Let X X', fig. 9, represent the axis of rotation; G, the centre of gravity of the rotating body or system, situated in that axis; so that the resultant centrifugal force is nothing.

Let W be any one of the parts of which the body or system is composed, so that, the weight of that part being denoted by W, the weight of the whole body or system may be denoted by W.

Let denote the perpendicular distance of the centre of W from the axis; then

Wa2 r
g

is the centrifugal force of W, pulling the axis in the direction x W.

W

y

Fig. 9.

Assume a pair of axes of co-ordinates, G Z, G Y, perpendicular to X X' and to each other, and fixed relatively to the rotating body or system—that is, rotating along with it.

From W let fall Wy perpendicular to the plane of G X and GY, and parallel to G Z; also Wz, perpendicular to the plane of GX and GZ, and parallel to GY; and make

xy=Wz

= y; xz = Wy=z; Gx = x. Then the centrifugal force which W exerts on the axis, and which is proportional to r, may be resolved into two components, in the direction of, and proportional to, y and z respectively, viz. :

Wa2 y g

Wa3 z

parallel to G Y, and

parallel to G Z;

g

and those two component forces, being both applied at the end of the lever G x=x, exert moments, or tendencies to turn the axis X X' about the point G, expressed as follows:

tending to turn G X about G Z towards G Y;

tending to turn G X about G Y towards G Z.

In the same manner are to be found the several moments of the

centrifugal forces of all the other parts of which the body or system consists; and care is to be taken to distinguish moments which tend to turn the axis towards G Y or G Z from those which tend to turn it from those positions, by treating one of these classes of quantities as positive, and the other as negative.

Then by adding together the positive moments and subtracting the negative moments for all the parts of the body or system, are to be found the two sums,

which represent the total tendencies of all the centrifugal forces to turn the axis in the planes of G Y and G Z respectively.

In fig. 10, lay down G Y to represent the former moment, and G Z, perpendicular to G Y, to represent the latter. Then the diagonal G M of the rectangle G Z MY will represent the resultant moment of what is called the CENTRIFUGAL COUPLE, and the direction of that line will indicate the direction in which that couple tends to turn the axis GX about the point G. Its value, and its angular position, are given by the equations,

=

√ ( G Y2 + G Z2);

GM = }

tan

YG M = GZ÷G Y

.(4.).

The condition which it is desirable to fulfil in all rapidly rotating pieces of machines, that the axis of rotation shall be a permanent axis, is fulfilled when each of the sums in the formula 3 is nothing; that is, when

The question, whether the axis of a rotating piece is a permanent axis or not, is tested experimentally by making the piece spin round rapidly with its shaft resting in bearings which are suspended by chains or cords, so as to be at liberty to swing to and fro. If the axis is not a permanent axis, it oscillates; if it is a permanent axis, it remains steady.

The practical application of those principles to locomotive engines will be explained in the sequel.

SECTION 3.-Of Effort, Energy, Power, and Efficiency.

23. Effort is a name applied to a force which acts on a body in the direction of its motion (A. M., 511).

If a force is applied to a body in a direction making an acute

A.

P

B

angle with the direction of the body's motion, the component of that oblique force along the direction of the body's motion is an effort. That is to say, in fig. 11, let A B represent the direction in which A is moving; let AF represent a force applied to A, obliquely to that direction; from F draw FP perpendicular to A B; then AP is the effort due to the force A F. The transverse component PF is a lateral force, like the transverse component of the oblique resisting force in Article 8.

Fig. li.

To express this algebraically, let the entire force A F = F, the effort A P=P, the lateral force P F = Q, and the angle of obliquity PAF. Then

24. Condition of Uniform Speed. (A. M., 510, 512, 537.)—According to the first law of motion, in order that a body may move uniformly, the forces applied to it, if any, must balance each other; and the same principle holds for a machine consisting of any number of bodies.

When the direction of a body's motion varies, but not the velocity, the lateral force required to produce the change of direction depends on the principles set forth in Section 2; but the condition of balance still holds for the forces which act along the direction of the body's motion, that is, for the efforts and resistances; so that, whether for a single body or for a machine, the condition of uniform velocity is, that the efforts shall balance the resistances.

