Compendium of Natural and Experimental Philosophy 5

Index

PREFACE TO THE REVISED AND ENLARGED EDITION 1871
INTRODUCTION
DIVISIONS OF THE SUBJECT
OF MATTER AND ITS PROPERTIES
OF GRAVITY
MECHANICS, on THE LAWS OF MOTION
THE MECHANICAL POWERS
REGULATORS OF MOTION
HYDROSTATICS
HYDRAULICS
PNEUMATICS
ACOUSTICS
PYRONOMICS
THE STEAM-ENGINE
OPTICS
ELECTRICITY
GALVANISM, OR VOLTAIC ELECTRICITY
MAGNETISM
ELECTRO MAGNETISM
THE ELECTROMAGNETIC TELEGRAPH
THE ELECTROTYPE PROCESS
MAGNETO-ELECTRICITY
THERMO-ELECTRICITY
ASTRONOMY

ON THE LAWS OF MOTION

What is Mechanics?

  1. MECHANICS. Mechanics treats of motion, and the moving powers, their nature and laws, with their effects in machines.

What is Motion?

  1. Motion is a continued change of place.
  2. On account of the inertia of matter, a body at rest cannot put itself in motion, nor can a body in motion stop itself.

What is meant by a Force?

  1. That which causes motion is called a Force.

What is meant by Resistance?

  1. That which stops or impedes motion is called Resistance.*

* A force is sometimes a resistance, and a resistance is sometimes a force. The two terms are used merely to denote opposition. [See Appendix, par. 1387]

What things are to be considered in relation to motion?

  1. In relation to motion, we must consider the force, the resistance, the time, the space, the direction, the velocity and the momentum.

What is the velocity, and to what is it proportional?

  1. The velocity is the rapidity with which a body moves; and it is always proportional to the force by which the body is put in motion.
  2. The velocity of a moving body is determined by the time that it occupies in passing through a given space. The greater the space and the shorter the time, the greater is the velocity. Thus, if one body move at the rate of six miles, and another twelve miles in the same time, the velocity of the latter is double that of the former.

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What is the rule for finding the velocity of a moving body?

  1. To find the velocity of a body, the space passed over must be divided by the time employed in moving over it.

Thus, if a body move 100 miles in 20 hours, the velocity is found by dividing 100 by 20. The result is five miles an hour.*

* Velocity is sometimes called absolute, and sometimes relative. Velocity is called absolute when the motion of a body in space is considered without reference to that of other bodies. When, for instance, a horse goes a hundred miles in ten hours, his absolute velocity is ten miles an hour.

Velocity is called relative when it is compared with that of another body. Thus, if one horse travel only fifty miles in ten hours, and another one hundred in the same time, the absolute velocity of the first horse is five miles an hour, and that of the latter is ten miles; but their relative velocity is as two to one.

  1. Questions for Solution.

(1.) If a body move 1000 miles in 20 days, what is its velocity 1 Ans. 60 miles a day.

(2.) If a horse travel 15 miles in an hour, what is his velocity? Ans. ¼ of a mile in a minute.

(3.) Suppose one man walk 300 miles in 10 days, and another 200 miles in the same time, – what are their respective velocities? Ans. 30 & 20.

(4.) If a ball thrown fiom a cannon strike the ground at the distance of 3 miles in 3 seconds from the time of its discharge, what is its velocity! A. 1.

(5.) Suppose a flash of lightning come from a cloud 3 miles distant from the earth, and the thunder be heard in 14 seconds after the flash is seen; how fast does sound travel? Ans. 1131 3/7 ft. per sec.

(6.) The sun is 95 millions of miles from the earth, and it takes 8. minutes for the light from the sun to reach the earth; with what velocity does light move! t Ans. 191919 + mi. per sec. **

** From the table here subjoined, the velocities of the objects enumerated may be ascertained in miles per hour and in feet per second, fractions omitted.

TABLE OF VELOCITIES

Miles per hour                                       Feet per second

A man walking                              3                           4

A horse trotting                            7                         10

Swiftest race-horse                   60                         88

Railroad train in England          32                         47

”          ”       America          18                         26

”          ”       Belgium          25                         36

”          ”       France             27                         40

”          ”       Germany         24                         35

English steamboats in

channels                              14                         26

American on the Hudson          18                         26

Fast-sailing vessels                    10                         14

 

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How is the time employed by a moving body ascertained?

  1. The time employed by a body in motion may be ascertained by dividing the space by the velocity.

Thus, if the space passed over be 100 miles. and the velocity 5 miles in in hour, the time will be 100 divided by 5. Ans. 20 hours.

  1. Questions for Solution.

(1.) If a cannon-ball, with a velocity of 3 miles in a minute, strike the ground at the distance of one mile, what is the time employed. Ans. 1/3 of a minute, or 20 seconds.

(2.) Suppose light to move at the rate of 192,000 miles in a second of time, how long will it take to reach the earth from the sun, which is 95 millions of miles distant. Ans. 8 min. 14.07 sec. +

(3.) If a railroad-car run at the rate of 20 miles an hour, how long will it take to go from Washington to Boston,-distance 432 miles? Ans. 21.6 hr.

(4.) Suppose a ship sail at the rate of 6 miles an hour, how long will it take to go from the United States to Europe, across the Atlantic Ocean, a distance of 2800’miles. Avs. 19 da. 10 hr. 40 min.

(5.) If the earth go round the sun in 365 days, and the distance travelled be 540 millions of miles, how fast does it travel? Ans. 1,479,452 4/73 Mil.

(6.) Suppose a carrier-pigeon, let loose at 6 o’clock in the morning from Washington, reach New Orleans at 6 o’clock at night, a distance of 1200 miles, how fast does it fly? Acts. 100 sni. per hr.

How may the space passed over by a body in motion be ascertained?

  1. The space passed over may be found by multiplying the velocity by the time.

Miles per hour         Feet per second

Slow rivers                                                                                 3                                   4

Rapid rivers                                                                               7                                 10

Moderate wind                                                                        7                                 10

A storm                                                                                    36                                 52

A hurricane                                                                             80                               117

Common musket-ball                                                         850                           1,240

Rifle-ball                                                                            1,000                           1,466

24 lb cannon-ball                                                             1,600                           2,316

Air rushing into a vacuum at 32°F                                    884                           1,296

Air gun bullet, air compressed

to -01 of its volume                                                        466                               683

Sound                                                                                     743                           1,142

A point on the surface of the earth                              1,037                           1,520

Earth in its orbit                                                             67,374                         98,815,

 

The velocity of light is 192,000 miles in a second of time.

The velocity of the electric fluid is said to be still greater, and some authorities state it to be at the rate of 288 000 miles in a second of time.

 

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Thus, if the velocity be 5 miles an hour, and the time 20 hours, the space will be twenty multiplied by 5. Ans. 100 miles.

  1. (1.) If a vessel sail 125 miles in a day for ten days, how far will it sail in that time? Ans. 1250 mil.

(2.) Suppose the average rate of steamers between New York and Albany be about 11 miles an hour, which they traverse in about 14 hours, what is the distance between these two cities by the river. Ands. 154 mi.

(3.) Suppose the cars going over the railroad between these two cities travel at the rate of 25 miles an hour and take 8 hours to go over the distance, how far is it from New York to Albany by railroad. Ans. 200 mi.

(4.) If a man walking from Boston at the rate of 24 miles in an hour reach Salem in 6 hours, what is the distance from Boston to Salem? Ans. 15 mi.

