Hence, we conclude in all cases, that the moment of the resultant of two forces is equal to the algebraic sum of the moments of the forces taken separately.

If we regard the force as the resultant of two others, and one of these in turn, as the resultant of two others, and so on, the principle may be extended to any number of forces lying in the same plane, and applied at the same point. This principle may, in the general case, be expressed by the equation

Rr

(Pp)

(9.)

That is, the moment of the resultant of any number of forces, lying in the same plane, and applied at the same point, is equal to the algebraic sum of the moments of the forces taken separately.

This is called the principle of moments.

The moment of the resultant is called the resultant moment; the moments of the components are called component moments; and the plane passing through the resultant and centre of moments, is the plane of moments.

When a force tends to turn its point of application about the centre of moments, in the direction of the motion of the hands of a watch, its moment is considered positive; consequently, when it tends to produce rotation in a contrary direction, the moment must be negative. If the resultant moment is negative, the tendency of the system is to produce rotation in a negative direction about the centre of moments. If the resultant moment is 0, there is no tendency to produce rotation in the system. The resultant moment may become 0, either in consequence of the lever arm becoming 0, or in consequence of the resultant itself being equal to 0. In the former case, the centre of moments lies upon the direction of the resultant, and the numerical value of the sum of the moments of the forces which tend to produce rotation in one direction, is equal to that of those which tend to produce motion in a contrary direction. In the latter case, the system of forces is in equilibrium.

Moments, with respect to an Axis.

B

Z

P

P

Fig. 16.

37. To form an idea of the moment of a force with respect to a straight line, taken as an axis of moments. Let P represent any force, and let the axis of Z be assumed so as to coincide with the axis of moments. Draw the straight line AB perpendicular, both to the direction of the force and to the axis of moments; at the point A, in which this perpendicular intersects the direction of the force, let the force P be resolved into two components, P" and P', the first parallel to the axis of Z, and the second at right angles to it. The former will have no tendency to produce rotation, the latter will tend to produce rotation, which tendency will be measured by P' × AB; this product is the moment of the force P with respect to the axis of moments, and is evidently equal to the moment of the projection of the force upon a plane at right-angles to the axis, taken with respect to the point in which this axis pierces the plane as a centre of moments.

If there are any number of forces situated in any manner in space, it is clear from the preceding principles that their resultant moment, with respect to any straight line taken as an axis of moments, is equal to the algebraic sum of the component moments with respect to the same axis.

Principle of Virtual Moments.

p' Op

P

38. Let P'represent a force applied to the material point; let the point O be moved by an extraneous force to some position, C, very near to 0; project the path OC upon the direction of the force; the projection Op, or Op', is called the

Fig. 17.

virtual velocity of the force, and is taken positively when it falls upon the direction of the force, as Op, and nega

R

S

tively when it falls upon the prolongation of the force, as Op'. The product obtained by multiplying any force by its virtual velocity is called the virtual moment of the force. Assume the figure and notation of Article 36. Op, Oq, and Or are the virtual velocities of the forces P, Q, and R. Let us denote the virtual velocity of o any force by the symbol of variation, followed by a small letter of the same name as that which designates the force.

pa

Fig. 18.

We have from the figure, as in Article 36, the relations,

Multiplying both numbers of the first by cos q, and of the second by sin o, and adding the resultant equations, we have,

R cos = P (cos a cos + sin a sin q) +

But, from the right-angled triangles COp, COq, and C'Or,

Substituting these in the preceding equation, and reducing, we have,

Ror Pop + Qoq.

Hence, the virtual moment of the resultant of two forces, is equal to the algebraic sum of the virtual moments of the two forces taken separately.

If we regard the force as the resultant of two other forces, and one of these as the resultant of two others, and so on, the principle may be extended to any number of forces, applied at the same point. This principle may be expressed by the following equation:

Hence, the virtual moment of the resultant of any number of forces applied at the same point, is equal to the algebraic sum of the virtual moments of the forces taken separately.

This is called the principle of virtual moments. If the resultant is equal to 0, the system is in equilibrium, and the algebraic sum of the virtual moments is equal to 0; conversely, if the algebraic sum of the virtual moments of the forces is equal to 0, the resultant is also equal to 0, and the forces are in equilibrium.

This principle, and the preceding one, are much used in discussing the subject of machines.

Resultant of parallel Forces.

Let P and Q be two forces lying in the same plane, and applied at points invariably

connected, for example, at the points M and N of a solid body. Their lines of direction being prolonged, will meet at some point O; and if we suppose the points of application to be transferred to O, their resultant may be deter

M

Fig. 19.

mined by the parallelogram of forces. The direction of the resultant will pass through O. (Art. 28.) Whether the forces be transferred to O or not, the direction of the resultant will always pass through O, and this whatever may be the value of the included angle. Now, supposing the points of application to be at M and N, let the force Q be turned about N as an axis. As it approaches parallelism with P,

the point will recede from M and N, and the resultant will also approach parallelism with M. Finally, when becomes parallel to P, the point O will be at an infinite distance from M and N, and the resultant will also be parallel to P and Q. In any position of P and Q, the value of the resultant, denoted by R, will be given by the equation (Art. 36),

R = Pcosa + Qcosß.

M

Fig. 19.

When the forces are parallel, and lying in the same direction, we shall have a =

0, and B

=

0; or, cos a

1, and

R = P+Q.

If the forces lie in opposite directions, we shall have

If we regard as the resultant of two parallel forces, and one of these as the resultant of two others, and so on, the, principle may be extended to any number of parallel forces. Denoting the resultant of a group of parallel forces, P, P', P", &c., by R, we have,

That is, the resultants of a group of parallel forces is equal in intensity to the algebraic sum of the forces. Its line of direction is also parallel to that of the given forces.