Fig. 35.
Next we will knock a tintack into any point B, and tie a string on to B. Then if I pull at the string in any direction B C the board tends to twist round the screw at A. What will the strength of the twisting force be? It will depend on the strength of the pull, and on the “leverage,” or distance of the line C B from A. We might imagine the string, instead of being attached at B, to be attached at D; then, if I put P as the strength of the pull, the twisting power would be represented by P × A D. This is called the “moment” of the force P round the centre A. It would be the same as if I had simply an arm A D, and pulled upon it with the force P. It is an experimental truth, known to the old Greek philosophers, that moments, or twisting powers, are equal when in each case the result of multiplying the arm by the power acting at right angles to it is equal.
Now suppose A B is a pendulum, with a bob B of 10 lbs. weight, and suppose it has been drawn aside out of the vertical so that the bob is in the position B. Then the weight of the bob will act vertically downwards along the line B C. The moment, or twisting power, of the weight will be equal to 10 lbs. multiplied by A D, A D being a line perpendicular to B C.
Fig. 36.
Now suppose that another string were tied to the bob B, and pulled in a direction at right angles to A B, with a force P just enough to hold the bob back in the position B. The pull along D B × A B would be the moment of that pull round the point A. But, because this moment just holds the pendulum up, it follows that the moment of the weight of the pendulum round A is equal to the moment of the pull of the string B D round A.
Whence P × A B = 10 lbs. × A D.
Whence P = 10 lbs. × (A D)/(A B).
But A B is always the same, whatever the side deflection or displacement of the pendulum may be. Whence then we see that when a pendulum is pulled aside a distance E B (which is always equal to A D), then the force tending to bring it back to E is always proportional to E B. But if the pendulum be fairly long, say 39-1/7 inches, and the displacement E B be small,—in other words, if we do not drag it much out of the vertical,—then we may say that the force tending to bring it back to F, its position of rest, is not very different from the force tending to bring it back to E. But F B is the “displacement” of the pendulum, and, therefore, we find that when a pendulum is displaced, or deflected, or pulled aside a little, the amount of the deflection is always very nearly proportional to the force which was used to produce the deflection. This important law can be verified by experiment. If C is a small pulley, and B C a string attached to a pendulum A B whose bob is B. Then if a weight D be tied to the string and passed over a pulley C, the amount F B by which the weight D will deflect the bob B is almost exactly proportional to D, so long as we only make the deflection E B small, that is two or three inches, where say 39-1/7 inches is the length A B of the pendulum.