(5.) The waters of a certain river, moving at the rate of 4 feet in a second, reach the sea in 6 days from the time of starting from the source of the river. What is the length of that river – Ans. 392 8/11 mi.

(6.) A cannon-ball, moving at the rate of 2400 feet in a second of time, strikes a target in 4 seconds. What is the distance of the target? A. 9600ft.

  1. The following formula embrace the several ratios of the time, space and velocity:

(1.) The space divided by the time equals the velocity, or s/t = v.

(2.) The space divided by the velocity equals the time, or s/v = t.

(3.) The velocity multiplied by the time equals the space, or v X t = s.

How many kinds of motion are there?

  1. There are three kinds of motion, namely, Uniform, Accelerated and Retarded.

What is Uniform Motion?

  1. Uniform Motion is that by which a body moves over equal spaces in equal times.

What is Accelerated Motion?

  1. Accelerated Motion is that by which the velocity increases while the body is moving.

What is Retarded Motion?

  1. Retarded Motion is that by which the velocity decreases while the body is moving.

How are uniform, accelerated and retarded motion respectively produced?

  1. Uniform Motion is produced by the – momentary action of a single force. Accelerated Motion is produced by the continued action of one or more forces. Retarded Motion is produced by some resistance.

 

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  1. A ball struck by a bat, or a stone thrown from the hand is in theory an instance of uniform motion; and, if the attraction of gravity and the resistance of the air could be suspended, it would proceed onwards in a straight line, with a uniform motion, forever. But, as the resistance of the air and gravity both tend to deflect it, it in fact becomes first an instance of retarded, and then of accelerated motion.
  2. A stone, or any other body, falling from a height, is an instance of accelerated motion. The force of gravity continues to operate upon it during the whole time of its descent, and constantly increases its velocity. It begins its descent with the first impulse of attraction, and, could the force of gravity which gave it the impulse be suspended, it would continue its descent with a uniform velocity. But, while falling it is every moment receiving a new impulse from gravity, and its velocity is constantly increasing during the whole time of its descent.
  3. A stone thrown perpendicularly upward is an instance of retarded motion; for, as soon as it begins to ascend, gravity immediately attracts it downwards, and thus its velocity is diminished. The retarding force of gravity acts upon it during every moment of its ascent, decreasing its velocity until its upward motion is entirely destroyed. It then begins to fall with a motion continually accelerated until it reaches the ground.

What time does a body occupy in its ascent and its descent?

  1. A body projected upwards will occupy the same time in its ascent and descent.

This is a necessary consequence of the effect of gravity, which uniformly retards it in the ascent and accelerates it in its descent.

How can perpetual motion be produced?

  1. PERPETUAL MOTION – Perpetual Motion is deemed an impossibility in mechanics, because action and reaction are always equal and in contrary directions.

What is meant by Action and Reaction?

  1. By the action of a body is meant the effect which it produces upon another body. By reaction is meant the effect which it receives from the body on which it acts.

Thus, when a body in motion strikes another body, it acts upon it, or produces motion; but it also meets with resistance from the body which is struck, and this resistance is the reaction of the body.

 

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Illustration of Action and Reaction by means of Elastic and Non-elastic Balls.

fig06 parkers compendium

(1.) Figure 6 represents two ivory * balls, A and B, of equal size, weight, &c., suspended by threads. If the ball A be drawn a little on one side and then let go, it will strike against the other ball B. and drive it off to a distance equal to that through which the first ball fell; but the motion of A will be stopped, because when it strikes B it receives in return a blow equal to that which it gave, but in a contrary direction, and its motion is thereby stopped, or, rather given to B. Therefore, when a body strikes against another the quantity of motion communicated to the second body is lost by the first; but this loss proceeds not from the blow given by the striking body, but from the reaction of the body which it struck.

* It will be recollected that ivory is considered highly elastic.

fig07 parkers compendium

(2.) Fig. 7 represents six ivory balls of equal weight, suspended by threads. If the ball A be drawn out of the perpendicular and let fall against B, it will communicate its motion to B, and receive a reaction from it which will stop its own motion. But the ball B cannot move without moving C; it will therefore communicate the motion which it received from A to C and receive from C a reaction, which will stop its motion.

In like manner the motion and reaction are received by each of the balls D, E, F; but, as there is no ball beyond F to act upon it, F will fly off.

  1. B. This experiment is to be performed with elastic balls only.

fig08 parkers compendium

(3). Fig. 8 represents two balls of clay (which are not elastic), of equal weight, suspended by strings. If the ball D be raised and let fall against E, only part of the motion of D will be destroyed by it (because the bodies are non-elastic), and the two balls will move on together to d and e, which are less distant from the vertical line than the ball D was before it fell. Still,

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however, action and reaction are equal, for the action on E is only enough to make it move through a smaller space, but so much of D’s motion is now also destroyed. *

* Figs. 6 and 7, as has been explained, show the effect of action and reaction in elastic bodies, and Fig. 8 shows the same effect in non-elastic bodies. When the elasticity of a body is imperfect, an intermediate effect will be produced; that is, the ball which is struck will rise higher than in case of non-elastic bodies, and less so than in that of perfectly elastic bodies; and the striking ball will be retarded more than in the former case, but not stopped completely, as in the latter. They will, therefore, both move onwards after the blow, but not together, or to the same distance; but -,this, as in the preceding cases, the whole quantity of motion destroyed in the striking ball will be equal to that produced in the ball struck. Connected with “the Boston school apparatus” is a stand with ivory balls, to give a visible illustration of the effects of collision.

  1. It is upon the principle of action and reaction that birds are enabled to fly. They strike the air with their wings, and the reaction of the air enables them to rise, fall, or remain stationary, at will, by increasing or diminishing the force of the stroke of their wings. **

** The muscular power of birds is much greater in proportion to their weight than that of man. If a man were furnished with wings sufficiently large to enable him to fly, he would not have sufficient strength or muscular power to put them in motion.

  1. It is likewise upon the same principle of action and reaction that fishes swim, or, rather, make their way through the water, namely, by striking the water with their fins. ***

*** The power possessed by fishes, of sinking or rising in the water, is greatly assisted by a peculiar apparatus furnished them by nature, called an air-bladder, by the expansion or contraction of which they rise or fall, on the principle of specific gravity.

 

  1. Boats are also propelled by oars on the same principle, and the oars are lifted out of the water, after every stroke, so, as completely to prevent any reaction in a backward direction.

How may Motion be caused?

  1. Motion may be caused either by action or reaction. When caused by action it is called Incident, and when caused by reaction it is called Reflected Motion. ****

**** The word incident implies falling upon, or directed towards. The word reflected implies turned back. Incident motion is motion directed towards any particular object, against which a moving body strikes. Reflected motion is that which is caused by the reaction of the body which is struck. Thus, when a ball is thrown against a surface, it rebounds or is turned back. This return of the ball is called reflected motion. As reflected motion is caused by reaction, and reaction is increased by elasticity, it follows that reflected motion is always greatest in those bodies which are most elastic. For this reason a ball filled with air rebounds better than one stuffed with bran or wool, because its elasticity is greater. For the same reason, balls made of caoutchouc, or India-rubber, will rebound more than those which are made of most other substances.

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What is an angle of Incidence?

  1. The angle * of incidence is the angle formed by the line which the incident body makes in its passage towards any object, with a line perpendicular to the surface of the object.

