178.] MOTION ON A FIXED CURVE. 95 



176. Taking R as origin and the axis of z vertically down- 

 wards, we have RQ=z=lcos6+k\ hence the force-function U~ 

 has the simple expression 



Umgz\ 



and the velocity v=^/2gz is seen to become zero when the 

 particle reaches the horizontal line MN, 



For the further treatment of the problem, three cases must 

 be distinguished according as this line of zero-velocity 'MN 

 intersects the circle, touches it, or does not meet it at all ; i.e. 

 according as 



A = / f or =2/sin 2 -. (11) 



> 2g > 2 



177. Equation (2), Art. 169, serves to determine the reaction 

 jVof the circle, or the pressure TV on the circle. We have 



whence 



Substituting for v* its value from (10), we find 



(12). 



The pressure on the curve has therefore its greatest value when 

 6 = 0, i.e. at the' lowest point A. It becomes zero for /cos X 

 = J//, which is easily constructed. 



178. If the constraint be complete as for a bead sliding along 

 a circular wire, or a small ball moving within a tube, the press- 

 ure merely changes sign at the point = r But if the con- 

 straint be one-sided, the particle may at this point leave the 

 circle. The one-sided constraint maybe such that OP^ /, as 

 when the particle runs in a groove cut on the inside of a ring, 

 or when it is joined to the centre by a string ; in this case the 



