i6i.] CONSTRAINED MOTION. 85 







IV. Constrained Motion. 



I. INTRODUCTION. 



160. It has been shown, in the preceding sections, that the 

 motion of a free particle is fully determined if all the forces 

 acting upon the particle, as well as the so-called initial con- 

 ditions, are given. The motion of a particle may, however, 

 depend not only on given forces, but on other conditions not 

 directly expressed in terms of forces. The motion is then 

 said to be constrained. 



Some of the more important forms of constraint have been 

 considered in Part II., Arts. 218-225. To mention some more 

 concrete examples : a heavy particle sliding down a smooth 

 inclined plane is subject not only to the force of gravity, but 

 also to the condition that it cannot pass through the plane ; a 

 railway train running on the rails, a piece of machinery slid- 

 ing in a groove or between guides, can, for many purposes, be 

 regarded as a particle constrained to a curve; the bob of a 

 pendulum, a stone attached to a cord and swung around by 

 the hand, may be regarded as constrained to a surface. 



161. Sometimes these constraining conditions can be easily 

 replaced by forces. Thus, in the first illustration above, the 

 condition that the particle cannot pass through the inclined 

 plane can be expressed by introducing the reaction of the 

 plane, i.e. a force acting on the particle at right angles to 

 the plane, so as to prevent it from passing through the plane. 

 Similarly, in the case of the stone attached to the cord, we 

 may imagine the cord cut and its tension introduced so as to 

 replace the condition by a force. 



Whenever the constraints to which a particle is subjected 

 can thus be expressed by means of forces, these forces can be 

 combined with the other impressed forces, and then, of course, 

 the equations of motion for a free particle can be applied. 



