CHAPTER VII.

COMPOUND FORCES.

THE

COMPOSITION AND RESOLUTION

OF

MOTION. -ROTATORY MOTION. -THE REVOLVING WATCHGLASS. THE SLING. THE CENTRIFUGAL AND CENTRIPETAL FORCES.-THEORY OF PROJECTILES.-THE CENTRIFUGAL RAILWAY. A GEOLOGICAL CONVERSATION BETWEEN MR. SEYMOUR AND THE VICAR, IN WHICH THE LATTER DISPLAYS HIS POWERS OF RIDICULE.

THE party having assembled around the table, Mr. Seymour commenced his lecture by reminding his young auditors, that the motion of a body actuated by a single force was always in a right line, in the direction in which it receives the impulse.

"Do you mean to say, papa, that a single force can never make a body move round, or in a crooked direction; if so, how does it happen that my marble or ball will frequently run along the ground in a curved direction; indeed, I always find it very difficult to make it go straight."

"Depend upon it, my dear, whenever the direction of a moving body deviates from a straight line, it has been influenced by some second force."

"Then I suppose that, whenever my marble runs in a curved line, there must be some second force to make it do so."

"Undoubtedly; the inequality of the ground may give it a new direction: which, when combined with the original force which it received from your hand, will fully explain the irregularity of its course. It is to the consideration of such compound motion that I am now desirous of directing your attention: the subject is termed the 'COMPOSITION OF FORCES.' Here is a block of wood, with two strings, as you may perceive, affixed to it: do you take hold of one of these strings, Louisa; and you, Tom, of the other. That is right. Now place the block at one of the corners or angles of the table and while Tom draws it along one of its sides, do you, Louisa, at the same time, draw it along the other.' The children obeyed their father's direction.

"See!" said Mr. Seymour; "the block obeys neither of

the strings, but picks out for itself a path which is intermediate. Can you tell me, Tom, the exact direction which it takes?"

Fig. 25.

"If we consider this table as a parallelogram, I should that the block described the diagonal."

say,

Well said, my boy; the ablest mathematician could not have given a more correct answer. The block was actuated by two forces at the same time; and, since it could not move in two directions at once, it moved under the compound force, in a mean or diagonal direction, proportioned to the influence of the joint forces acting upon it. You will, therefore, be pleased to remember, it is a general law, that where a body is actuated by two forces at the same time, whose directions are inclined to each other, at any angle whatever, it will not obey either of them, but move along

Fig. 26.

Y

B

X

the diagonal. In determining, therefore, the course which a body will describe under the influence of two such forces, we have nothing more to do than to draw lines which show the direction and quantity of the two forces, and then to complete the parallelogram by parallel lines, and its diagonal will be the path of the body. I have here a diagram which may render the subject more intelligible. Suppose the ball B were, at the same moment, struck by two forces x and y in the directions B A and B D. It is evident

that the ball would not obey either of such forces, but would move along the oblique or diagonal lines B c."

"But," said Tom, "why have you drawn the line B D So much longer than BA?"

"I am glad you have asked that question. Lines are intended, not only to represent the direction, but the momenta or quantities of the forces: the line в D, is, as you observe, twice as long as B A; it consequently denotes that the force y acting in the direction B D, is twice as great as the force x acting in the direction B A. Having learned the direction which the body will take when influenced by joint forces of this kind, can you tell me the relative time which it would require for the performance of its diagonal journey?"

Tom hesitated; and Mr. Seymour relieved his embarrassment by informing him, that it would pass along the diagonal in exactly the same space of time that it would have required to traverse either of the sides of the parallelogram, had but one force been applied. Thus, the ball в would reach c in the same time that the force x would have sent it to A, or the force Y to D. “I will endeavour to prove this fact beyond all doubt. It is, I think, evident, that the force which acts in the direction B A can neither accelerate nor retard the approach of the body to the line D C, which is parallel to it; hence it will arrive at c in the same time that it would have done had no motion been communicated to it in the direction B A. In like manner, the motion in the direction B D can neither make the body approach to nor recede from A c; and it therefore follows, that, in consequence of the two motions, the body will be found both in A c and c D, and will therefore be found in c, the point of intersection."

