Page:Elementary Text-book of Physics (Anthony, 1897).djvu/58

44 of the force, is called the moment of the force about that axis. Since, if several forces act on the body, they each contribute their share to the angular acceleration produced, we have also $$\sum F \,. \, OP = I\alpha.$$.

40. Principle of Moments.— If the angular velocity of the body be constant, we have $$\alpha = 0$$, and hence $$\sum F \,. \, OP = 0$$. The body is then said to be in equilibrium about the fixed axis. Hence a body free to rotate about a fixed axis is in equilibrium if the sum of the moments of the forces which tend to turn it in one sense is equal to the sum of the moments of the forces which tend to turn it in the opposite sense. This theorem is called the principle of moments.

41. Principle of Work.— If the body rotate uniformly about a fixed axis, it does not gain angular velocity, and we have $$\omega = 0$$, and therefore $$\sum F \,. \, OP \,. \, \phi = 0.$$ Now $$OP \,. \, \phi$$ is the distance traversed by the point of application of the force, and this is proportional to the velocity $$v$$ with which that point of application moves. Therefore $$\sum Fs = 0$$, or $$\sum Fv = 0$$. The body is in equilibrium about a fixed axis when the positive work done upon it by some of the forces applied to it during any small displacement is equal to the negative work done by the other forces upon it. The expression $$Fv = \frac{Fs}{t}$$ measures the rate at which work is done by the force, and the condition of equilibrium may be otherwise stated by saying that the rotating body is in equilibrium when the rate at which positive work is done upon the body equals the rate at which negative work is done upon it.

42. Couples.— A combination of two forces which are equal and oppositely directed, but not in the same straight line, is called a couple. The sum of their moments (Fig. 14) is $$F \,. \, OQ - F \,. \, OP = F \,. \, PQ$$, and is manifestly the same wherever the forces are applied in the body, provided the distance $$PQ$$ remains the same. $$PQ$$ is called the arm of the couple. Since the effect of different forces in producing rotation is the same if the sum of their moment's is the same, it is also clear that the