve must sustain a pressure equal to

that force, which would thus be capable of continually deflecting the

body from the rectilinear path in which it would move in virtue of its

inertia. This pressure entirely depends on the curvature of the path

in which the body is constrained to move, and on its inertia, and is

therefore altogether independent of the weight, and would, in fact,

exist if the weight were without effect.

(137.) This pressure has been denominated _centrifugal force_, because

it evinces a tendency of the moving body to _fly from_ the centre of

the curve in which it is moved. Its quantity depends conjointly on the

velocity of the motion and the curvature of the path through which

the body is moved. As circles may be described with every degree of

curvature, according to the length of the radius, or the distance from

their circumference to their centre, it follows that, whatever be the

curve in which the body moves, a circle can always be assigned which

has the same curvature as is found at any proposed point of the given

curve. Such a circle is called “the circle of curvature” at that point

of the curve; and as all curves, except the circle, vary their degrees

of curvature at different points, it follows that different parts of

the same curve will have different circles of curvature. It is evident

that the greater the radius of a circle is, the less is its curvature:

thus the circle with the radius A B, _fig. 29._, is more

curved than that whose radius is C D, and that in the exact

proportion of the radius C D to the radius A B. The radius of

the circle of curvature for any part of a curve is called “the radius

of curvature” of that part.

(138.) The centrifugal pressure increases as the radius of curvature

increases; but it also has a dependence on the velocity with which the

moving body swings round the centre of the circle of curvature. This

velocity is estimated either by the actual space through which the body

moves, or by the _angular velocity_ of a line drawn from the centre of

the circle to the moving body. That body carries one end of this line

with it, while the other remains fixed at the centre. As this angular

swing round the centre increases, the centrifugal pressure increases.

To estimate the rate at which this pressure in general varies, it is

necessary to multiply the square of the number expressing the angular

velocity by that which expresses the radius of curvature, and the force

increases in the same proportion as the product thus obtained.

(139.) We have observed that the same causes which produce pressure

on a body restrained, will produce motion if the body be free.

Accordingly, if a body be moved by any efficient cause in a curve, it

will, by reason of the centrifugal force, _fly off_, and the moving

force with which it will thus retreat from the centre round which

it is whirled will be a measure of the centrifugal force. Upon this

principle an apparatus called a _whirling table_ has been constructed,

for the purpose of exhibiting experimental illustrations of the laws

of centrifugal force. By this machine we are enabled to place any

proposed weights at any given distances from centres round which they

are whirled, either with the same angular velocity, or with velocities

having a certain proportion. Threads attached to the whirling weights

are carried to the centres round which they respectively revolve, and

there, passing over pulleys, are connected with weights which may be

varied at pleasure. When the whirling weights fly from their respective

centres, by reason of the centrifugal force, they draw up the weights

attached to the other ends of the threads, and the amount of the

centrifugal force is estimated by the weight which it is capable of

raising.

With this instrument the following experiments may be exhibited:--

Exp. 1. Equal weights whirled with the same velocity at equal distances

from the centre raise the same weight, and therefore have the same

centrifugal force.

Exp. 2. Equal weights whirled with the same angular velocity at

distances from the centre in the proportion of one to two, will raise

weights in the same proportion. Therefore the centrifugal forces are in

that proportion.

Exp. 3. Equal weights whirled at equal distances with angular

velocities which are as one to two, will raise weights as one to four,

that is, as the squares of the angular velocities. Therefore the

centrifugal forces are in that proportion.

Exp. 4. Equal weights whirled at distances which are as two to three,

with angular velocities which are as one to two, will raise weights

which are as two to twelve; that is, as the products of the distances

two and three, and the squares one and four, of the angular velocities.

Hence, the centrifugal forces are in this proportion.

The centrifugal force must also increase as the mass of the body moved

increases; for, like attraction, each particle of the moving body is

separately and equally affected by it. Hence a double mass, moving

at the same distance, and with the same velocity, will have a double

force. The following experiment verifies this:--

Exp. 5. If weights, which are as one to two, be whirled at equal

distances with the same velocity, they will raise weights which are as

one to two.

The law which governs centrifugal force may then be expressed in

general symbols briefly thus:--

Let _c_ = the centrifugal force with which a weight of one lb.

revolving in a circle in one second, the radius of which is one foot,

would act on a string connecting it with the centre. The force with

which it would act on a string, the length of which is R feet, would

be _c_ × R; and if instead of revolving in one second it revolved in T

seconds, the force would be

(_c_ × R)/T^2;

and if the revolving mass were W lbs. the force would be