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and

Direct

rolling, Ý i (direct-rolling component)...... р

9 9 1

[(A' – A)** + 2 (C' C) #j+(BB) ]......(7).

9
Choose OX, OY so that C-C'= 0, and put A' – A=a, B'-B=B
(6) and (7) become
(twisting component)

(8),
9

1 (direct-rolling component)

p= -(acc*+Bijo)......(9).

9 9 9 [Compare below, § 124 (2) and (1).]

And for o, the angular velocity of spinning, the obvious proposition stated in the preceding large print gives

C

w +
9 9

(B-a) ký

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if and

o=9

...(10),
1 1

be the curvatures of the projections on the tangent
y
plane of the fixed and moveable traces. [Compare below, $ 124
(3).]

From (1) and (2) it follows that
When one of the surfaces is a plane, and the trace on the
other is a line of curvature ($ 130), the rolling is direct.

When the trace on each body is a line of curvature, the rolling is direct. Generally, the rolling is direct when the twists of infinitely narrow bands ($ 120) of the two surfaces, along the traces, are equal and in the same direction.

112. Imagine the traces constructed of rigid matter, and all the rest of each body removed. We may repeat the motion with these curves alone. The difference of the circumstances now supposed will only be experienced if we vary the direction of the instantaneous axis. In the former case, we can only do this by introducing more or less of spinning, and if we do so we alter the trace on each body. In the latter, we have always the same moveable curve rolling on the same fixed curve; and therefore a determinate line perpendicular to their common tangent for one component of the rotation ; but along with this we may give arbitrarily any velocity of twisting round the common tangent. The consideration of this case is very in

Curve rolling on curve,

1
+

structive. It may be roughly imitated in practice by two stiff wires bent into the forms of the given curves, and prevented from crossing each other by a short piece of elastic tube clasping them together,

First, let them be both plane curves, and kept in one plane. We have then rolling, as of one cylinder on another.

Let p be the radius of curvature of the rolling, p of the fixed, cylinder ; w the angular velocity of the former, V the linear velo

city of the point of contact. We have O'

1

V.

P р
For, in the figure, suppose P to be at any time

the point of contact, and Q and (the points which Р

are to be in contact after an infinitely small Q

interval t; 0, O the centres of curvature; POQ = 0, PO'Q' = 6.

Then PQ = PQ = space described by point of contact. In symbols po = p'O' = Vt.

Also, before O'Q' and OQ can coincide in direction, the former must evidently turn through an angle 0 + 6'.

Therefore wt = 0 + 6'; and by eliminating 0 and 0', and dividing by t, we get the above result.

It is to be understood, that as the radii of curvature have been considered positive here when both surfaces are convex, the negative sign must be introduced for either radius when the

corresponding curve is concave. Angular Hence the angular velocity of the rolling curve is in this velocity of rolling in a case equal to the product of the linear velocity of the point of plano.

contact by the sum or difference of the curvatures, according as the curves are both convex, or one concave and the other

convex,

Plane curves not in same plane.

113. When the curves are both plane, but in different planes, the plane in which the rolling takes place divides the angle between the plane of one of the curves, and that of the other produced through the common tangent line, into parts whose sines are inversely as the curvatures in them respectively; and the angular velocity is equal to the linear velocity

curves not

pR=

P

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+ 2

cos a).

2

of the point of contact multiplied by the difference of the pro

Plane jections of the two curvatures on this plane. The projections of lasame the circles of the two curvatures on the plane of the common tangent and of the instantaneous axis coincide.

For, let PQ, Pp be equal arcs of the two curves as before, and let PR be taken in the common tangent (i.e., the intersection of the planes of the curves) equal to each. Then QR, pR are ultimately perpendicular to PR.

PR
Hence
20'

AR
PR
QR=

2p
Also, 2 QRp=a, the angle between the planes of the curves.

PR4 / 1 1 2
We have Qp

4

p

op
Therefore if w be the velocity of rotation as before,

1

2 cos a
p"

op
Also the instantaneous axis is evidently perpendicular, and there-
fore the plane of rotation parallel, to Qp. Whence the above.

