Notes on the Theory of the Reversible Pendulum. (Part II.)

Extract

We may now turn our attention to the questions which arise specially in connection with a reversible pendulum, such as Kater’s. Of course the object in view is so to place two knife-edges on opposite sides of the C.G. that the period shall be the same from either, the two positions being selected according to the rule at the end of § 4. But the usual procedure will be to decide on the distance between the knife-edges at the start, say one metre; then to clamp them to the more or less uniform bar of the pendulum, using a standard distance-piece to secure the proper interval between them, and finally to adjust a sliding weight in such a position as to make the period the same for both. We have seen in § 2 how the period, or rather the length of the equivalent S.P., varies with the position of the sliding weight, but it will be convenient now to change the notation. Let I₁, I₂, ... denote the moments of inertia of the various masses making up the pendulum, each about its own C.G., M₁, M₂.. the masses; x₁, x₂... the co-ordinates of the centres of gravity, referred to one of the knife-edges as origin; let the co-ordinate of the other knife-edge be d, and let the letters I, M, x without suffixes refer to a moveable mass which is being adjusted. We shall at once see how much simpler it is to discuss the graph whose ordinate is the length of the S.P., than the one where the period is used, for the position which the moveable mass should occupy of course corresponds to the intersection of the two graphs relating respectively to the two knife-edges: and it is surely better to have to deal with the intersections of conics than of cubics.