A uniform rod of mass M and length L is hinged at one end to a vertical wall. It is held in a horizontal position by a massless string attached to its other end. The string makes an angle of 30^ with the rod. The magnitude of the net reaction force exerted by the hinge on the rod is:

Step-by-step solution

Taking torque about the hinge to be zero for rotational equilibrium: T (30^ ) L = Mg L 2 T ( 1 2 ) = Mg 2 T = Mg For translational equilibrium, balancing the horizontal forces: H_x = T (30^ ) = Mg 3 2 Balancing the vertical forces: H_y + T (30^ ) = Mg H_y + Mg ( 1 2 ) = Mg H_y = Mg 2 The net reaction force exerted by the hinge is the vector sum of its components: R = H_x^2 + H_y^2 = ( 3 Mg 2 )^2 + ( Mg 2 )^2 = 3 4 M^2g^2 + 1 4 M^2g^2 = Mg Answer: Mg