94 



KINETICS OF A PARTICLE. 



['75- 



circle about the hole so as to stretch the cord and just prevent the 

 mass of 2 Ibs. from descending, (a) How many revolutions must it 

 make? (b) If only one-fourth of the cord lie on the table while three- 

 fourths hang down, how many revolutions must be made ? 



(14) Show that, when a particle moves with constant velocity in a 

 vertical circle, the constraining force F* (Art. 167) is always directed 

 towards a fixed point on the vertical diameter. 



175. A particle of mass m subject to gravity alone is con- 

 strained to move in a vertical circle of radius 1. If there be 

 no friction on the curve and the constraint be produced by a 

 weightless rod or string joining the particle to the centre of 

 the circle, we have the problem of the simple mathematical 

 pendulum. 



Equation (i), Art. 169, is readily seen to reduce in this cas 

 (see Fig. 25) to the form 



: 



(8) 



A first integration gives, as shown in kinematics (Part I., Arts. 

 215, 216), 



o 



-/cos ), 



(9) 



where V Q is the velocity which the particle has at the time /=o 

 B when its radius makes the angle 



AOP = 6 Q with the vertical 



M S !R NS V~ - Multiplying by m, we have, for the 



kinetic energy of the particle, 



where h = v/2glco$6 Q is 

 constant. If the horizontal line 

 MN t drawn at the height v^/ 

 above the initial point P Q , inter 

 sect the vertical diameter AB at 

 R, it appears from the figure that 



