I 4 STATICS. [227. 



The first three of these equations serve to determine the 

 reaction of the fixed point ; the last three are the actual con- 

 ditions of equilibrium corresponding to the three degrees of 

 freedom of a body with a fixed point. 



Hence, a rigid body having a fixed point is in equilibrium if 

 the sum of the moments of all the forces vanishes for any three 

 axes passing through the fixed point and not -situated in the same 

 plane. 



227. Rigid Body with a Fixed Axis, A body with a fixed axis 

 has but one degree of freedom ; indeed, the only possible motion 

 consists in rotation about this axis. 



An axis is fixed as soon as two of its points, say A, B, are 

 fixed. Hence, introducing the reactions A t , A y , A gt B# B y , B z 

 of these points, the body can be regarded as free. If the point B 

 be taken as origin, the line BA as axis of z (Fig. 73), the equa- 

 tions of equilibrium become 



B 



IA, 



where a = BA. 



The last of the six equations is the only independent condi- 

 tion of equilibrium of the con- 

 strained body ; the first five 

 determine A x , B& A y , B y , A x 

 + B g . The two ^-components 



_ , ; cannot be found separately, 



/ "R "7 A 

 jf ' jf since they act in the same 



^/D 'Ay 



/ straight line. 



Fig. 73. 



Hence, a rigid body having- 



a fixed axis is in equilibrium if the sum of the moments of all the 

 forces vanishes for the fixed axis. 



228. If, in the preceding article, the axis be not absolutely 

 fixed, but only fixed in direction so that the body can rotate about 

 the axis and also slide along it, we have evidently 





