20, 21] CONSTRAINED MOTION. 41 



If it starts with no velocity the constant is zero, consequently F the 

 velocity at a distance R is given by 



Therefore we may state the result by saying that the path will be 

 an ellipse, parabola, or hyperbola according as the body is projected 

 in any direction with a velocity less than, equal to, or greater than 

 the velocity that it would acquire in falling from an infinite distance 

 to the point of projection. 



21. Constrained Motion. We have so far considered the 

 moving particle as free to move in any direction. This is however 

 by no means usually the case, since in the majority of cases with 

 which we have to deal the particle forms part of a body which is 

 possibly itself a part of a machine, and is guided by contact with 

 other bodies to travel in certain definite paths, although the velocities 

 with which it travels may be left undetermined. Such limitations to 

 the freedom of movement of a body are known as constraints, and 

 they are specified by certain equations having a geometrical significance. 

 In the case of a single particle, the simplest constraint is that in 

 which the particle is constrained to move upon a certain surface. 

 For instance, if the surface is a material one, the particle may, 

 during the whole motion, press against its inner, or concave side, 

 the material preventing the particle from passing across the geometrical 

 surface. The surface may itself be in motion, in this case the 

 constraint is said to be varying, and the equation of the surface will 

 contain the time. Let the equation expressing the constraint be 



33) <p (x, y, z, t) = 0. 



It is evident that a particle cannot move subject to constraints 

 without calling into play certain reactions due to the constraints. 

 In other words the acceleration experienced by the particle under 

 the influence of given forces will no longer be the same as if the 

 particle were free, but there will be a certain action and reaction 

 between the surface and particle which may be represented by an 

 extra force whose components are X , Y 19 Z lf applied to the particle. 

 The equations of motion may then be written 



34) m = X+X lr m = Y+Y 1 , m = Z+Z i , 



where X, Y, Z are the components of the given forces and X 1; Y lf Z^ 

 are the components of the force exercised by the surface upon the 

 particle, that is the reaction of the surface. These are to be found 

 by means of the equation of condition, y = 0, which holds for all 

 values of t. Differentiating by t, 



