and
L = Σ(P)·(ny − mz), M = Σ(P)·(lz − nx), N = Σ(P)·(mx − ly),
(13)
provided
| x = | Σ(Px) | , y = | Σ(Py) | , z = | Σ(Pz) | . |
| Σ(P) | Σ(P) | Σ(P) |
(14)
These are the same as if we had a single force Σ(P) acting at the point (x, y, z), which is the same for all directions (l, m, n). We can hence derive the theory of the centre of gravity, as in § 4. An exceptional case occurs when Σ(P) = 0.
If we imagine a rigid body to be acted on at given points by forces of given magnitudes in directions (not all parallel) which are fixed in space, then as the body is turned about the resultant wrench will assume different configurations in the body, and will in certain positions reduce to a single force. The investigation of such questions forms the subject of “Astatics,” which has been cultivated by Möbius, Minding, G. Darboux and others. As it has no physical bearing it is passed over here.
| Fig. 45. |
§ 9. Work.—The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e. it is equal to F·δs cos θ, where F is the force, δs the displacement, and θ is the angle between the directions of F and δs. In the language of vector analysis (q.v.) it is the “scalar product” of the vector representing the force and the displacement. In the same way, the work done by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action. This product is the same whatever point on the line of action be taken, since the lengthwise components of the displacements of any two points A, B on a line AB are equal, to the first order of small quantities. To see this, let A′, B′ be the displaced positions of A, B, and let φ be the infinitely small angle between AB and A′B′. Then if α, β be the orthogonal projections of A′, B′ on AB, we have