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tending to produce rotation about a perpendicular axis, the result will be a change of direction of the axis about which the body revolves, but no change in the angular velocity.
109. If a pyramid or cone of any form roll on a similar pyramid (the image in a plane mirror of the first position of the first) all round, it clearly comes back to its primitive position. This (as all rolling of cones) is exhibited best by taking the intersection of each with a spherical surface. Thus we see that if a spherical polygon turns about its angular points in succession, always keeping on the spherical surface, and if the angle through which it turns about each point is twice the supplement of the angle of the polygon, or, which will come to the same thing, if it be in the other direction, but equal to twice the angle itself of the polygon, it will be brought to its original position.
110. The method of $ 100 also applies to the case of g 106; and it is thus easy to show that the most general motion of a spherical figure on a fixed spherical surface is obtained by the rolling of a curve fixed in the figure on a curve fixed on the sphere. Hence as at each instant the line joining C and O contains a set of points of the body which are momentarily at rest, the most general motion of a rigid body of which one point is fixed consists in the rolling of a cone fixed in the body upon a cone fixed in space—the vertices of both being at the fixed point.
111. To complete our kinematical investigation of the motion of a body of which one point is fixed, we require a solution of the following problem :-From the given angular velocities of the body about three rectangular axes attached to it to determine the position of the body in space after a given time. But the general solution of this problem demands higher analysis than can be admitted into the present treatise.
112. We shall next consider the most general possible motion of a rigid body of which no point is fixed-and first we must prove the following theorem. There is one set of parallel planes in a rigid body which are parallel to each other in any two positions of the body. The parallel lines of the body perpendicular to these planes are of course parallel to each other in the two positions.
Let C and C be any point of the body in its first and second positions. Move the body without rotation from its second position to a third in which the point at C' in the second position shall occupy its original position C. The preceding demonstration shows that there is a line CO common to the body in its first and third positions. Hence a line C'Ớ of the body in its second position is parallel to the same line CO in the first position. This of course clearly applies to every line of the body parallel to CO, and the planes perpendicular to these lines also remain parallel.
113. Let S denote a plane of the body, the two positions of which are parallel. Move the body from its first position, without rotation, in a direction perpendicular to S, till S comes into the plane of its
second position. Then to get the body into its actual position, such a motion as is treated in § 91 is farther required. But by § 91 this may be effected by rotation about a certain axis perpendicular to the plane S, unless the motion required belongs to the exceptional case of pure translation. Hence this case excepted), the body may be brought from the first position to the second by translation through a determinate distance perpendicular to a given plane, and rotation through a determinate angle about a determinate axis perpendicular to that plane. This is precisely the motion of a screw in its nut.
114. To understand the nature of this motion we may commence with the sliding of one straight-edged board on another.
Thus let GDEF be a plane board whose edge, DE, slides on the edge, AB, of anotherboard, ABC, of which for convenience
we suppose the edge, AC, to be horiG
zontal. By $30, if the upper board
move horizontally to the right, the
B constraint will give it, in addition, EL
vertically upward motion, and the rates D
of these motions are in the constant
ratio of AC to CB. Now, if both A
planes be bent so as to form portions of the surface of a vertical right cylinder, the motion of DF parallel to AC will become a rotation about the axis of the cylinder, and the necessary accompaniment of vertical motion will remain unchanged. As it is evident that all portions of AB will be equally inclined to the axis of the cylinder, it is obvious that the thread of the screw, which corresponds to the edge, DE, of the upper board, must be traced on the cylinder so as always to make a constant angle with its generating lines ($128). A hollow mould taken from the screw itself forms what is called the nut—the representative of the board, ABC-and it is obvious that the screw cannot move without rotating about its axis, if the nut be fixed. If a be the radius of the cylinder, w the angular velocity, a the inclination of the screw thread to a generating line, u the linear velocity of the axis of the screw, we see at once from the above construction that
aw:U:: AC:CB :: sin a: cos a, which gives the requisite relation between w and u,
115. In the excepted case of $ 113, the whole motion consists of two translations, which can of course be compounded into a single one: and thus, in this case, there is no rotation at all, or every plane of it fulfils the specified condition for S of $ 113.
116. We may now briefly consider the case in which the guiding cones ($ 110) are both circular, as it has important applications to the motion of the earth, the evolutions of long or flattened projectiles, the spinning of tops and gyroscopes, etc.
The motion in this case may be called Precessional Rotalion. The plane through the
instantaneous axis and the axis of the fixed cone passes through the axis of the rolling cone. This plane turns round the axis of the fixed cone with an angular velocity 12, which must clearly bear a constant ratio to the angular velocity w of the rigid body about its instantaneous axis.
117. The motion of the plane containing these axes is called the precession in any such case. What we have denoted by 1 is the angular velocity of the precession, or, as it is sometimes called, the rate of precession.
The angular motions W, 12 are to one another inversely as the distances of a point in the axis of the rolling cone from the instantaneous axis and from the axis of the fixed cone.