In a machine, this condition must be fulfilled for each of the single moving pieces of which it consists.

It can be shown from the principles of statics (that is, the science of balanced forces), that in any boc, system, or machine, that condition is fulfilled when the sum of the products of the efforts into the velocities of their respective points of action is equal to the sum of the products of the resistances into the velocities of the points where they

are overcome.

Thus, let v be the velocity of a driving point, or point where an effort P is applied; v' the velocity of a working point, or point where a resistance R is overcome; the condition of uniform velocity for any body, system, or machine is

If there be only one driving point, or if the velocities of all the

driving points be alike, then P being the total effort, the single product P v may be put in in place of the sum · P v; reducing the above equation to

Referring now to Article 9, let the machine be one in which the comparative or proportionate velocities of all the points at a given instant are known independently of their absolute velocities, from the construction of the machine; so that, for example, the velocity of the point where the resistance R is overcome bears to that of the driving point the ratio

then the condition of uniform speed may be thus expressed :—

Р

= Σ n R...........

...(3.) that is, the total effort is equal to the sum of the resistances reduced to the driving point.

25. Energy—Potential Energy. (A. M., 514, 517, 593, 660.)— Energy means capacity for performing work, and is expressed, like work, by the product of a force into a space.

The energy of an effort, sometimes called "potential energy" (to distinguish it from another form of energy to be afterwards referred to), is the product of the effort into the distance through which it is capable of acting. Thus, if a weight of 100 pounds be placed at an elevation of 20 feet above the ground, or above the lowest plane to which the circumstances of the case admit of its descending, that weight is said to possess potential energy to the amount of 100 × 20 2,000 foot-pounds; which means, that in descending from its actual elevation to the lowest point of its course, the weight is capable of performing work to that amount.

=

To take another example, let there be a reservoir containing 10,000,000 gallons of water, in such a position that the centre of gravity of the mass of water in the reservoir is 100 feet above the lowest point to which it can be made to descend while overcoming resistance. Then as a gallon of water weighs 10 lbs., the weight of the store of water is 100,000,000 lbs., which being multiplied by the height through which that weight is capable of acting, 100 feet, gives 10,000,000,000 foot-pounds for the potential energy of the weight of the store of water.

26. Equality of Energy Exerted and Work Performed.—When an effort actually does drive its point of application through a certain distance, energy to the amount of the product of the effort into that distance is said to be exerted; and the potential energy,

or energy which remains capable of being exerted, is to that amount diminished.

When the energy is exerted in driving a machine at an uniform speed, it is equal to the work performed.

To express this algebraically, let t denote the time during which the energy is exerted, v the velocity of a driving point at which an effort P is applied, s the distance through which it is driven, v' the velocity of any working point at which a resistance R is overcome, s the distance through which it is driven; then

8 = = vt; s=v't;

and multiplying equation 1 of Article 24 by the time t, we obtain the following equation:

2. Pvt Rv't · P s = 2 · R s';......... (1.)

=

which expresses the equality of energy exerted, and work performed, for constant efforts and resistances.

When the efforts and resistances vary, it is sufficient to refer to Article 11, to show that the same principle is expressed as follows:

where the symbol expresses the operation of finding the work performed against a varying resistance, or the energy exerted by a varying effort, as the case may be; and the symbol Σ expresses the operation of adding together the quantities of energy exerted, or work performed, as the case may be, at different points of the machine.

27. Various Factors of Energy.-A quantity of energy, like a quantity of work, may be computed by multiplying either a force into a distance, or a statical moment into an angular motion, or the intensity of a pressure into a volume. These processes have already been explained in detail in Articles 5 and 6.

28. The Energy Exerted in Producing Acceleration (A. M., 549) is equal to the work of acceleration, whose amount has been investigated in Articles 14 A and 15.

29. The Accelerating Effort (A. M., 554) by which a given increase of velocity in a given mass is produced, and which is exerted by the driving body against the driven body, is equal and opposite to the resistance due to acceleration which the driven body exerts against the driving body, and whose amount has been given in Articles 14 A and 15. Referring, therefore, to equations 4 and 8 of Article 14 A, we find the two following expressions, the first of which gives the accelerating effort required to produce a given

D