229.] CONSTRAINTS. 



hence, by the third equation of equilibrium, 



as an additional condition of equilibrium. 



The body has in this case two degrees of freedom. 



229. Rigid Body with a Fixed Plane. A body constrained to 

 slide on a fixed plane has three degrees of freedom. At every 

 point of contact between the body and the plane, the latter 

 exerts a reaction. As all these reactions are parallel, they can 

 be combined into a single resultant N. Taking the fixed plane 

 as the plane xy, TV will be parallel to the axis of z\ hence, if a, 

 b, o be the co-ordinates of its point of application, the six 

 equations of equilibrium are 



The third, fourth, and fifth equations determine the reaction 

 N and the co-ordinates a, b of its point of application. The 

 three other equations are the actual conditions of equilibrium ; 

 they agree, of course, with the three conditions of equilibrium 

 of a plane system as found in Art. 143. 



If there be not more than three points of contact (or supports) 

 between the body and the fixed plane, the reactions of these 

 points can be found separately. Let A v A 2 , A 3 be the three 

 points of contact ; N v N^ N 3 the required reactions ; a v b^ 

 # 2 , b v a s , 3 the co-ordinates of A v A 2 , A 3 ; then N must be 

 resolved into three parallel forces passing through these points, 

 and the conditions are 



a^ 4- 



