55] 



TRANSLATIONS AND ROTATIONS. 



199 



CHAPTER VI. 



SYSTEMS OF VECTORS. DISTRIBUTION OF MASS. 

 INSTANTANEOUS MOTION. 



55. Translations and Rotations. A rigid body or system 

 of material particles is one in which the distance of each point of 

 the system from every other is invariable. Its position is known 

 when the positions of any three of its points are known, for every 

 point is determined by its distances from three given points. These 

 three points have each three coordinates, but, since there are three 

 conditions between them, defining their mutual distances, there are 

 only six independent coordinates. Thus, a rigid body has six coordinates. 



A rigid body may evidently be displaced in such a manner that 

 the displacement of every point is represented by equal vectors, that 

 is equal in length and parallel. Such a dis- 

 placement is called a translation, and, being 

 represented by a free vector, has three coordinates. 



A rigid body may also evidently be displaced, 

 so that two given points in it, A and IB, remain 

 fixed. Since any point P must move on a sphere 

 of radius BP about B, and also on a sphere of 

 radius AP about A, the locus of its positions is 

 the intersection of the two spheres, that is a circle 

 whose plane is perpendicular to the line AB, and 

 whose radius CP is the perpendicular distance 

 from P to the line AS. If this is zero, the 

 point does not move, therefore all points on the 

 line AB remain fixed. The displacement is called 

 a rotation and the line AB, the axis of rotation. The rotation is 

 specified if we know the situation of the line AB and the magnitude 

 of the angle POP', or the angle of rotation. 



A line may be specified by giving the two pairs of coordinates 

 of the points in which it intersects two of the coordinate planes. 

 A line has thus four coordinates, and a rotation, five, the four 

 of the axis together with the magnitude of the angle. 



Fig. 39. 



