Johnston — Initial Motion. 9 



The condition that the motion may be initially a pure translation is 

 Z = 0, and that it may be a pure rotation is X = 0. 



"We will now investigate the initial reaction when there is a single 

 smooth constraint, taking as axes of reference, axes through the centre 

 of inertia such that the axis of x is parallel to the line in which the 

 constrained point is prevented from moving.^ "We have already con- 

 sidered this question when the principal axes through the centre of 

 inertia have been taken as the axes of reference. 



If, as usual, Xi denotes the initial reaction, we have, from equa- 

 tions (3), 



flScox — h^ooy — gdwz = £St, 



- hSojx + bSuy -fSwa ={M+ zXi) St, 



— ffSoDx —fduiy + cSooz = (iV — 1/Xl) St, 



mSu = {X+ Xi) St. 



Since there is no motion of the point xyz in the direction of the 

 axis of a;, 



Su — ySwz + zScDy = 0. 

 Therefore 



m {ySaij - zSaiy) = {X + Xi) St. 



Therefore, eliminating 8w^, Swy, Sco^ from this equation and the first 

 three, we get the following equation to determine Xi : — 



= 0. 



It is an interesting problem to find, when the system of applied 

 forces is given, what point in the body may be fixed in space in order 

 that initially the centre of inertia may remain at rest, or, in other 

 words, that the initial motion may be a rotation round the line join- 

 ing the point to the centre of inertia. Take, as axes of reference, the 

 principal axes through the centre of inertia, and let cc, y, s be the co- 

 ordinates of the fixed point. Then, since Su = 0, 8v = 0, 8'W = 0, 

 our dynamical equations are 



Z + Zi = 0, ASwx ={Z + Ji) St ; 

 r+Yi = 0, BSwy ={M+ Mi)St; 

 Z-\-Z\ =0, CSoiz ={N+ Ni)St. 



1 This question is solved in Williamson and Tarleton's "Dynamics," p. 356, 

 when gravity is the applied force. 



a 



- h 



-9 



L 



-h 



b 



-f 



M^zXy 



-9 



-f 



c 



N-yXY 







— mz 



my 



Z + Xi 



