DISPLACEMENT OF RIGID BODY.

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Every displacement of a rigid body may be produced by rotation about a fixed axis together with translation parallel to the axis (screw motion). Let a be any point of the body whose new position is a'; then we can produce the whole displacement by first giving the body a translation aa', and then turning it about a' as a fixed point. The latter step can be effected by rotation about an axis through d. Now consider those points of the body which lie in a plane perpendicular to this axis. By the rotation they are merely turned round in that plane; while by the translation the plane was moved parallel to itself. Hence the new position of this plane is parallel to its original position. Let then the body have first a translation perpendicular to the plane, so as to bring the plane into its new position; then the remaining displacement consists of a sliding of this plane on itself, which may be produced by rotation about a fixed point of it, or, which is the same thing, about an axis perpendicular to the plane. Thus the whole displacement is produced by rotation about that axis, together with translation parallel to it.

If two plane polygons, which are perversions of one another, be rolled symmetrically along a straight line, one on each side, until the same two corresponding sides come into contact, the result will be merely a translation of each along the line through a distance equal to its perimeter. Hence successive finite rotations through angles equal to the exterior angles of a polygon about successive vertices (taken the same way round) are equivalent to a translation of length equal to the perimeter. By supposing one polygon fixed, and the other to roll round it, we find that successive rotations about the vertices through twice the exterior angles will bring the plane back to its original position.

The corresponding theorems for a spherical surface are easily stated.

CHAPTER II. VELOCITY-SYSTEMS.

SPINS.

WHEN a body is rotating about a fixed axis with angular velocity w, every point in the body is describing a circle in a plane perpendicular to the axis, whose radius is the perpendicular distance of the point from the axis. Hence the velocity of the point is in magnitude times its distance from the axis, and its direction is perpendicular to the plane which contains the axis and the point.

m

If ab be the axis, pm perpendicular to it, the velocity of p is a times mp perpendicular to the plane pab. If, therefore, we represent the angular velocity w by means of a length ab marked off on the axis, the velocity of p is ab multiplied by mp, which is the moment of ab about p, being twice the area pab.

P

In the case of a plane figure, the rotation being about an axis perpendicular to the plane, or say about a point m in the plane (where it is cut by the axis), the velocity of any point p is w.mp in magnitude, but perpendicular to mp; that m is, it is io.mp, the angular velocity being reckoned positive when it goes round counter clockwise. When a body has a motion of translation, the velocity of every point in it is the same, and that is called the velocity of the rigid body. But in the case of rotation, the

velocity of different points of the body is different, and we can only speak of the system of velocities, or velocitysystem, of its different points. Still, the velocity-system due to a definite angular velocity about a definite axis is spoken of as the rotation-velocity, or simply the velocity of a rigid body which has that motion. To specify it completely we must assign its magnitude and the position of the axis; it is thus represented by a certain length marked off anywhere on a certain straight line. For it clearly does not matter on what part of the axis the length ab is marked off; its moment in regard to p will always be the velocity of p. A rotation-velocity, so denoted, shall be called a spin.

Such a quantity, which has not only magnitude and direction, but also position, is called a rotor (short for rotator) from this simplest case of it, the rotation-velocity of a rigid body. A rotor is a localised vector. While the length representing a vector may be moved about anywhere parallel to itself, without altering the vector, the length representing a rotor can only be slid along its axis without the rotor being altered.

Two velocity-systems are said to be compounded into a third, when the velocity of every point in the third system is the resultant of its velocities in the other two.

COMPOSITION OF SPINS.

The resultant of two spins l, m about the points a,

in a plane, is a spin (l+m) about a point c, such that l.ca+m.cb = 0. For the velocities of p due to the two spins are il. ap and im. bp, and their resultant is consequently i (l+m) cp ;

that is, it is the velocity due to a spin l+m about c.

b

P

It should be observed that the result holds good whatever be the signs of l, m; but that, if their signs are different, the point c will be in the line ab produced. There is one very important exception, when the spins are equal but of opposite signs; the resultant is then a

il. ap-il. bp=il (ap — bp) = il. ab. Thus the velocity of every point p is the same, namely it is of the magnitude l. ab and is perpendicular to ab.

1+m

Translating these results into language relating to axes perpendicular to the plane, we find that the resultant of two parallel spins l, m is a spin of magnitude equal to their sum, about an axis which divides any line joining them in the inverse ratio of their magnitudes. But the resultant of two equal and opposite parallel spins is a trans

lation-velocity, perpendicular to the plane containing them, of magnitude equal to either multiplied by the distance between them.

It follows that if we compound a spin 7 with a translation-velocity v perpendicular to its axis, the effect is to shift the axis parallel to itself through a distance v : l in a direction perpendicular to the plane containing it and the velocity.

A translation-velocity may be regarded as a spin about an infinitely distant axis perpendicular to it. Hence all theorems about the composition of translation-velocities with spins are special cases of theorems about the composition of spins.

The resultant of two spins about axes which meet is a spin about the diagonal of the parallelogram whose sides are their representative lines, of the magnitude represented by that diagonal. In other

words, spins whose axes meet are compounded like vectors. For if ab, ac represent the two spins, and ad is the diagonal of the parallelogram acdb, the velocities of any point p due to the two spins are the moments of ab and ac about p, and the resultant of

TWIST-VELOCITIES.

125

them is the moment of ad about p, that is, it is the velocity due to a spin ad.

It follows from this that the resultant of any number of spins whose axes meet in a point is also a spin whose axis passes through that point. And that if i, j, k are spins of unit angular velocity about axes oX, oŸ, oZ at right angles to one another, any spin about an axis through o may be represented by xi+yj + zk, where x, y, z are magnitudes of the component spins about the axes oX, oY, OZ.

If a rigid body have an angular velocity w about a certain axis, combined with a translation-velocity v along that axis, the whole state of motion is described as a twistvelocity (or more shortly, a twist) about a certain screw. We may in fact imagine the motion of the body to be produced by rigidly attaching it to a nut which is moving on a material screw. The ratio vw is called the pitch of the screw; it is a linear magnitude (of dimension [L] simply), and we may cut a screw of given pitch upon a cylinder of any radius. The pitch is the amount of translation which goes with rotation through an angle whose arc is equal to the radius. For our present purpose it is convenient to regard the axis of the rotation as a cylinder of very small radius, on which a screw of pitch p is cut. The screw is entirely described when its axis is given, and the length of the pitch. The angular velocity w is called the magnitude of the twist.

The velocity of a point at distance k from the axis is ko perpendicular to the plane through the axis, due to the rotation, and v parallel to the axis, due to the translation. If the resultant-velocity makes an angle with the axis, we shall have tan 0 = kw : v=k: p. Thus for points very near to the axis, the velocity is nearly parallel to it; for points very far off, nearly perpendicular to it; and for points whose distance is equal to the pitch of the screw, it is inclined at an angle of 45o.

A quantity like a twist-velocity, which has magnitude, direction, position, and pitch, is called a motor, from the