Final answer:
To calculate the mass of the arm and its center of mass, a free body diagram is drawn showing forces at the scales and the gravitational force at the arm's center of mass. Equilibrium equations for forces and torques are used in conjunction with scale readings and distances to solve for the desired quantities.
Step-by-step explanation:
To find the mass of the arm and its center of mass using scales, we will construct a free body diagram to aid our analysis. The setup involves the subject lying on their back with one scale under the elbow and another under the hand. The forces to consider are the gravitational force acting at the center of mass of the arm and the normal forces exerted by each scale (Fe at the elbow and Fh at the hand). The arm is in static equilibrium, which means the sum of forces and sum of torques (moments) are zero.
By using the scale readings and the distances from the shoulder joint to each scale, we apply Newton's second law. Considering a pivot at the shoulder, we calculate the torque produced by the weight of the arm and equate it to the sum of torques produced by the scale readings. From these equations, we can solve for the mass of the arm and the position of its center of mass. If needed, repositioning the scale under the hand can provide additional data points for a more accurate determination of the center of mass.
The free body diagram for the arm must include vectors representing the force of gravity acting at the center of mass (downward) and the normal forces exerted by the scales pointing upward (Fe and Fh). For additional clarity, distances from the shoulder joint to the point of force application by each scale should also be included in the diagram.