167.] MOTION ON A FIXED CURVE. 87- 



In many cases, however, the constraint is not complete, but 

 only partial, or one-sided. Thus, the rails compelling the train 

 to move in a definite curve do not prevent its being lifted verti- 

 cally out of this curve, nor does the cord that confines the 

 motion of the stone to a sphere prevent it from moving towards 

 the inside of the spherical surface. 



While complete constraints are generally expressed by equa- 

 tions, one-sided constraints should properly be expressed by 

 inequalities. Thus, in the case of the stone, the condition is 

 really that its distance r from the hand is not greater than the 

 length / of the cord, i.e. 



but as soon as r becomes less than /, the constraining action 

 -ceases, and the stone becomes free. It is, therefore, in general 

 :sufficient to consider conditional equations ; but the nature of 

 the constraint, whether complete or partial, must be taken into 

 account to determine when and where the constraint ceases to 

 exist. 



166. We now proceed to consider separately the motion of 

 a particle constrained to a fixed curve and that of a particle 

 constrained to a fixed surface. After these special cases, the 

 general problem of motion on a movable curve or surface will 

 -be discussed. 



/ 



2. MOTION ON A FIXED CURVE. 



167. The condition that a particle should move on a given 

 fixed curve can always be replaced by introducing a single addi- 

 tional force F' called the constraining force, or the constraint. 

 An example will best show how this force can be determined. 



Let us consider a particle of mass m, subject to the force of 

 gravity F=mg alone; in general it will describe a parabola 

 whose equation can be found if the initial conditions are known. 

 To compel the particle to describe some other curve, say a verti- 



