Page:EB1911 - Volume 14.djvu/146

Rh If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate about a line parallel to the resultant momentum F at an angle, the velocity of the vector of angular momentum, as in the case of a top, is

C1p sin − C22 sin cos ; (4)

and equating this to the impressed couple (multiplied by g), that is, to

(5)

and dividing out sin, which equated to zero would imply perfect centring, we obtain

(6)

The least admissible value of p is that which makes the roots equal of this quadratic in, and then

(7)

the roots would be imaginary for a value of p smaller than given by

(8)

(9)

Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving the Angle of Rifling, and n the Pitch of Rifling in Calibres.

If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle of the rifling is given by

tan = d / nd =  dp / u, (10)

which is the ratio of the linear velocity of rotation dp to u, the velocity of advance,

·   .   W   (11)

For a shot in air the ratio W′/W is so small that the square may be neglected, and formula (11) can be replaced for practical purpose in artillery by

(12)

if then we can calculate, , or −  for the external shape of the shot, this equation will give the value of and n required for stability of flight in the air.

The ellipsoid is the only shape for which and  have so far been determined analytically, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b = c,

(13)

A0 + 2B0 = 1, (14)

(15)

The length of the shot being denoted by and the calibre by d, and the length in calibres by x

l / d = 2a / 2b = x, (16)

(17)

(18)

(19)

If denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction ƒ of the linear scale; then the volume of a round shot being  d3, and d3 x of a shot x calibres long

W = d3 x (i − ƒ3) , (20)

(21)

(22)

If denotes the density of the air or medium

W′ = d3 x, (23)

(24)

(25)

(26)

in which / 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 =, or n = 180, when  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

(1)

and transverse velocity

(2)

and the time of completing a turn of the spiral is 2/.

When has the critical value in (7),

(3)

which makes the circumference of the cylinder on which the helix is wrapped

= nd ( − ) (x2 + 1) sin cos , (4)

and the length of one turn of the helix

(5)

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, , , and angular ,, , required to start the circulation.