28. COR. Conversely, if P and Q be inversely as the perpendiculars from the fulcrum upon their lines of action, they will balance each other.

For suppose to be the force which applied at B in BN balances P at A; then, by what has been proved,

PQ CN: CM.

But, by supposition, P: Q :: CN : CM;

. P Q

P : Q, or Q'=Q;

but Q'balances P, :. Q also balances P.

In this Proposition the arms of the Lever may be either straight or crooked, since nothing in the proof is made to depend upon the particular form of CA, or CB. The rigidity, however, of the arms, whatever be their form, is a necessary condition.

It may also be noticed here, that every possible case of two forces balancing on a Lever has now been discussed. In Prop. I. the Lever is straight, and the forces act perpendicularly to the arms on opposite sides of the fulcrum. Prop. III. is the same as Prop. II., except that the forces act on the same side of the fulcrum. In Prop. v. the forces still act perpendicularly to the arms, but the Lever is bent. In Prop. vI. the forces act at any angles to the arms, and the Lever is either straight or bent. But in every case

PxCM=QxCN;

from which equation, any three of the quantities being given, the fourth may be found.

29. PROP. VII. If two weights balance each other on a straight Lever when it is horizontal, they will balance each other in every position of the Lever.

Let P and Q be two weights, which balance each other round. the fulcrum C on the straight Lever ACB, when it is horizontal. They will balance each A other on the Lever, when it is made to take any other position, as A'CB'.

Produce QB' to cut AB in N, and A'P, if necessary, to cut AC in M.

M

NB

B'

Q

Since weights act perpendicularly to the horizon, A'P and QB'N are both perpendicular to the horizontal line ACB;

.. angle CMA'= right angle = angle CNB',

and angle A'CM= opposite angle B'CN, Euc. 1. 15. .. also angle CA'M= angle CB'N, and the triangles CA'M, CB'N, are equiangular, and therefore similar. Hence, by EUC. VI. 4,

CN: CB':: CM: CA';

alternando, CN: CM:: CB': CA',

:: CB CA.

But, since Pand Q balance on ACB, CB: CA :: P: Q; .. CN CM :: P: Q.

But CM and CN are the perpendiculars from C on the lines in which P and Q act, when they are hung at A' and B'; therefore, by Prop. VI., Cor., P and Q will balance on A'CB'; and since A'CB' is the Lever in any position, the above proof applies to every position of the Lever.

COR. 1. Hence, if two weights do not balance each other on a straight Lever, when the Lever is horizontal, they cannot balance each other in an inclined position of the Lever.

For if they did balance in an inclined position, it would follow from Prop. VI., that

PQ

CN : CM,

.. PQ

CB': CA', by what has been proved,

:: CB : CA;

and.. P and Q balance in the horizontal position of the lever, which is contrary to the supposition.

COR. 2. Hence, also, if two weights, acting freely, balance each other on a straight Lever in any one position of the Lever, except the vertical, they will balance in every other position of the Lever.

For the ratio CN: CM is independent of the angle at which the Lever is inclined; therefore if it once satisfies the conditions of equilibrium, it will do so always.

QUESTIONS ON CHAP. II.

(1) What is a Lever? Is there any such Lever in practice as that which is assumed in this chapter?

(2) What is the fulcrum, and what are the arms, of a Lever? Must the arms necessarily lie on opposite sides of the fulcrum?

(3) In Axiom II., if the Lever itself be supposed to have weight, how will the result be affected?

(4) In Axiom III., if the weight be placed exactly half-way between the fulcrums, what is the pressure on each?

(5) In Prop. I., if the prism, or cylinder, were not horizontal, how would the proof be affected?

(6) In Prop. I., if the prism, or cylinder, were not of uniform density, how would the proof be affected?

(7) What is the meaning of "effect" in the enunciation of Prop. I.

(8) In Prop. II., would P and Q balance, if they were to exchange places? Are there any other points between M and N at which they would balance?

(9) In Prop. II., if Q were doubled, where must P act to maintain the equilibrium?

(10) If 2 cwt., acting at a distance from the fulcrum of 1 foot, is balanced on a horizontal straight Lever by a power of 28 lbs. acting perpendicularly, what is the length of arm at which the power acts?

(11) In Prop. III., where P and Q balance each other, acting on the same side of the fulcrum, would the equilibrium be disturbed, if P were doubled, and CM halved? Also would the pressure on the fulcrum remain the same?

(12) There are three classes of Levers; what is it which distinguishes one class from another?

(13) Is power "lost" or "gained" in the use of fire-tongs?

(14) Where would you place the nut in a pair of nut-crackers to produce the greatest effect; and why?

(15) How does the contrivance of placing the row-locks outside the boat affect the efforts of the rower?

(16) In Prop. VI. is it necessary that the angles at which the forces act should be equal to one another?

If the forces once balanced acting at equal angles, would the same forces balance on the same Lever acting at any other equal angles?

CHAPTER III.

COMPOSITION AND RESOLUTION OF FORCES.

30. Definition of COMPONENT and RESULTANT FORCES.

It is found, by experiment, that a body which is acted on by two forces applied, at the same instant and in different lines, to the same point of it, instead of moving, or hav ing a tendency to move, in either of the lines in which the' forces act, moves, or has a tendency to move, in a line lying between them. Whence it appears, that the two original forces by their combined action produce the effect of a' single third force, which third force is called, from the circumstance of its resulting from the actions of the original forces, their "Resultant" with respect to them; while they are called, with respect to it, its "Components".

The Resultant (R), which produces the same effect as the compound action of the original forces P and Q applied at the same point at the same instant, is said to be "compounded" of P and Q. This Resultant (R) also, if conceived to be the sole original force, may be supposed to be "resolved" into the two forces P and Q ; since those two forces, acting in the manner described (namely, at the same point, and at the same instant), produce exactly the same effect on the body as the single force R does.

Similarly, if there be more than two forces, acting at the same point, and at the same instant, the resulting action is found to be such as can be produced by a certain single force, which latter force is therefore called the Resultant of all the other forces, whilst those other forces are called the Components of such Resultant.

31. PROP. VIII. If the adjacent sides of a parallelogram represent the component forces in direction and magnitude, the diagonal will represent the resultant force in direction and magnitude.

Let AB and AC represent, in direction and magnitude, the two component forces which act at A. Complete the parallelogram ACDB, and draw the diagonal AD. Then AD will represent the Resultant of AB and AC, (1) in direction, and (2) in magnitude.

M

B

From D draw DM and DN perpendiculars to AB and AC, produced if necessary.

(1) Then in the triangles DBM, DCN,

LDMB a right angle = DNC,

=

and DBM= <BAC, (since BD, AC are parallel, and MBA cuts them), = ≤DCN;

.. the third angle, BDM, of the one triangle = the third angle, CDN, of the other; and the triangles are equiangular and similar;

hence, CD: DN :: BD : DM,

and alternately, CD: BD :: DN : DM.

Now, if there be a Lever AD whose fulcrum is D, which is acted on by the forces AB, AC, applied at A, since

Force in the line AM: force in the line AN

the two forces acting on the Lever AD are inversely as the perpendiculars from the fulcrum on their lines of action, and therefore the Lever will be kept at rest about D by them (Art. 28). Wherefore the Lever will also be kept at rest by the Resultant of those forces; because that single force produces the same effect as they do, when they act at the same point and at the same instant.