| Wh tan θ = N = (c2 − c1) | c1 | u2 | tan θ, | |
| c2 | g |
(6)
| h = | c2 − c1 | c1 | u2 | = (β − α) | 1 + α | u2 | . | |||
| W | c2 | g | 1 + β | g |
(7)
51. An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the investigation proceeds in the same way for the two problems (see [Gyroscope]).
The effective angular inertia of the body in the medium is now required; denote it by C1 about the axis of the figure, and by C2 about a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in motion, so that C1 is the moment of inertia of the body about the axis, denoted by Wk12. But if Wk22 is the moment of inertia of the body about a mean diameter, and ω the angular velocity about it generated by an impulse couple M, and M′ is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k′,
Wk22ω = M − M′, W′k′2ω = M′,
(1)
(Wk22 + W′k′2) ω = M,
(2)