equivalent to an incessant couple
N = FV sin ([theta] - [phi])
= (F sin [theta] cos [phi] - F cos [theta] sin [phi])V
= (c2 - c1)(V²/g) sin [phi] cos [phi]
= W´([beta] - [alpha]uv/g). (5)
This N is the couple in foot-pounds changing the momentum of the
medium, the momentum of the body alone remaining the same; the medium
reacts on the body with the same couple N in the opposite direction,
tending when c2-c1 is positive to set the body broadside to the
advance.
An oblate flattened body, like a disk or plate, has c2 - c1 negative,
so that the medium steers the body axially; this may be verified by a
plate dropped in water, and a leaf or disk or rocket-stick or piece of
paper falling in air. A card will show the influence of the couple N
if projected with a spin in its plane, when it will be found to change
its aspect in the air.
An elongated body like a ship has c2-c1 positive, and the couple N
tends to disturb the axial movement and makes it unstable, so that a
steamer requires to be steered by constant attention at the helm.
Consider a submarine boat or airship moving freely with the direction
of the resultant momentum horizontal, and the axis at a slight
inclination [theta]. With no reserve of buoyancy W = W´, and the
couple N, tending to increase [theta], has the effect of diminishing
the metacentric height by h ft. vertical, where
c1 u²
Wh tan[theta] = N = (c2 - c1) -- --- tan [theta], (6)
c2 g
c2 - c1 c1 u² 1 + [alpha] u²
h = ------- -- --- = ([beta] - [alpha] ----------- ---. (7)
W c2 g 1 + [beta] g
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 [omega] 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´,