such a manner as to stretch it, otherwise the cord would bend under the action of the forces. In the second place, the intensities of the forces must be equal, otherwise the greater force would prevail, and motion would ensue. Hence, in order that two forces applied at the extremities of a cord may be in equilibrium, the forces must be equal and directly opposed.

The measure of the tension of the cord, or the force by which any two of its adjacent particles are urged to separate, is the intensity of one of the equal forces, for it is evident that the middle point of the cord might be fixed and either force withdrawn, without diminishing or increasing the tension. When a cord is solicited in opposite directions by unequal forces directed along the cord, the tension will be measured by the intensity of the lesser force.

Let AB represent a cord solicited by two groups of forces applied at its two extrem

ities. In order that these forces may be in equilibrium, the resultant of the group applied at A and the resultant of

Fig. 60.

the group at B must be equal and directly opposed. Hence, if we suppose all of the forces at each point to be resolved into components respectively coinciding with, and at right angles to AB, the normal components at each of the points must be such as to maintain each other in equilibrium, and the resultants of the remaining components at each of the points A and B must be equal and directly opposed.

Let ABCD represent a cord, at the different points A, B, C, D, of which are applied groups of forces. If these forces are in equilibrium through the intervention of the cord, there must necessarily be an equilibrium at each point of ap

P

t

B

t"

P

A

Р

Fig. 61.

plication. Denote the tension of AB, BC, CD, by t, t', t",

and the forces applied by P, P', P", &c., as shown in the figure. The forces in equilibrium about the point A are P, P', P", and t, directed from A to B; the forces in equili brium about B are P'"', P, t, directed from B to A, and t', directed from B to C. The tension t is the same at all points of the branch AB, and, since it acts at A in the direction AB, and at B in the direction BA, it follows that these two forces exactly counterbalance each other. If, therefore, the forces P' and P" were transferred from A to B, unchanged in direction and intensity, the equilibrium at that point would be undisturbed. In like manner, it may be shown that, if all the forces now applied at B be transferred to C, without change of direction or intensity, the equilibrium at C would be undisturbed, and so on to the last point of the cord. Hence we conclude, that a system of forces applied in any manner at different points of a cord will be in equilibrium, when, if applied at a single point without change of intensity or direction, they will maintain each other in equilibrium.

Hence, we see that cords in machinery simply serve to transmit the action of forces, without in any other manner modifying their effects.

The Lever.

78. A lever is an inflexible bar, free to turn about an axis. This axis is called the fulcrum.

Levers are divided into three classes, according to the relative positions of the points of application of the power and resistance.

In the first class, the resistance is beyond both the power and fulcrum, and on the side of the fulcrum. The common weighing-scale is an example of this class of levers. The matter to be weighed is the resistance, the counterpoising weight is the power, and the axis of suspension is the fulcrum.

1ST CLASS.

F

R

Fig. 62.

In the second class, the resistance is between the power and the fulcrum. The oar used in rowing a boat is an example of this class of levers. The end of the oar in the water is the fulcrum, the point at which the oar is fastened to the boat is the point of application of the resistance, and the remaining end of the oar is the point of application of the power.

In the third class, the resistance is beyond both the fulcrum and the

2ND CLASS.

Fig. 63.

3RD CLASS.

power, and on the side of the power. The treadle of a lathe is an example of a lever of this kind. The point at which it is fastened to the floor is the fulcrum, the point at which the foot is applied is the point of application of the power, and the point where it is attached to the crank is the point of application of the resistance.

Fig. 64.

Levers may be either curved or straight, and the directions of the power and resistance may be either parallel or oblique to each other. We shall suppose the power and resistance to be situated in planes at right angles to the fulcrum; for, if they were not so situated, we might conceive each to be resolved into two components-one at right angles, and the other parallel to the axis. The latter component would be exerted to bend the lever laterally, or to make it slide along the axis, developing only hurtful resistance, whilst the former only would tend to turn the lever about the fulcrum.

The perpendicular distances from the fulcrum to the lines of direction of the power and resistance, are called the lever arms of these forces. In the bent lever MFN, the perpen

dicular distances FA and FB are, respectively, the lever arms of P and R.

To determine the conditions of equilibrium of the lever, let us denote the power by P, the resistance by R, and their respective lever arms by p and r. We have the case of a body restrained by an axis, and if we take this as

M

A

T

P

Fig. 65.

B

the axis of moments, we shall have for the condition of equilibrium (Art. 49),

Pp = Rr; or, P: R::r: p (36.)

.

That is, the power is to the resistance, as the lever arm of the resistance is to the lever arm of the power.

This relation holds good for every kind of lever.

The ratio of the power to the resistance when in equili brium, either statical or dynamical, is called the leverage, or mechanical advantage.

When the power is less than the resistance, there is said to be a gain of power, but a loss of velocity; that is, the space passed over by the power in performing any work, is as many times greater than that passed over by the resistance, as the resistance is greater than the power. When the power is greater than the resistance, there is said to be a loss of power, but a gain of velocity. When the power and resistance are equal, there is neither gain nor loss of power, but simply a change of direction.

In levers of the first class, there may be either a gain or a loss of power; in those of the second class, there is always a gain of power; in those of the third class, there is always a loss of power. A gain of power is always attended with a corresponding loss of velocity, and the reverse.

If several forces act upon a lever at different points, all being perpendicular to the direction of the fulcrum, they will be in equilibrium, when the algebraic sum of their moments, with respect to the fulcrum, is equal to 0.

This principle enables us to take into account the weight of the lever, which may be regarded as a vertical force applied at the centre of gravity.

The pressure on the fulcrum is equal to the resultant of the power and resistance, together with the weight of the lever, when that is considered, and it may be found by the rule for finding the resultant of forces applied at points of a rigid body.

The Compound Lever.

R"

79. A compound lever consists of a combination of simple levers AB, BC, CD, so arranged that the resis tance in one acts as a power in the next, throughout the combination. Thus, a power P produces at B a resis tance R', which, in turn, produces at Ca resistance R", and so on. Let us as

B

D

p'

A r

R"

R

Fig. 66.

sume the notation of the figure. From the principle of the simple lever, we shall have the relations,

Pp R'r', R'p' R'r', R'p" Rr.

=

=

=

Multiplying these equations together, member by member, and striking out the common factors, we have,

Ppp'p" Rrr'r''; or, P: R :: rr'r" : pp'p". (37.)

=

We might proceed in a similar manner, were there any number of levers in the combination.

Hence, in the compound lever, the power is to the resis tance as the continued product of the alternate arms of lever, commencing at the resistance, is to the continued product of the alternate arms of lever, commencing at the power.

By suitably adjusting the simple levers, any amount of mechanical advantage may be obtained.