* As this book may fall into the hands of some who are unacquainted with geometrical figures, a few explanations are here subjoined:

  1. An angle is the opening made by two lines which meet each other in a point. The size of the angle depends upon the opening, and not upon the length of the lines.
  2. A circle is a perfectly round figure, every part of the outer edge of which, called the circumference, is equally distant from a point within, called the centre. [See Fig. 9]

fig09 parkers compendium

  1. The straight lines drawn from the centre to the circumference are called radii. [The singular of this word is radius.] Thus, in Fig. 9, the lines C D, C O, C R, and C A, are radii.
  2. The lines drawn through the centre, and terminating in both ends at the circumference, are called diameters. Thus, in the same figure, D A is a diameter of the circle.
  1. The circumference of all circles is divided into 360 equal parts, called degrees. The diameter of a circle divides the circumference into two equal parts, of 180 degrees each.
  2. All angles are measured by the number of degrees which they contain.

Thus, in Fig. 9, the angle R C A, as it includes one-quarter of the circle, is an angle of 90 degrees, which is a quarter of 360. And the angles R C O and O C D are angles of 45 degrees.

  1. Angles of 90 degrees are right angles; angles of less than 90 degrees, acute angles; and angles of more than 90 degrees are called obtuse angles. Thus, in Fig. 9, R C A is a right angle, O C R an acute, and O C A an obtuse angle.
  2. A perpendicular line is a line which makes an angle of 90 degrees on each side of any other line or surface; therefore, it will incline neither to the one side nor to the other. Thus, in Fig. 9, R C is perpendicular to D A.
  3. The tangent of a circle is a line which touches the circumference, without cutting it when lengthened at either end. Thus, in Fig. 9, the line RT is a tangent.
  4. A square is a figure having four equal sides, and four equal angles. These will always be right angles. [See Fig. 11.]
  5. A parallelogram is a figure whose opposite sides are equal and parallel [See Figs. 12 and 13.] A square is also a parallelogram.
  6. A rectangle is a parallelogram whose angles are right angles.

[N. B. It will be seen by these definitions that both a square and a rectangle are parallelograms, but all parallelograms are not rectangles nor squares. A square is both a parallelogram and a rectangle. Three things are essential to a square; namely, the four sides must all be equal, they must also be parallel, and the angles must all be right angles. Two things only are essential to a rectangle; namely, the angles must all be right angles, and the opposite sides must be equal and parallel. One thing only is essential to a parallelogram; namely, the opposite sides must be equal and parallel.]

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  1. The diagonal of a square, of a parallelogram, or a rectangle, is a line drawn through either of them, and terminating at the opposite angles. Thus, in Figs. 11, 12, and 13, the line A C is the diagonal of the square, parallelogram, or rectangle.

fig10 parkers compendium

Explain Fig. 10

  1. Thus, in Fig. 10, the line A B C represents a wall, and P B a line perpendicular to its surface. O is a

ball moving in the direction of the dotted line, O B. The angle O B P is the angle of incidence.

What is the angle of reflection?

  1. The angle of reflection is the angle formed the angle by the perpendicular with the line made by the reflected body as it leaves the surface against which it struck. Thus, in Fig. 10, the angle P B R is the angle of reflection.

What is the proportion of the angle of incidence to the angle of reflection?

  1. The angles of incidence and reflection are always equal to one another. *

* An understanding of this law of reflected motion is very important, because it is a fundamental law, not only in Mechanics, but also in Pyronomics, Acoustics and Optics.

(1.) Thus, in Fig. 10, the angle of incidence, O B P, and the angle of reflection, P B R, are equal to one another; that is, they contain an equal number of degrees.

What will be the course of a body in motion which strikes against another fixed body?

  1. From what has now been stated with regard to the angles of incidence and reflection, it follows, that when a ball is thrown perpendicularly against an object which it cannot penetrate, it will return in the same direction; but, if it be thrown obliquely, it will return obliquely on the opposite side of the perpendicular. The more obliquely the ball is thrown, the more obliquely it will rebound. **

** It is from a knowledge of these facts that skill is acquired in many different sorts of games, as Billiards, Bagatelle, &c. A ball may also, on the same principle, be thrown from a gun against a fortification so as to reach an object out of the range of a direct shot.

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What is the Momentum of a body?

  1. MOMENTUM. The Momentum* of a body is its quantity of motion **, and it expresses the force with which it would strike against another body.

How is the Momentum of a body calculated?

The Momentum of a body is ascertained by multiplying its weight by its velocity.

  1. Thus, if the velocity of a body be represented by 5 and its weight by 6, its momentum will be 30.

How can a small or a light body be made to do as much damage as a large one?

  1. A small or a light body may be made to strike against another body with a greater force than a heavier body simply by giving it sufficient velocity, that is, by making it have greater momentum.

Thus, a cork weighing ¼ of an ounce, shot from a pistol with the velocity of 100 feet in a second, will do more damage than a leaden shot weighing 1/8 of an ounce, thrown from the hand with a velocity

of 40 feet in a second, because the momentum of the cork will be the greater.

The momentum of the cork is 1/4 X 100 = 25.

That of the leaden shot is 1/8 X 40 = 5.

  1. Questions for Solution.

(1.) What is the momentum of a body weighing 5 pounds, moving with the velocity of 50 feet in a second? Ans. 250.

(2.) What is the momentum of a steam-engine, weighing 3 tons, moving with the velocity of 60 miles in an hour? Ans. 180.

[N. B. It must be recollected that, in comparing the momenta* of bodies, the velocities and the time of the bodies compared must be respectively of the same denomination. If the time of one be minutes and of the other be hours, they must both be considered in minutes, or both in hours. So, with regard to the spaces and the weights, if one be feet all must be expressed in feet; if one be in pounds, all must be in pounds. It is better, however, to express the weight, velocities and spaces, by abstract numbers, as follows:]

(3.) If a body whose weight is expressed by 9 and velocity by 6 is in motion, what is its momentum? Ans. 54.

(4.) A body whose momentum is 63 has a velocity of 9; what is its weight? Ans. 7.

* The plural of this word is momenta.

** The quantity of motion communicated to a body does not affect the duration of the motion. If but little motion be communicated, the body will move slowly. If a great degree be imparted, it will move rapidly. But in both cases the motion will continue until it is destroyed by some external force.

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[N. B. The momentum being the product of the weight and velocity, the weight is found by dividing the momentum by the velocity, and the velocity is found by dividing the momentum by the weight.]

(5.) The momentum is expressed by 12, the weight by 2; what is the velocity? Ans. 6.

(6.) The momentum 9, velocity 9, what is the weight? Ans. 1.

(7.) Momentum 36, weight 6, required the velocity. Ans. 6.

(8.) A body with a momentum of 12 strikes another with a momentum of 6; what will be the consequence? Ans. Both have mom. of 6.

[N. B. When two bodies, in opposite directions, come into collision, they each lose an equal quantity of their momenta.]

(9.) A body weighing 15, with a velocity of 12, meets another coming in the opposite direction, with a velocity of 20, and a weight of 10; what will be the effect 3. Ans. Both move with mom. of 20.

(10.) Two bodies meet together in opposite directions. A has a velocity of 12 and a weight of 7, B has a momentum expressed by 84. What will be the consequence 3 Ans. Both mom. destroyed.

(11.) Suppose the weight of a comet be represented by 1 and its velocity by 12, and the weight of the earth be expressed by 100 and its velocity by 10, what would be the consequence of a collision, supposing them to be moving in opposite directions 3 Ans. Both have mom. of 988.

(12.) If a body with a weight of 75 and a velocity of 4 run against a man whose weight is 150, and who is standing still, what will be the consequence, if the man uses no effort but his weight? Ans. Man has vel. of 1 1/3.