In the case of a = 7, this agrees with the result of § 112.
A good example of this is the case of a coin spinning on a
table (mixed rolling and spinning motion), as its plane becomes
gradually horizontal. In this case the curvatures become more
and more nearly equal, and the angle between the planes of the
curves smaller and smaller. Thus the resultant angular velo-
city becomes exceedingly small, and the motion of the point
of contact very great compared with it.

114. The preceding results are, of course, applicable to tor- Curve rolltuous as well as to plane curves ; it is merely requisite to sub-curve : two stitute the osculating plane of the former for the plane of the freedom. latter.

1

w = V

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+

ing on surface: three degrees of

115. We come next to the case of a curve rolling, with or Curve rollwithout spinning, on a surface.

It may, of course, roll on any curve traced on the surface, freedom. When this curve is given, the moving curve may, while rolling along it, revolve arbitrarily round the tangent. But the com

degrees of freedom,

Curve roll- ponent instantaneous axis perpendicular to the common tanface: thrre gent, that is, the axis of the direct rolling of one curve on the

other, is determinate, $ 113. If this axis does not lie in the surface, there is spinning. Hence, when the trace on the surface is given, there are two independent variables in the motion; the space traversed by the point of contact, and the inclination of the moving curve's osculating plane to the tangent plane of the fixed surface.

Trace pre- 116. If the trace is given, and it be prescribed as a condiscribed and no spinning tion that there shall be no spinning, the angular position of the permitted.

rolling curve round the tangent at the point of contact is determinate. For in this case the instantaneous axis must be in the tangent plane to the surface. Hence, if we resolve the rotation into components round the tangent line, and round an axis

perpendicular to it, the latter must be in the tangent plane. Thus the rolling, as of curve on curve, must be in a normal plane to

the surface; and therefore (SS 114, 113) the rolling curve must Two degrees be always so situated relatively to its trace on the surface that of freedom.

the projections of the two curves on the tangent plane may be of coincident curvature.

The curve, as it rolls on, must continually revolve about the tangent line to it at the point of contact with the surface, so as in every position to fulfil this condition.

Let a denote the inclination of the plane of curvature of the trace, to the normal to the surface at any point, a' the same for

1 1 the plane of the rolling curve; their curvatures. We

p'p reckon a as obtuse, and a acute, when the two curves lie on opposite sides of the tangent plane. Then

1

1
sin a'=
p

р
which fixes a' or the position of the rolling curve when the point

of contact is given. Angular ve

Let w be the angular velocity of rolling about an axis perpenlocity of di. rect rolling. dicular to the tangent, w that of twisting about the tangent, and let

1 V be the linear velocity of the point of contact. Then, since - cosa'

sin ay

1 and cos a (each positive when the curves lie on opposite sides Angular veр

locity of di.

rect rolling. of the tangent plane) are the projections of the two curvatures on a plane through the normal to the surface containing their common tangent, we have, by $ 112,

1

1 cos a'

р a' being determined by the preceding equation. Let 7 and 1 denote the tortuosities of the trace, and of the rolling curve, respectively. Then, first, if the curves were both plane, we see that one rolling on the other about an axis always perpendicular to their common tangent could never change the inclination of their planes. Hence, secondly, if they are both tortuous, such rolling will alter the inclination of their osculating planes by an indefinitely small amount (T-7') ds during rolling which shifts Angular ve.

locity round the point of contact over an arc ds. Now a is a known function tangent. of s if the trace is given, and therefore so also is a'. But a-a is the inclination of the osculating planes, hence

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surface.

117. Next, for one surface rolling and spinning on another. Surface on First, if the trace on each is given, we have the case of g 113 or $ 115, one curve rolling on another, with this farther condition, that the former must revolve round the tangent to the two curves so as to keep the tangent planes of the two surfaces coincident. It is well to observe that when the points in contact, and the Both traces

: two traces, are given, the position of the moveable surface is one degree

of freedom. quite determinate, being found thus :-Place it in contact with the fixed surface, the given points together, and spin it about the common normal till the tangent lines to the traces coincide.

Hence when both the traces are given the condition of no spinning cannot be imposed. During the rolling there must in general be spinning, such as to keep the tangents to the two traces coincident. The rolling along the trace is due to rotation round the line perpendicular to it in the tangent plane. The whole rolling is the resultant of this rotation and a rotation about the tangent line required to keep the two tangent planes coincident

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