IIO KINETICS OF A PARTICLE. [205. 



Differentiating again, we find 



+ 2X<l> tx + 2>(, y + (/> = O. (20) 



If in this last equation the values of Jr, y be substituted from 

 (17), we have a linear equation for X. The value of X thus 

 obtained can then be introduced into the equations of motion 

 (17) ; and it only remains to integrate these equations. 



This integration will often be facilitated by introducing new 

 variables for x, y. 



205. A particle moves without friction in a straight tube which 

 revolves uniformly in a horizontal plane about one of its points. De- 

 termine its motion. 



To illustrate the application of the general methods, we shall solve 

 this problem completely, first without the use of indeterminate multi- 

 pliers, and then with their aid, although the problem is so simple that it 

 might be solved without applying these general methods, as will be 

 pointed out below. 



As the weight of the particle is balanced by the vertical reaction of 

 the tube, we have a case of plane motion with X = o, Y o. Hence 

 d'Alembert's equation (14) becomes 



x8x + y8y=o. (21) 



If we take as origin the point O about which the tube rotates, the con- 

 straining curve is a straight line through the origin y = x tan 0, where 

 6 = to/, to being the constant angular velocity of the tube and the axis of 

 x coinciding with the initial position of the tube at the time /= o. Hence 



x = r cos to/, y = r sin to/ ; ^ 



8x = Br cos to/, By = Br sin to/ ; 



(22) 

 x = r cos to/ tor sin to/, y= r sin to/ + tor cos to/; 



x=r cos <o/ 2 tor sin to/ toV cos to/, y= r sin to/+ 2 tor cos to/ to 2 r sin to/. 



Substituting these values of x, y and 8x, By into the equation of motion 

 (21), we find after reduction 



r wV=o. (23) 



As mentioned above, this equation might have been derived directly 

 by considering that the acceleration along the tube is due to the cen- 

 trifugal force alone (see Art. 1 70) . 



