§ 17. Two-dimensional Problems.—In the case of rotation about a fixed axis, the principles take a very simple form. The position of the body is specified by a single co-ordinate, viz. the angle θ through which some plane passing through the axis and fixed in the body has turned from a standard position in space. Then dθ/dt, = ω say, is the angular velocity of the body. The angular momentum of a particle m at a distance r from the axis is mωr·r, and the total angular momentum is Σ(mr2)·ω, or Iω, if I denote the moment of inertia (§ 11) about the axis. Hence if N be the moment of the extraneous forces about the axis, we have
| d | (Iω) = N. |
| dt |
(1)
This may be compared with the equation of rectilinear motion of a particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation. If N = 0, ω is constant.
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| Fig. 72. | Fig. 73. |
As a first example, suppose we have a flywheel free to rotate about a horizontal axis, and that a weight m hangs by a vertical string from the circumferences of an axle of radius b (fig. 72). Neglecting frictional resistance we have, if R be the tension of the string,
Iω̇ = Rb, mu̇ = mg − R,
whence
| bω̇ = | mb2 | g. |
| 1 + mb2 |