in the equations above formula_51 and formula_52 are reaction forces at the joint. formula_53 is the normal force applied to the cart. this second equation only depends on the vertical reaction force thus the equation can be used to solve for the normal force. the first equation can be used to solve for the horizontal reaction force. in order to complete the equations of motion, the acceleration of the point mass attached to the pendulum must be computed. the position of the point mass can be given in inertial coordinates as