(13.) With what velocity must a 64 pound cannon-ball fly to be equally effective with a battering-ram of 12,000 pounds propelled with a velocity of 16 feet in a second?. Ans. 3000 ft.

  1. ATTRACTION LAW OF FALLING BODIES. When one body strikes another it will cause an effect proportional to its own weight and velocity (or, in other words, its momentum); and the body which receives the blow will move on with a uniform velocity (if the blow be sufficient to overcome its inertia) in the direction of the motion of the blow. But, when a body moves by the force of a constant attraction, it will move with a constantly accelerated motion.
  2. This is especially the case with falling bodies. The earth attracts them with a force sufficient to bring them down through a certain number of feet during the first second of time. While the body is thus in motion with a velocity, say of sixteen feet, the earth still attracts it, and during the second second it communicates an additional velocity, and every successive second of time the attraction of the earth adds to the velocity in a similar proportion, so that during any given time, a falling body will acquire a velocity which, in the same time, would carry it over twice the space through which it has already fallen. Hence we deduce the following law:

What is the law of falling bodies?

  1. A body falling from a height will fall sixteen feet in the first second of time,* three times that distance in the second, five times in the third, seven in the fourth, its velocity increasing during every successive second, as the odd numbers 1, 3, 5, 7, 9, 11, 13, &c.*

* This is only an approximation to the truth; it actually falls sixteen feet and one inch during the first second, three times that distance in the second, &c. The questions proposed to be solved assume sixteen feet only.

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The laws of falling bodies are clearly demonstrated by a mechanical arrangement known by the name of “Attwood’s Machine,” in which a small weight is made to communicate motion to two others attached to a cord passing over friction-rollers (causing one to ascend and the other to descend), and marking the progress of the descending weight by the oscillations of a pendulum on a graduated scale, attached to one of the columns of the machine. It has not been deemed expedient to present a cut of the machine, because without the machine itself the explanation of its operation would be unsatisfactory, with the machine itself in view the simplicity of its construction would render an explanation unnecessary.

* The entire spaces through which a body will have fallen in any given number of seconds increase as the squares of, the times. This law was discovered by Galileo, and may thus be explained. If a body fall sixteen feet in one second, in two seconds it will have fallen four times as far, in three seconds nine times as far, in four seconds sixteen times as far, in the fifth second twenty-five times, &c., in the sixth thirty-six times, &c.

ANALYSIS OF THE MOTION OF A FALLING BODY

Number of Seconds         Spaces                 Velocities            Total Space.

1                                                 1                                 2                              1

2                                                 3                                 4                              4

3                                                 5                                 6                              9

4                                                 7                                 8                            16

5                                                 9                               10                            25

6                                               11                               12                            36

7                                               13                               14                            49

8                                               15                               16                            64

9                                               17                               18                            81

10                                            19                               20                          100

From this statement it appears that the spaces passed through by a falling body, in any number of seconds, increase as the odd numbers 1, 3, 5, 7, 9, 11, &c.; the velocity increases as the even numbers 2, 4, 6, 8, 10, 12, &c.; and the total spaces passed through in any given number of seconds increase as the squares of the numbers indicating the seconds, thus, 1, 4, 9, 16, 25, 36, &c.

Aristotle maintained that the velocity of any falling body is in direct proportion to its weight; and that, if two bodies of unequal weight were let fall from any height at the same moment, the heavier body would reach the ground in a shorter time, in exact proportion as its weight exceeded that of the lighter one. Hence, according to his doctrine, a body weighing two pounds would fall in half the time required for the fall of a body weighing only one pound. This doctrine was embraced by all the followers of that distinguished philosopher, until the time of Galileo, of Florence, who flourished about the middle of the sixteenth century. Ile maintained that the velocity of a falling body is not affected by its weight, and challenged the adherents of the Aristotelian doctrine to the test of experiment. The leaning tower of Pisa was selected for the trial, and there the experiment was tried which proved the truth of Galileo’s theory.

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A distinguished writer thus describes the scene. “On the appointed day the disputants repaired to the tower of Pisa, each party, perhaps, with equal confidence. It was a crisis in the history of human knowledge. On the one side stood the assembled wisdom of the universities, revered for age and science, venerable, dignified, united and commanding. Around them thronged the multitude, and about them clustered the associations of centuries. On the other there stood an obscure young man (Galileo), with no retinue of followers, without reputation, or influence, or station. But his courage was equal to the occasion; confident in the power of truth, his form is erect and his eye sparkles with excitement. But the hour of trial arrives. The balls to be employed in the experiments are carefully weighed and scrutinized, to detect deception. The parties are satisfied. The one ball is exactly twice the weight of the other. The followers of Aristotle maintain that, when the balls are dropped from the tower, the heavy one will reach the ground in exactly half the time employed by the lighter ball. Galileo asserts that the weights of the balls do not affect their velocities, and that the times of descent will be equal; and here the disputants join issue. The balls are conveyed to the summit of the lofty tower. The crowd assemble round the base; the signal is given; the balls are dropped at the same instant; and, swift descending, at the same moment they strike the earth. Again and again the experiment is repeated, with uniform results: Galileo’s triumph was complete; not i shadow of a doubt remained.” [“The Orbs of Heaven.”]

 

  1. The height of a building, or the depth of a well, may thus be estimated very nearly by observing the length of time which a stone takes in falling from the top to the bottom.
  2. Exercises for Solution.

(1.) If a ball, dropped from the top of a steeple, reaches the ground in 5 seconds, how high is that steeple.

16+48+80+ —112+144 —400 feet; or, 5X5 —25, square of the number of seconds, multiplied by the number of feet it falls through in one second, namely, 16 feet; that is, 25X16=400 feet.

(2.) Suppose a ball, dropped from the spire of a cathedral, reach the ground in 9 seconds, how high is that spire.

16 —48 — 80+- 112+- 144+-176-208 +2L0+272129 6 feet.

Or, squaring the time in seconds, 92=81, multiplied by 16l-1296. Ans. [It will hereafter be shown that this law of falling bodies applies to all bodies, whether falling perpendicularly or obliquely. Thus, whether a stone be thrown from the top of a building horizontally or dropped perpendicularly downwards, in both cases the stone will reach the ground in the same time; and this rule applies equally to a ball projected from a cannon, and to a stone thrown from the hand.]

(3.) If a ball, projected from a cannon from the top of a pyramid, reach the ground in 4 seconds, how high is the pyramid? Anss. 256ft.

(4.) How deep is a well, into which a stone being dropped, it reaches the water 6 feet from the bottom of the well in 2 seconds. Ames. 70et.

(5.) The light of a meteor bursting in the air is seen, and in 45 seconds a meteoric stone falls to the ground. Supposing the stone to have proceeded from the explosion of the meteor perpendicularly, how far from the earth, in feet, was the meteor? 452X16 = — 32,400 feet.

(6.) What is the difference in the depth of two wells, into one of which a stone being dropped, is heard to strike the water in 5 seconds, and into the other in 9 seconds, supposing that the water be of equal depth in both, and making no allowance for the progressive motion of sound 3 A. 896.ft,

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(7.) A boy raised his kite in the night, with a lantern attached to it. Unfortunately, the string which attached the lantern broke, and the lantern fell to the ground in 6 seconds. How high was the kite? Ans. 576ft.

  1. RETARDED MOTION OF BODIES PROJECTED UPWARDS. – All the circumstances attending the accelerated descent of falling bodies are exhibited when a body is projected upwards, but in a reversed order.

