360 RELATIVE MOTION AND UNIVERSAL GRAVITATION. [CH. XIII. 



7. A particle being observed to move, relatively to a certain frame, with 

 a simple harmonic motion of period STT/W in a line, which turns uniformly 

 about the mean position of the particle in a plane fixed relatively to the 

 frame with angular velocity o&amp;gt;, prove that the acceleration of the particle 

 when at distance r from its mean position is compounded of a radial 

 acceleration (^ 2 -f&amp;lt;o 2 )r, and a transverse acceleration Za&amp;gt;r in the sense in 

 which the line turns. 



*285. Relativity of Force. Passing now from this par 

 ticular set of facts connected with the rotation of the Earth 

 about its polar axis and their dynamical formulation, we consider 

 next the influence of the doctrine of the relativity of motion in 

 respect of the notion of force, and we say at once that force, 

 like motion and position, is relative, meaning thereby that the 

 force acting on a body is force relative to a frame , in other words, 

 just as we hold that no meaning can be attached to any phrase of 

 the kind &quot; a body is moving with such a velocity &quot; or &quot; a body 

 is at rest &quot; or &quot; a body is rotating about an axis &quot; until the 

 frame of reference is specified, so no meaning can be attached 

 to any phrase of the kind &quot;a body is acted upon by a force 

 of such and such a magnitude &quot; or &quot; the force acting on a body 

 is directed to a certain other body &quot; until the frame of reference 

 is specified. This conclusion is at once established when we 

 reflect that (1) the force on a body is the resultant of the forces 

 on its particles, (2) the force on a particle is made up of com 

 ponents each of which is the product of the mass of the particle 

 and the acceleration induced in it by some other particle, and 

 (3) the acceleration of a particle is the rate of change of its 

 velocity relative to a frame; in fact &quot;acceleration,&quot; and by 

 consequence &quot; force,&quot; have no meaning except as dependent on a 

 frame ; &quot; acceleration &quot; means &quot; acceleration relative to a frame,&quot; 

 and similarly with force. 



But now it is important to observe that, whereas the position 

 of a point, or the velocity or acceleration of a particle, can be 

 described by reference to any frame determined by parts of 

 natural bodies, this is not the case for forces; but the notion 

 of force requires that the frame of reference should be properly 

 chosen. Thus there are frames of reference for which the phrase 

 &quot; force on a particular body &quot; would have no meaning. We shall 

 now illustrate and enforce this theory by considering the choice 

 of the origin and of the lines of reference of a frame of reference. 