Louisa seemed to express by her looks the irksomeness of such demonstrations; and which did not pass unobserved.

"This may appear tedious and uninteresting," said Mr. Seymour, "but the information is absolutely essential to our future progress; if you would reap, you must sow."

Tom and Louisa both expressed themselves willing to receive whatever instruction their father might consider necessary; and they farther declared, that they understood the demonstration he had just offered them.

"Is it not then evident," proceeded Mr. Seymour, "that the composition of forces must always be attended with loss of power; since the diagonal of a parallelogram can never,

Fig. 27.

under any circumstances, be equal to two of its sides? and is it not also evident, that the length of the diagonal must diminish as the angles of the sides increase: so that the more acute the angle at which the forces act, the less must be the loss by composition? But I shall be better able to explain this law by a diagram. If B A, A c be the sides of a parallelogram representing the direction of two forces, and A D the diagonal path of the body, is it not evident that the line A D will shorten as the angle B A C increases?" "We see that at once," cried Tom, "from the diagram before us."

A

B

Fig. 28.

"Then we will proceed to another fact connected with the same subject. Look at this diagram; is not the diagonal A D common to both the parallelograms inscribed about it, viz. of A B C D, and A E F D?" "To be sure it is."

B

E

"Then it is equally clear, that a body may be made to traverse the same path A D, by any pair of forces represented by the adjacent sides, of either

of such parallelograms." "Undoubtedly."

"I request you to keep that fact in your recollection." "I have now to inform you," continued he, "that a single force may be resolved into any number of forces, and may, in fact, be regarded as compounded of innumerable oblique ones. In order, however, to render this fact more intelligible, I must refer you to the same figure, from which it will appear that the motion of a body, along the line A D, will be the same whether it arise from one single force acting in that direction, or from two forces impressed upon it in the directions A B, A C, or in those of AE, A F; and, consequently, although the motion may, in reality, be the effect of a single force, yet it may be considered as compounded of two or more in other directions, since the very same motion would arise from such a composition."

Tom acknowledged the truth of this statement; and Mr. Seymour assured him, that, when they came to play at ball and marbles, he should be able to give him a practical demonstration of the fact; for he would show him, that when

ever a body strikes a surface obliquely, or in an inclined direction, such a resolution of force will actually take place: "and now, Tom," said his father, “give me a marble; for I wish to explain the reason why it turns round, or revolves on its axis, as it proceeds forward."

"I suppose, ," said Tom, "it depends upon the action which I give to it by my thumb and finger when I shoot it out of my hand."

"You are, undoubtedly, capable of thus giving to your marble a certain spinning motion, the effect of which we shall have to consider hereafter; but I fancy you would be greatly puzzled to make it proceed without revolving, give it what impulse you might by your hand.”

"I have sometimes tried," said Tom, "to make it do so by pushing it along with a flat ruler, but it always rolled in spite of me."

Then it is clear, from your own experiment, that its rotation cannot arise from the

father, I will endeavour to
explain the operation.
is evident that, as the marble
moves along the ground B D,
the motion of the point B
will be retarded by the re-
sistance occasioned by its

rubbing on the ground; while the point c, which does not meet with any such resistance, is carried forward without opposition, and it consequently must move faster than the point B; but since all the parts of the marble cohere or stick together, the point c cannot move faster than B, unless the marble revolves from c to E; and as the several points of the marble which are successively applied to the floor are retarded in their motion, while the opposite points move freely, the marble during its progressive motion must continue to revolve."

"But you said, papa, that whenever a body moved in any direction, except that of a straight line, it must have been acted upon by more than one force; and yet the marble not only runs along the ground, but turns round, at the same time, by the simple force of my hand."

"The revolution of the marble, my dear boy, is brought