How can we determine the height to which body, projected upwards with a given velocity, will ascend?

  1. To determine the height to which a body with a given velocity, will rise with a given velocity, it is only necessary to determine the height from which a body would fall to acquire the same velocity.
  2. Thus, if it be required to ascertain how high a body would rise when projected upwards with a force sufficient to carry it 144 feet in the first second of time, we reverse the series of numbers 16+ 48+80+112 144 [see table on page 52], and, reading them backward, 144 + 112 + 80 + 48 + 16, we find their sum to be 400 feet, and the time employed would be 5 seconds.

How does the time of the ascent of a body compare with the time of its descent?

  1. The time employed in the ascent and descent of a body projected upwards will, therefore, always be equal.

Questions for Solution.

(1.) Suppose a cannon-ball, projected perpendicularly upwards, returned to the ground in 18 seconds; how high did it ascend, and what is the velocity of projection? Ans. 1296ft.; 272ft. lst sec.

(2.) How high will a stone rise which a man throws upward with a force sufficient to carry it 48 feet during the first second of time. Ans. 64ft.

(3.) Suppose a rocket to ascend with a velocity sufficient to carry it 176 feet during the first second of time; how high will it ascend, and what time will it occupy in its ascent and descent? Ans. 576ft.; 12 sec.

(4.) A musket-ball is thrown upwards until it’reaches the height of 400 feet. How long a time, in seconds, will it occupy in its ascent and descent, and what space does it ascend in the first second?. Ans. 10 see.; 144ft.

(5.) A sportsman shoots a bird flying in the air, and the bird is 3 seconds in falling to the ground. How high up was the bird when he was shot? Ans. 144ft.

(6.) How long time, in seconds, would it take a ball to reach an object 6000 feet above the surface of the earth, provided that the ball be projected, with a force sufficient only to reach the object? Ants. 17.67 sec. +

  1. COMPOUND MOTION. -Motion may be produced either by a single force or by the operation of two or more forces.

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In what direction is the motion of a body impelled by a single force?

  1. Simple Motion is the motion of a body impelled by a single force, and is always in a straight line in the same direction with the

force that acts.

What is Compound Motion?

  1. Compound Motion is caused by the operation of two or more forces at the same time.

When a body is struck by two equal forces, in opposite directions, how will it move?

  1. When a body is struck by two equal forces, in opposite directions it will remain at rest.
  2. If the forces be unequal, the body will move with diminished force in the direction of the greater force: Thus, if a body with a momentum of 9 be opposed by another body with a momentum of 6, both will move with a momentum of 3 in the direction of the greater force.

How will a body move when struck by two forces in different directions?

  1. A body, struck by two forces in different directions, will move in a line between them, in the direction of the diagonal of a parallelogram, having for its sides the lines through which the body would pass if urged by each of the forces separately.

How will the body move, if the forces are equal and at right angles to each other?

  1. When the forces are equal and at right angles to each other, the body will move in the diagonal of a square.

fig11 parkers compendium

  1. Let Fig. 11 represent a ball struck by the two equal forces X and Y. In this figure the forces are inclined to each other at an angle of 90°, or a right angle. Suppose that the force X would send it from C to B, and the force Y from C to D. As it cannot obey both, it will go between them to A and the line C A through which it passes, is the diagonal of the square, A B C D.

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This line also represents the resultant of the two forces. The time occupied in its passage from C to A will be the same as the force X would require to send it to B, or the force Y to send it to D.

How will a body move under the influence of two unequal forces at right angles each other?

  1. If two unequal forces act at right angles to each other on a body, the body will move in the direction of the diagonal of a rectangle.

fig12 parkers compendiumFig. 12

Explain Fig. 12.

  1. Illustration.-In Fig. 12 the ball C is represented as acted upon by two unequal forces, X and Y. The force X would send it to B, and the force Y to D. As it cannot obey both, it will move in the direction C A, the diagonal of the rectangle A B C D. B

How will the body move if the forces act in the direction of any other than a right angle?

  1. When two forces act in the direction of an acute or an obtuse angle, the body will move in the direction of the diagonal of a parallelogram.

fig13 parkers compendiumFig. 13

Explain Fig. 13.

  1. Illustration. — In figure 13 the ball C is supposed to be influenced by two forces, one of which would send it to B and the other to D, the forces acting in the direction of an acute angle. The ball will, therefore, move between them in the line C A, the longer diagonal of the parallelogram A B C D.
  2. The same figure explains the motion of a ball when the two forces act in the direction of an obtuse angle.

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  1. Illustration. – The ball D, under the influence of two forces, one of which would send it to C, and the other to A, which, it will be observed, is in the direction of an obtuse angle, will proceed in this case to B, the shorter diagonal of the parallelogram A B C D.

[N. B. A parallelogram containing acute and obtuse angles has two diagonals, the one which joins the acute angles being the longer.]

What is Resultant Motion?

  1. Resultant Motion is the effect or result of two motions compounded into one.
  2. If two men be sailing in separate boats, in the same direction, and at the same rate, and one toss an apple to the other, the apple would appear to pass directly across from one to the other. in a line of direction perpendicular to the side of each boat. But its real course is through the air in the diagonal of a parallelogram, formed by the lines representing the course of each boat, and perpendiculars drawn to those lines from the spot where each man stands as the one tosses and the other catches the apple.

Explain Fig. 14.

fig14 parkers compendiumFig. 14.

In Fig. 14. the lines A B and C D represent the course of each boat. E the spot where the man stands who tosses the apple; while the apple is in its passage, the boats have passed from E and G to H and F respectively. But the apple, having a motion, with the man, that would carry it from E to H, and likewise a projectile force which would carry it from E to G, cannot obey them both, but will pass through the dotted line E F, which is the diagonal of the parallelogram E G F H.*

* On the principle of resultant motion, if two ships in an engagement be sailing before the wind, at equal rates, the aim of the gunners will be exactly as though they both stood still. But, if the gunner fire from a ship standing still at another under sail, or a sportsman fire at a bird on the wing, each should take his aim a little forward of the mark, because the ship and the bird will pass a little forward while the shot is passing to them.

How can we ascertain the direction of the motion when the body is influenced by three or more forces?

  1. When a body is acted upon by three or of the more forces at the same time, we may take any two of them alone, and ascertain the resultant of those two, and then employ the resultant as a new force, in conjunction with the third,* &c.

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* The resultant of two forces is always described by the third side of a triangle, of which the two forces may be represented, in quantity and direction, by the other two sides. When three forces act in the direction of the three sides of the same triangle, the body will remain at rest.

When two forces act at right angles, the resultant will form the hypothenuse of a right-angled triangle, either of the sides of which may be found, when the two others are given, by the common principles of arithmetic or geometry.

From what has now been stated, it will easily be seen, that if any number of forces whatever act upon a body, and in any directions whatever, the resultant of them all may easily be found, and this resultant will be their mechanical equivalent. Thus, suppose a body be acted upon at the same time by six forces, represented by the letters A, B, C, D, E, P. First find the resultant of A and B by the law stated in No. 184, and call this resultant G. In the same manner, find the resultant of G and C, calling it H. Then find the resultant of H and D, and thus continue until each of the forces be found, and the last resultant will be the mechanical equivalent of the whole.

 

What is Circular motion?

  1. CIRCULAR MOTION. — Circular Motion is motion around a central point.

What causes Circular Motion?

