g KINETICS OF A PARTICLE. [171. 



centre of curvature. A force equal and opposite to this cen- 

 tripetal force, i.e. = miP/r, is called centrifugal force. It 

 should be noticed that this is a force exerted not on the 

 moving particle, but by it. 



It appears from equation (2) that the normal reaction N is 

 the resultant of the centripetal force mv^/p and the reversed 

 normal component of. the given forces, F n . Changing all the 

 signs, we can express the same thing by saying that the pressure 

 on the curve, N, is the resultant of the centrifugal force, mv 2 /^, 

 and of the normal component F n of the given forces. 



If, in particular, this normal component F n is zero, the press- 

 ure on the curve is equal to the centrifugal force. This case 

 is of frequent occurrence. Thus, if a small stone attached to 

 a cord be whirled around rapidly, the action of gravity on the 

 stone can be neglected in comparison with the centripetal force 

 due to rotation ; hence the centrifugal force measures approxi- 

 mately the tension of the string, and may cause it to break. 

 Again, when a railway train runs in a curve, the centrifugal 

 force produces the horizontal pressure on the rails, which tends 

 to displace and deform the rails. 



171. It may happen that at a certain time t the pressure N 

 vanishes. If the constraint be complete (Art. 165), this would 

 merely indicate that the pressure in passing through zero. 

 inverts its sense. If, however, the constraint be one-sided, the 

 consequence will be that the particle at this time leaves the 

 constraining curve ; for at the next moment the pressure will 

 be exerted in a direction in which the particle is free to move. 



Now Evanishes when its components F n and witf/p become 

 equal and opposite. The conditions under which the particle 

 would leave the curve are, therefore, that the resultant Fot the 

 given forces should lie in the osculating plane of the path, and 

 that F=z 



172. To obtain the equations of motion expressed in rec- 

 tangular Cartesian co-ordinates, let X, Y, Z be the components of 