For, let OA be the axis of the fixed cone, OB that of the rolling cone, and of the instantaneous axis. From any point P in OB draw PN perpendicular tool, and PQ perpendicular to 0A. Then we perceive that P moves always in the circle whose centre
is Q, radius PQ, and plane perpendicular A
Hence the actual velocity of I the point P is 2.QP. But, by the prinN
ciples explained above ( 110) the velocity B
of P is the same as that of a point moving Q
in a circle whose centre is N, plane perpendicular to ON, and radius NP, which, as this radius revolves with angular velocity w, is w.NP. Hence
w:12::QP: NP. 118. Suppose a rigid body bounded by any curved surface to be touched at any point by another such body. Any motion of one on the other must be of one or more of the forms sliding, rolling, or spinning. The consideration of the first is so simple as to require no comment.
Any motion in which the bodies have no relative velocity at the point of contact, must be rolling or spinning, separately or combined.
Let one of the bodies rotate about successive instantaneous axes, all lying in the common tangent plane at the point of instantaneous contact, and each passing through this point—the other body being fixed. This motion is what we call rolling, or simple rolling, of the movable body on the fixed.
On the other hand, let the instantaneous axis of the moving body be the common normal at the point of contact. This is pure spinning, and does not change the point of contact.
Let it move, so that the instantaneous axis, still passing through the point of contact, is neither in, nor perpendicular to, the tangent plane. This motion is combined rolling and spinning.
119. As an example of pure rolling, we may take that of one cylinder on another, the axes being parallel.
Let p be the radius of curvature of the rolling, o of the fixed,
cylinder; w the angular velocity of the former, V the linear velocity of the point of contact. We have
For, in the figure, suppose P to be at any time the point of contact, and Q and p the points which are to be in contact after a very small interval t; 0, 0 the centres of curvature; POp=0, POQ=0.
OI Then PQ=Pp=space described by point of contact. In symbols po=of=VT.
R Also, before 0 Q and OP can coincide in direc
G tion, the former must evidently turn through an angle 0 +0. Therefore
wr=0+0; and by eliminating and o, and dividing by get the above result.
It is to be understood here, that as the radii of curvature have been considered positive when both surfaces are convex, the negative sign must be introduced for either radius when the corresponding surface is concave.
Hence the angular velocity of the rolling curve is in this case equal to the product of the linear velocity of the point of contact into the sum or difference of the curvatures, according as the curves are both convex, or one concave and the other convex.
120. We may now take up a few points connected with the curvature of surfaces, which are useful in various parts of our subject.
The tangent plane at any point of a surface may or may not cut it at that point. In the former case, the surface bends away
from the tangent plane partly towards one side of it, and partly towards the other, and has thus, in some of its normal sections, curvatures oppositely directed to those in others. In the latter case, the surface on every side of the points bends away from the same side of its tangent plane, and the curvatures of all normal sections are similarly directed. Thus we may divide curved surfaces into Anticlastic and Synclastic. A saddle gives a good example of the former class; a ball of the latter. Curvatures in opposite directions, with reference to the tangent plane, have of course different signs. The outer portion of the surface of an anchor-ring is synclastic, the inner anticlastic.
121. Meunier's Theorem.—The curvature of an oblique section of a surface is equal to that of the normal section through the same tangent line multiplied by the secant of the inclination of the planes of the sections. This is evident from the most elementary considerations regarding projections.
122. Euler's Theorem.—There are at every point of a synclastic surface two normal sections, in one of which the curvature is a
maximum, in the other a minimum; and these are at right angles to each other.
In an anticlastic surface there is maximum curvature (but in opposite directions) in the two normal sections whose planes bisect the angles between the lines in which the surface cuts its tangent plane. On account of the difference of sign, these may be considered as a maximum and a minimum.
Generally the sum of the curvatures at a point, in any two normal planes at right angles to each other, is independent of the position of these planes.
1 If and be the maximum and minimum curvatures at any
р point, the curvature of a normal section making an angle 0 with the normal section of maximum curvature is
123. Let P, p be two points of a surface indefinitely near to each other, and let r be the radius of curvature of a normal section passing through them. Then the radius of curvature of an oblique section through the same points, inclined to the former at an angle a, is r cos a ($ 121). Also the length along the normal section, from P to p, is less than that along the oblique section—since a given chord cuts off an arc from a circle, longer the less is the radius of that circle,
124. Hence, if the shortest possible line be drawn from one point of a surface to another, its osculating plane, or plane of curvature, is everywhere perpendicular to the surface.
Such a curve is called a Geodetic line. And it is easy to see that it is the line in which a flexible and inextensible string would touch the surface if stretched between those points, the surface being supposed smooth.
125. A perfectly flexible but inextensible surface is suggested, although not realized, by paper, thin sheet-metal, or cloth, when the surface is plane; and by sheaths of pods, seed-vessels, or the like, when not capable of being stretched flat without tearing. The process of changing the form of a surface by bending is called 'developing.' But the term 'Developable Surface' is commonly restricted to such inextensible surfaces as can be developed into a plane, or, in common language, ‘smoothed flat.'
126. The geometry or kinematics of this subject is a great contrast to that of the flexible line (§ 16), and, in its merest elements, presents ideas not very easily apprehended, and subjects of investigation that have exercised, and perhaps even overtasked, the powers of some of the greatest mathematicians.
127. Some care is required to form a correct conception of what