  1. Circular motion is caused by the continued operation of two forces, by one of which the body is projected forward in a straight line, while the other is constantly deflecting it towards a fixed point. [See No. 184.]
  2. The whirling of a ball, fastened to a string held by the hand, is an instance of circular motion. The ball is urged by two forces, of which one is the force of projection, and the other the string which confines it to the hand. The two forces act at right angles to each other, and (according to No. 184) the ball will move in the diagonal of a parallelogram. But, as the force which confines it to the hand only keeps it within a certain distance, without drawing it nearer to the hand, the motion of the ball will be through the diagonals of an indefinite number of minute parallelograms, formed by every part of the circumference of the circle.

How many centres require to be noticed in Mechanics?

  1. There are three different centres which require to be distinctly noticed; namely, the Centre of Magnitude, the Centre of Gravity, and the Centre of Motion.

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What is the Centre of Magnitude?

  1. The Centre of Magnitude is the central point of the bulk of a body.

What is the Centre of Gravity?

  1. The Centre of Gravity is the point about which all the parts balance each other.

What is the Centre of Motion?

  1. The Centre of Motion is the point around which all the parts of a body move.

What is the Axis of Motion?

  1. When the body is not of a size nor shape to allow every point to revolve in the same plane, the line around which it revolves is called the Axis of Motion.*

* Circles may have a centre of motion; spheres or globes have an axis of motion. Bodies that have only length and breadth may revolve around their own centre, or around axes; those that have the three dimensions of length, breadth and thickness, must revolve around axes.

Does the centre or the axis of motion revolve?

  1. The centre or the axis of motion is generally supposed to be at rest.
  2. Thus the axis of a spinning top is stationary, while every other part is in motion around it. The axis of motion and the centre of motion are terms which relate only to circular motion.

What are Central Forces?

  1. The two forces by which circular motion is produced are called Central Forces. Their names are the Centripetal Force and the Centrifugal Force. **

** The word centripetal means seeking the centre, and centrifugal means flying from the centre. In circular motion these two forces constantly balance each other; otherwise the revolving body will either approach the centre, or recede from it, according as the centripetal or centrifugal force is the stronger.

What is the Centripetal Force?

  1. The Centripetal Force is that which confines a body to the centre around which it revolves.

What is the Centrifugal Force?

  1. The Centrifugal Force is that which impels the body to fly off from the centre.

 

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What follows if the centripetal or centrifugal force be destroyed?

  1. If the centrifugal force of a revolving body be destroyed, the body will immediately approach the centre which attracts it; but if the centripetal force be destroyed, the body will fly off in the direction of a tangent to the curve which it describes in its motion.*

* The centrifugal force is proportioned to the square of the velocity of a moving body. Hence, a cord sufficiently strong to hold a heavy body revolving around a fixed centre at the rate of fifty feet in a second, would require to have its strength increased four-fold, to hold the same ball, if its velocity should be doubled.

  1. Thus, when a mop filled with water is turned swiftly round by the handle, the threads which compose the head will fly off from the centre; but, being confined to it at one end, they cannot part from it; while the water they contain, being unconfined, is thrown off in straight lines.

When a body is revolving around its centre what parts move with the greatest velocity?

  1. The parts of a body which are furthest from the centre of motion move with the greatest velocity; and the velocity of all the parts diminishes as their distance from the axis of motion diminishes.

fig15 parkers compendiumFig. 15.

Explain Fig. 15

  1. Fig. 15 represents the vanes of a windmill. The circles denote the paths in which the different parts of the vanes move. M is the centre or axis of motion around which all the parts revolve. The outer part revolves in the circle D E F G, another part revolves in the circle H I J K, and the inner part in the circle L N O P. Consequently, as they all revolve around M in the same time, the velocity of the parts which revolve in the outer circle is as much greater than the velocity of the parts which revolve in the inner circle, L N O P, as the diameter of the outer circle is greater than the diameter of the inner.

 

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In the daily revolution of the earth around its own axis, what parts of the earth move most slowly, and what parts most rapidly?

  1. As the earth revolves round its axis, it follows, from the preceding illustration, that the portions of the earth which move most rapidly are nearest to the equator, and that the nearer any portion of the earth is to the poles the slower will be its motion.

What is required in order to produce curvilinear motion? and why?

  1. Curvilinear motion requires the action of two forces; for the impulse of one single force always produces motion in a straight line.

What effect has the centrifugal force on a body revolving around its longer axis?

  1. A body revolving rapidly around its longer axis, if suspended freely, will gradually change the direction of its motion, and revolve around its shorter axis.

This is due to the centrifugal force, which, impelling the parts from the centre of motion, causes the most distant parts to revolve in a larger circle.*

* This law is beautifully illustrated by a simple apparatus, in which a hook is made to revolve rapidly by means of multiplying wheels. Let an oblate spheroid, a double cone, or any other solid having unequal axes, be suspended from the hook by means of a flexible cord attached to the extremity of the longer axis. If, now, it be caused rapidly to revolve, it will immediately change its axis of motion, and revolve around the shorter axis.

The experiment will be doubly interesting if an endless chain be suspendedc from the hook, instead of a spheroid. So soon as the hook with the chain suspended is caused to revolve, the sides of the chain are thrown outward by the centrifugal force, until a complete ring is formed, and then the circular chain will commence revolving horizontally. This is a beautiful illustration of the effects of the centrifugal force. An apparatus, with a chain and six bodies of different form, prepared to be attached to the multiplying wheels in the manner described, accompanies most sets of philosophical apparatus.

Attached to the same apparatus is a thin hoop of brass, prepared for connexion with the multiplying wheels. The hoop is made rapidly to revolve around a vertical axis, loose at the top and secured below. So soon as the hoop begins to revolve rapidly, the horizontal diameter of the ring begins to increase and the vertical diameter to diminish, thus exhibiting the manner in which the equatorial diameter of a revolving body is lengthened, and the polar diameter is shortened, by reason of the centrifugal force. The daily revolution of the earth around its axis has produced this effect, so that the equatorial diameter is at least twenty-six miles longer than the polar. In those planets that revolve faster than the earth the effect is still more striking, as is the case with the planet Jupiter, whose figure is nearly that of an oblate spheroid.

The developments of Geology have led some writers to the theory that the earth, during one period of its history, must have had a different axis of motion; but it will be exceedingly difficult to reconcile such a theory to the law of rotations which has now been explained, especially as a much more rational explanation can be given to the phenomena on which the theory was built.

 

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What is Projectiles?

  1. PROJECTILES. – Projectiles is a branch of Mechanics which treats of the motion of bodies thrown or driven by an impelling force above the surface of the earth.

What is a Projectile?

  1. A Projectile is a body thrown into the air, — as a rocket, a ball from a gun, or a stone from the hand.

How are projectiles affected in their motion?

The force of gravity and the resistance of the air cause projectiles to form a curve both in their  ascent and descent; and, in descending, their motion is gradually changed from an oblique towards a perpendicular direction.

fig 16 parkers compendiumExplain Fig. 16.

  1. In Fig. 16 the force of projection would carry a ball from A to D, while gravity would bring it to C. If these two forces alone prevailed, the ball would proceed in the dotted line to B. But, as the resistance of the air operates in direct opposition to the force of projection, instead of reaching the ground at B, the ball will fall somewhere about E. *

* It is calculated that the resistance of the air to a cannon-ball of two pounds’ weight, with the velocity of two thousand feet in a second, is more than equivalent to sixty times the weight of the ball.

What is the course of a body thrown obliquely in a horizontal direction?

