W´ 1 [rho]
--- = ------ -------, (24)
W 1 - f³ [sigma]
k1² 1 1 - f^5 k2² x² + 1
--- = --- -------, ---- = ------, (25)
d² 10 1 - f³ k1² 2
[rho] x² + 1
tan² [delta] = ------- ([beta] - [alpha])-------------, (26)
[sigma] 1/5 (1 - f^5)
in which [sigma]/[rho] may be replaced by 800 times the S.G. of the
metal, taking water as 800 times denser than air on the average, in
round numbers, and formula (10) may be written n tan [delta] = [pi],
or n[delta] = 180, when [delta] is a small angle, and given in
degrees.
From this formula (26) the table following has been calculated by A.
G. Hadcock, and the results are in agreement with practical
experience.
52. In the steady motion the centre of the shot describes a helix,
with axial velocity
/ c1 \
u cos [theta]= v sin [theta] = ( l + -- tan² [theta] ) u cos [theta] [asympt] u sec [theta], (1)
\ c2 /
and transverse velocity
/ c1 \
u sin [theta] - v cos [theta] = ( l - -- ) u sin [theta] [asympt] ([beta] - [alpha]) u sin [theta]; (2)
\ c2 /
and the time of completing a turn of the spiral is 2[pi]/[mu].
When [mu] has the critical value in (7),
2[pi] 4[pi] C2 2[pi]
----- = ----- -- cos [theta] = ----- (x² + 1) cos [theta], (3)
[mu] p C1 p
which makes the circumference of the cylinder on which the helix
is wrapped
2[pi] 2[pi]u
-----(u sin [theta] - v cos [theta] = ------ ([beta] - [alpha]) (x² + 1) sin² [theta] cos [theta]
[mu] p
= nd ([beta] - [alpha]) (x² + 1) sin [theta] cos [theta], (4)
and the length of one turn of the helix
2[pi]
----- (u cos [theta] + v sin [theta] ) = nd(x² + 1); (5)
[mu]
thus for x = 3, the length is 10 times the pitch of the rifling.
53. _The Motion of a Perforated Solid in Liquid._--In the preceding
investigation, the liquid stops dead when the body is brought to rest;
and when the body is in motion the surrounding liquid moves in a
uniform manner with respect to axes fixed in the body, and the force
experienced by the body from the pressure of the liquid on its surface
is the opposite of that required to change the motion of the liquid;
this has been expressed by the dynamical equations given above. But if
the body is perforated, the liquid can circulate through a hole, in
reentrant stream lines linked with the body, even while the body is at
rest; and no reaction from the surface can influence this circulation,
which may be supposed started in the ideal manner described in § 29,
by the application of impulsive pressure across an ideal membrane
closing the hole, by means of ideal mechanism connected with the body.
The body is held fixed, and the reaction of the mechanism and the
resultant of the impulsive pressure on the surface are a measure of
the impulse, linear [xi], [eta], [zeta], and angular [lambda], [mu],
[nu], required to start the circulation.
This impulse will remain of constant magnitude, and fixed relatively
to the body, which thus experiences an additional reaction from the
circulation which is the opposite of the force required to change the
position in space of the circulation impulse; and these extra forces
must be taken into account in the dynamical equations.
An article may be consulted in the _Phil. Mag._, April 1893, by G. H.
Bryan, in which the analytical equations of motion are deduced of a
perforated solid in liquid, from considerations purely hydrodynamical.
The effect of an external circulation of vortex motion on the motion
of a cylinder has been investigated in § 29; a similar procedure will
show the influence of circulation through a hole in a solid, taking as
the simplest illustration a ring-shaped figure, with uniplanar motion,
and denoting by [xi] the resultant axial linear momentum of the
circulation.
As the ring is moved from O to O´ in time t, with velocity Q, and
angular velocity R, the components of liquid momentum change from
[alpha]M´U + [xi] and [beta]M´V along Ox and Oy
to
[alpha]M´U´ + [xi] and [beta]M´V´ along O´x´ and O´y´, (1)
the axis of the ring changing from Ox to O´x´; and
U = Q cos [theta], V = Q sin [theta],