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Understanding the forces acting on an object is crucial for analyzing real-world problems in physics. A free-body diagram is a graphical representation that shows all the forces acting on an object. It helps to visually simplify complex scenarios. For objects in equilibrium, these diagrams are particularly useful as they set the stage for calculating net forces and torques.

When constructing a free-body diagram, each force is represented by an arrow pointing in the direction in which it acts, with its tail at the point of application on the object. The length of the arrow reflects the magnitude of the force. In the case of the uniform plank, we identify the gravitational forces due to the weights of the plank and the painter, as well as the reactive forces from the scaffold's bars. To solve for equilibrium problems, it's also helpful to specify the pivot point, which is where we'll assess the torques.

Torque is a measure of the force causing an object to rotate about an axis or pivot. It is calculated as the product of the force and the perpendicular distance from the pivot point to the line of action of the force. Mathematically, it's expressed as \( \tau = r \times F \), where \( r \) is the distance to the pivot and \( F \) is the force applied.

Torque and equilibrium are closely linked in physics problems involving rotational motion. An object is in rotational equilibrium when the sum of all torques acting on it is zero, meaning that it will not start rotating spontaneously. In our plank scenario, the torques due to the plank's weight and the applied weight of the painter about the scaffold's second bar must balance for the plank not to tip. By setting the sum of these torques to zero, we can determine the maximum distance the painter can venture out on the overhanging part of the plank.

The center of mass is a point representing the mean location of the mass distribution of an object or system. For uniform, symmetrical objects, such as the plank described in the exercise, the center of mass is at the geometric center. This concept is crucial because the weight of the object acts as though it's concentrated at this point.

If we understand where the center of mass is, we can predict how an object will balance or tip. The plank will remain in equilibrium as long as its center of mass remains over the area supported by the scaffold. Once the painter's weight adds enough counter-torque to shift the plank's center of mass off the supporting scaffold, the plank will tip. Knowing the location of the center of mass allows us to use its distance from the pivot point to calculate the torques and solve equilibrium problems accurately.