  1. When a body is thrown in a horizontal direction, or upwards or downwards, obliquely, its course will be in the direction of a curve-line, called a parabola **

** The science of gunnery is founded upon the laws relating to projectiles.

 

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fig17 parkers compendium

(see Fig. 17); but when it is thrown perpendicularly upwards or downwards, it will move perpendicularly, because the force of projection and that of gravity are in the same line of direction.

The force of gunpowder is accurately ascertained, and calculations are predicated upon these principles, which enable the engineer to direct his guns in such a manner as to cause the fall of the shot or shells in the very spot where he intends. The knowledge of this science saves an immense expenditure of ammunition, which would otherwise be idly wasted, without producing any effect. In attacks upon towns and fortifications, the skilful engineer knows the means he has in his power, and can calculate, with great precision, their effects. It is in this way that the art of war has been elevated into a science, and much is made to depend upon skill which, previous to the knowledge of these principles, depended entirely upon physical power.

The force with which balls are thrown by gunpowder is measured by an instrument called the Ballistic pendulum. It consists of a large block of wood, suspended by a rod in the manner of a pendulum. Into this block the balls are fired, and to it they communicate their own motion. Now, the weight of the block and that of the ball being known, and the motion or velocity of the block being determined by machinery or by observation, the elements are obtained by which the velocity of the ball may be found for the weight of the ball is to the weight of the block as the velocity of the block is to the velocity of the ball. By this simple apparatus many facts relative to the art of gunnery may be ascertained. If the ball be fired from the same gun, at different distances, it will be seen how much resistance the atmosphere opposes to its force at such distances. Rifles and guns of smooth bores may be tested, as well as the various charges of powder best adapted to different distances and different guns. These, and a great variety of other experiments, useful to the practical gunner or sportsman, may be made by this simple means.

The velocity of balls impelled by gunpowder from a musket with a common charge has been estimated at about 1650 feet in a second of time, when first discharged. The utmost velocity that can be given to a cannonball is 2000 feet per second, and this only at the moment of its leaving the gun.

In order to increase the velocity from 1650 to 2000 feet, one-half more powder is required; and even then, at a long shot, no advantage is gained, since, at the distance of 500 yards, the greatest velocity that can be obtained is only 1200 or 1300 feet per second. Great charges of powder are, therefore, not only useless, but dangerous; for, though they give little additional force to the ball, they hazard the lives of many by their liability to burst the gun.

Experiment has also shown that, although long guns give a greater velocity to the shot than short ones, still that, on the whole, short ones are preferable; and, accordingly, armed ships are now almost invariably furnished with short guns, called carronades.

The length of sporting guns has also been greatly reduced of late years. Formerly, the barrels were from four to six feet in length; but the best fowling-pieces of the present day have barrels of two feet or two and a half only in length. Guns of about this length are now universally employed for such game as woodcocks, partridges, grouse, and such birds as are taken on the wing, with the exceptions of ducks and wild geese, which require longer and heavier guns

 

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What forces affect a horizontal projectile, and what effect do they produce?

  1. A ball thrown in a horizontal direction is influenced by three forces; namely, first, the force of projection (which gives it a horizontal direction); second, the resistance of the air through which it passes, which diminishes its velocity, without changing its direction; and third, the force of gravity, which finally brings it to the ground.

How is the force of gravity affected by the force of projection?

 

  1. The force of gravity is neither increased nor diminished by the force of projection.*

* The action of gravity being always the same, the shape of the curve of every projectile depends on the velocity of its motion; but, whatever this velocity be, the moving body, if thrown horizontally from the same elevation, will reach the ground at the same instant. Thus, a ball from a cannon, with a charge sufficient to throw it half a mile, will reach the ground at the same instant of time that it would had the charge been sufficient to throw it one, two, or six miles, from the same elevation. The distance to which a ball will be projected will depend entirely on the force with which it is thrown, or on the’ velocity of its motion. If it moves slowly, the distance will be short; if more rapidly, the space passed over in the same time will be greater;. but in both cases the descent of the ball towards the earth, in the same time, will be the same number of feet, whether it moves fast or slow, or even whether it move forward at all, or not.

fig18 parkers compendiumExplain Fig. 18.

  1. Fig. 18 represents a cannon, loaded with a ball, and placed on the top of a tower, at such a height as to require just three seconds for another ball to descend perpendicularly. Now, suppose the cannon to be fired in a horizontal direction, and at the same instant the other ball to be dropped towards the ground. They will both reach the horizontal line at the base of the tower at the same instant. In this figure C a represents the perpendicular line of the filling ball. C b is the curvilinear path of the projected ball, 3 the horizontal line at the base of the tower. During the first second of time, the falling ball reaches 1, the next second 2, and at the end of the third second it strikes the ground.

 

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Meantime, that projected from the cannon moves forward with such velocity as to reach 4 at the same time that the falling ball reaches 1. But the projected ball falls downwards exactly as fast as the other, since it meets the line 1 4, which is parallel to the horizon, at the same instant. During the next second the ball from the cannon reaches 5, while the other falls to 2, both having an equal descent. During the third second the projected ball will have spent nearly its whole force, and therefore its downward motion will be greater, while the motion forward will be less than before.

What effect has the projectile force on gravity?

  1. Hence it appears that the horizontal motion does not interfere with the action of gravity, but that a projectile descends with the same rapidity while moving forward that it would if it were acted on by gravity alone. This is the necessary result of the action of two forces.

What is the Random of a projectile?

  1. The Random of a projectile is the horizontal distance from the place whence it is thrown to the place where it strikes.

At what angle does the greatest random take place?

  1. The greatest random takes place at an angle of 45 degrees; that is, when a gun is pointed at this angle with the horizon, the ball is thrown to the greatest distance.

What will be the effect if a ball be thrown at any angle above 45 degrees?

fig19 parkers compendiumFig. 19.

Let Fig. 19 represent a gun or a carronade, from which a ball is thrown at any angle is thrown at an angle of 45 degrees with the horizon. If the ball be thrown at any angle above 45 degrees, the random will be the same, as it would be at the same number of degrees below 45 degrees.*

* A knowledge of this fact, and calculations predicated on it, enables the engineer so to direct his guns as to reach the object of attack when within the range of shot.

 

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What is the Centre of Gravity of a body?

  1. CENTRE OF GRAVITY. – It has already been stated [see Nos. 109 & 110] that the Centre of Gravity of a body is the point around which all the parts balance each other. It is, in other words, the centre of the weight of a body.

What is the Centre of Magnitude?

  1. The Centre of Magnitude is the central point of the bulk of a body.

Where is the Centre of gravity of a body?

  1. When a body is of uniform density, the centre of gravity is in the same point with the centre of magnitude. But when one part of the body is composed of heavier materials than another part, the centre of gravity (being the centre of the weight of the body) no longer corresponds with the centre of magnitude.

Thus the centre of gravity of a cylinder plugged with lead is not in the same point as the centre of magnitude.

If a body be composed of different materials, not united in chemical combination, the centre of gravity will not correspond with the centre of magnitude, unless all the materials have the same specific gravity.

When will a body stand and when will it fall?

  1. When the centre of gravity of a body is supported, the body itself will be supported; but when the centre of gravity is unsupported, the body will fall.*

* The Boston School Apparatus contains a set of eight Illustrations for the purpose of giving a clear idea of the centre of gravity, and showing the difference between the centre of gravity and the centre of magnitude.

What is the Line of Direction?

  1. A line drawn from the centre of gravity, perpendicularly to the horizon, is called the Line of Direction.
  2. The line of direction is merely a line indicating the path which the centre of gravity would describe, if the body were permitted to fall freely.

