But the change of the resultant momentum F of the medium as well as of the body from the vector OF to O′F′ requires an impulse couple, tending to increase the angle FOO′, of magnitude, in sec. foot-pounds
F·OO′·sin FOO′ = FVt sin (θ − φ),
(4)
equivalent to an incessant couple
| N = FV sin (θ − φ) = (F sin θ cos φ − F cos θ sin φ) V = (c2 − c1) (V2 / g) sin φ cos φ = W′ (β − α) 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 θ. With no reserve of buoyancy W = W′, and the couple N, tending to increase θ, has the effect of diminishing the metacentric height by h ft. vertical, where