Class Workshop Exercise: Consider a person strengthening the shoulder muscles by means of dumbbell exercises. The figure bellow illustrates the position of the left arm when the arm is abducted to horizontal. The free$-$body diagram of the arm is shown in the bellow figure along with a mechanical model of the arm. Also the forces acting on the arm are resolved into their rectangular components along the horizontal and vertical directions. O corresponds to the axis of the shoulder joint, A is where the deltoid muscle is attached to the humerus, $B$ is the center of gravity of the entire arm, and $C$ is the center of gravity of the dumbbell. Wis the weight of the arm, $w_0$ is the weight of the dumbbell, $FM$ is the magnitude of the tension in the deltoid muscle, and $F_j$ is the joint reaction force at the shoulder. The resultant of the deltoid muscle force makes an angle $$ with the horizontal. The distances between $O$ and $A,B$ and $C$ are measured as $a,b,$ and $c,$ respectively. Determine the magnitude $F_M$ of the force exerted by the deltoid muscle to hold the arm at the position shown. Also determine the magnitude and direction of the reaction force at the shoulder joint in terms of specified parameters.

Arm Abducted to horizontal position

Free Body Diagram Acting in Arm