 

 

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When will a body stand and when will it fall?

  1. When the line of direction falls within the base * of any body, the body will stand; but when that line falls outside of the base, the body will fall, or be overset.

fig 20 parkers compendiumFig. 20.

* The base of a body is its lowest side. The base of a body standing on wheels or legs is represented by lines drawn from the lowest part of one wheel or leg to the lowest part of the other wheel or leg.

Thus, in Figs. 20 and 21, D E represents the base of the wagon and of the table.

fig 21 parkers compendiumFig. 21

Explain Fig. 21.

  1. (1.) Fig. 21 represents a loaded wagon on the declivity of a hill. The line C F represents a horizontal line, D E the base of the wagon. If the wagon be loaded in such a manner that the centre of gravity be at B, the perpendicular B D will fall within the base, and the wagon will stand. But if the load be altered so that the centre of gravity be raised to A, the perpendicular A C will fall outside of the base, and the wagon will be overset. From this it follows that a wagon, or any carriage, will be most firmly supported when the line of direction of the centre of gravity falls exactly between the wheels; and that is the case on a level road. The centre of gravity in the human body is between the hips, and the base is the feet.
  2. So long as we stand uprightly, the line of direction falls within this base. When we lean on one side, the centre of gravity not being supported, we no longer stand firmly.

How does a rope-dancer perform his feats of agility?

  1. A rope-dancer performs all his feats of agility by dexterously supporting the centre of gravity. For this purpose, he carries a heavy pole in his hands, which he shifts from side to side as he alters his position, in order to throw the weight to the side which is deficient; and thus, in changing the situation of the centre of gravity, he keeps the line of direction within the base, and he will not fall.**

** The shepherds in the south of France afford an interesting instance of the application of the art of balancing to the common business of life. These men walk on stilts from three to four feet high, and their children, when quite young, are taught to practise the same art. By means of these odd additions to the length of the leg, their feet are kept out of the water, or the heated sand, and they are also enabled to see their sheep at a greater distance. They use these stilts with great skill and care, and run, jump and even dance on them, with great ease.

 

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  1. A spherical body will roll down a slope, because the centre of gravity is not supported.*

* A cylinder can be made to roll up a slope, by plugging one side of it with lead; the body being no longer of a uniform density, the centre of gravity is removed from the middle of the body to some point in the lead as that substance is much heavier than wood. Now, in order that the cylinder may roll down the plane, as it is here situated, the centre of gravity must rise, which is impossible; the centre of gravity must always descend in moving, and will descend by the nearest and readiest means, which will be by forcing the cylinder up the slope, until the centre of gravity is supported, and then it stops.

A body also in the shape of two cones united at their bases can be made to roll up an inclined plane formed by two bars with their lower ends inclined towards each other. This is illustrated by a simple contrivance in the “Boston School Set” and the fact illustrated is called “the mechanical paradox.”

  1. Bodies, consisting of but one kind of substance, as wood, stone or lead, and whose densities are consequently uniform, will stand more firmly than bodies composed of a variety of substances of different densities, because the centre of gravity in such cases more nearly corresponds with the centre of magnitude.
  2. When a body is composed of different materials, it will stand most firmly when the parts whose specific gravity is the greatest are placed nearest to the base.

When will a body stand most firmly?

  1. The broader the base and the nearer the centre of gravity to the ground, the more firmly a body will stand.
  2. For this reason, high carriages are more dangerous than low ones.
  3. A pyramid also, for the same reason, is the firmest of all structures, because it has a broad base, and but little elevation.

Fig. 22.

fig 22 parkers compendium

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  1. A cone has also the same stability; but, mathematically considered, a cone is a pyramid with an infinite number of sides.
  2. Bodies that have a narrow base are easily overset, because, if they are but slightly inclined, the line of direction will fall outside of the base, and consequently their centre of gravity will not be supported.

Why can a person carry two pails of water more easily than one?

  1. A person can carry two pails of water easily than one, because the pails balance each other, and the centre of gravity remains supported by the feet. But a single pail throws the centre of gravity on one side, and renders it more difficult to support the body.

Where is the centre of gravity of two bodies connected together?

  1. COMIMON CENTRE OF GRAVITY OF TWO BODIES. -When two bodies are connected, they are to be considered as forming but one body, and have but one centre of gravity. If the two bodies be of equal weight, the centre of gravity will be in the middle of the line which unites them. But, if one be heavier than the other, the centre of gravity will be as much nearer to the heavier one as the heavier exceeds the light one in weight.

fig23 parkers compendiumFig. 23

Explain. Figures 23, 24 and 25.

  1. Fig. 23 represents a bar with an equal weight fastened at each end; the centre of gravity is at A, the middle of the bar, and whatever supports this centre will support both the bodies and the pole.

fig24 parkers compendiumFig. 24.

  1. Fig. 24 represents a bar with an unequal weight at each end. The centre of gravity is at C, nearer to the larger body.

 

fig25 parkers compendium

Fig. 25.

  1. Fig. 25 represents a bar with unequal weights at each end, but the larger weight exceeds the less in such a degree that the centre of gravity is within the larger body at C.*

* There are no laws connected with the subject of Natural Science so grand and stupendous as the laws of attraction. Long before the sublime fiat, “Let there be light” was uttered, the Creator’s voice was heard amid the expanse of universal emptiness, calling matter into existence, and subjecting it to these laws. Obedient to the voice of its Creator, matter sprang from “primeval nothingness” and, in atomic embryos, prepared to cluster into social unions. Spread abroad in the unbounded fields of space, each particle felt that it was “not good to be alone”. Invested with the social power, it sought companionship. The attractive power, thus doubled by the anion, compelled the surrounding particles to join in close embrace, and thus were worlds created. Launched into regions of unbound space, the new-created worlds found that their union was but a part of a great social system of law and order. Their bounds were set. A central point controls the Universe, and in harmonious revolution around this central point for ages have they rolled. Nor can one lawless particle escape. The sleepless eye of Nature’s law, vicegerent of its God, securely binds them all.

“Could but one small, rebellious atom stray,
Nature itself would hasten to decay.”

With this sublime view of Creation, how can we escape the conclusion that the very existence of a law necessarily implies a Law-giver, and that Law-giver must be the Creator? Shall we not then say, with the Psalmist, “It is the FOOL who hath said in his heart that there is no God”?

Who, then, will not see and admire the beautiful language of Mr. Alison, while his heart burns with the rapture and gratitude which the sentiments are so well fitted to kindle:

“When, in the youth of Moses, the Lord appeared to him in Horeb,’ a voice was heard, saying,’ Draw nigh hither, and put off thy shoes from off thy feet, for the place where thou standest is holy ground.’ It is with such a reverential awe that every great or elevated mind will approach to the study of nature, and with such feelings of adoration and gratitude that he will receive the illumination that gradually opens upon his soul.”

“It is not the lifeless mass of matter, he will then feel, that he is examining; it is the mighty machine of Eternal Wisdom, the workmanship of Him’ in whom everything lives, and moves, and has its being.’ Under an aspect of this kind, it is impossible to pursue knowledge without mingling with it the most elevated sentiments of devotion — it is impossible to perceive the laws of nature without perceiving, at the same time, the presence and the providence of the Law-giver:-and thus it is that, in every age, the evidences of religion have advanced with the progress of true philosophy; and that SCIIENCE, IN ERECTING A MONUMSENT TO HERSELF, HAS, AT THE SAME, ERECTED AN ALTAR TO THE DEITY.”

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