Page:Elementary Text-book of Physics (Anthony, 1897).djvu/59

§ 44] couple may be replaced by any other couple if the product $$F \,. \, PQ$$ is the same for both. The couple will have the same moment whatever point be chosen as the fixed point. Since the two forces which constitute the couple are equal and opposite, their resultant is zero, and therefore no single force can be found which will equilibrate a couple.

43. Movement of a Free Body.— Whatever forces act on a free body, they may always be reduced to a single force applied at the centre of mass and to a single couple which will produce rotation, about the centre of mass. Let $$F$$ (Fig. 15) be any one of these forces. Apply the equal and opposite forces $$F$$ and $$-F$$ at the centre of mass $$O$$. These two forces, having no resultant, will not affect the motion of the body. The force $$F$$ applied at the centre of mass will determine the effect of the original force $$F$$ in producing translation of the centre of mass. The original force $$F$$ and the force $$-F$$ constitute a couple, the arm of which is $$OP$$ and which produces rotation about the centre of mass. Treating all other forces applied to the body in a similar way, we have finally an assemblage of forces applied at the centre of mass, the resultant of which determines the acceleration of the centre of mass or the translation of the body, and a set of couples, equivalent to a single couple of which the moment is equal to the sum of their moments, which produces rotation about the centre of mass.

A free body is in equilibrium, or undergoes no change in the velocity of its centre of mass or in its rotation, when the resultant $$P$$ of the forces applied to it vanishes, and the moment of couple to which the moments of these forces about the centre of mass may be reduced also vanishes, that is, the body is in equilibrium when $$R = 0$$ and $$\sum F \,. \, OP = 0$$.

44. Centre of Percussion.— We will illustrate the foregoing principles by considering the motion of a free body to which a force is applied for a very short time, during which it may be considered constant. The force is supposed to act in the plane containing the centre of mass of the body. Then, as has just been shown, the