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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302, 303] THE GEOMETRICAL THEORY. 323
position of rest by an impulsive system of forces of the most general type.
This is the object of the present chapter.
303. One Pair of Impulsive and Instantaneous Screws.
Let it be supposed that nothing is known of the position, mass, or other
circumstances of an unconstrained rigid body save what can be deduced
from the fact that, when struck from a position of rest by an impulsive
wrench on a specified screw y, the effect is to make tbe body commence to
move by twisting around a specified screw a.
As a, like every other screw, is defined by five coordinates, the knowledge
of this screw gives us five data towards the nine data that are required for
the complete specification of the rigid body and its position.
We have first to prove that the five elements which can be thence
inferred with respect to the rigid body are in general—
(1) A diameter of the momental ellipsoid.
This is clearly equivalent to two elements, inasmuch as it restricts
the position of the centre of gravity to a determinate straight line.
(2) The radius of gyration about this diameter.
This is, of course, one element.
(3) A straight line in the plane conjugate to that diameter.
A point in the plane would have been one element, but a straight
line in the plane is equivalent to two. If the centre of gravity were
also known, we should at once be able to draw the conjugate plane.
Draw a plane through both the instantaneous screw a and the common
perpendicular to a and y. Then the centre of gravity of the rigid body
must lie in that plane (§ 301). It was also shown that if pa be the pitch of
a, and if (a,;) represent the angle between a and y, then the perpendicular
distance of the centre of gravity from a will be expressed by pa tan (ay)
(§ 301). This expression is completely known since a and y are known.
Thus we find that the centre of gravity must lie in a determinate ray
parallel to a. There will be no ambiguity as to the side on which this
straight line lies if it be observed that it must pass between a and the point
in which y is met by the common perpendicular to y and a. In this manner
from knowing a and y we discover a diameter of the momental ellipsoid.
If a be the twist velocity with which a rigid body of mass M is twisting
about any screw a. If y be the corresponding impulsive screw, and if
denote as usual the virtual coefficient of a and y, then it is proved in § 279
that the kinetic energy of the body
Ma? —
cos (a?;)
21—2