A Treatise on the Theory of Screws
Forfatter: Sir Robert Stawell Ball
År: 1900
Forlag: The University Press
Sted: Cambride
Sider: 544
UDK: 531.1
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298 THE THEORY OF SCREWS. [280-
Unless in this exceptional case where is infinite it is always true that
when pa is zero, a and t) are at right angles.
It is universally true that when the impulsive screw and the instan-
taneous screw are at right angles (the body being quite free), the pitch of
the instantaneous screw must be zero.
For if were not zero when cos(a^) was zero then å must be zero. As
some motion must result from the impulse (the mass of the body being
finite) we must have pa infinite. The initial motion is thus a translation.
Therefore the impulse must have been merely a force through the centre
of gravity ; a and 77 must be parallel and cos (a>?) could not be zero.
The expression for the kinetic energy in § 279,
assumes an indeterminate form when the impulsive wrench reduces to a
couple. For we then have pa = 0, but as cos (a?;) is not zero the expression
for mari, i.e.
4 {(p* + Pv) C0S (ai?) - sin (a7?)]>
becomes infinite.
The expression for the kinetic energy arising from an impulsive wrench
of unit intensity on a screw ■»; applied to a free body of unit mass which
thereupon begins to twist with an instantaneous movement about a screw a
has the concise form
cos (arj)
^0.7]’
Pa
281. Conditions to be fulfilled by two pairs of Impulsive and
Instantaneous Screws.
Let a be a screw about which a free rigid body is made to twist in
consequence of an impulsive wrench administered on some other screw ry.
Let ß be another instantaneous screw corresponding in like manner to £ as
an impulsive screw. Then we have to prove that the two following formulae
are satisfied*:
—cos (/fy) + cos (a£) =
cos (oh?) cos(/3f)
Pa _ pß
cos (a?/) cos (ߣ)
* Proceedings of the Camb. Phil. Soo., Vol. ix. Part iii. p. 193.