will give a possible state of motion of the axis of the body; and the
motion of the centre may then be inferred from (22).
50. The theory preceding is of practical application in the
investigation of the stability of the axial motion of a submarine boat,
of the elongated gas bag of an airship, or of a spinning rifled
projectile. In the steady motion under no force of such a body in a
medium, the centre of gravity describes a helix, while the axis
describes a cone round the direction of motion of the centre of gravity,
and the couple causing precession is due to the displacement of the
medium.
In the absence of a medium the inertia of the body to translation is the
same in all directions, and is measured by the weight W, and under no
force the C.G. proceeds in a straight line, and the axis of rotation
through the C.G. preserves its original direction, if a principal axis
of the body; otherwise the axis describes a cone, right circular if the
body has uniaxial symmetry, and a Poinsot cone in the general case.
But the presence of the medium makes the effective inertia depend on the
direction of motion with respect to the external shape of the body, and
on W´ the weight of fluid medium displaced.
Consider, for example, a submarine boat under water; the inertia is
different for axial and broadside motion, and may be represented by
c1 = W + W´[alpha], c2 = W + W´[beta], (1)
where [alpha], [beta] are numerical factors depending on the external
shape; and if the C.G. is moving with velocity V at an angle [phi]
with the axis, so that the axial and broadside component of velocity
is u = V cos [phi], v = V sin [phi], the total momentum F of the
medium, represented by the vector OF at an angle [theta] with the
axis, will have components, expressed in sec. lb.,
u V v V
F cos [theta] = c1 --- = (W + W´[alpha]) --- cos [phi], F sin [theta] = c2 --- = (W + W´[beta]) --- sin [phi]. (2)
g g g g
Suppose the body is kept from turning as it advances; after t seconds
the C.G. will have moved from O to O´, where OO´ = Vt; and at O´ the
momentum is the same in magnitude as before, but its vector is
displaced from OF to O´F´.