When it comes to studying the behavior of objects in mechanics, one of the most important tools available is the free-body diagram. Consider a simple example of a system of two blocks coupled by a massless string over a frictionless pulley. Block 1 is sliding over a table pulled by block 2 as block 2 falls under gravity.

To find the acceleration of the system, it is first necessary to calculate the net force on the system. In order to accomplish this, a free-body diagram can be created to represent the two blocks, which is considered as a single system isolated from the environment. The external forces are then identified and represented by vector arrows on the diagram.

When looking at block 1, several forces are acting on it. These include its weight, the normal force, the tension force due to the rope, and the friction force that opposes the tension force, thus causing the motion. For block 2, its weight and the tension forces are the only forces acting on it.

Once the free-body diagram has been created, it is possible to apply Newton's second law of motion to the blocks. By doing this, an equation for the net force acting on the system can be derived. This equation will include the masses of the blocks, the acceleration of the system, and the forces acting on each block.

With the equation in hand, it is then possible to use algebraic operations to rearrange the terms and solve the magnitude of the acceleration of the system. The result will be a numerical value that represents how the system will behave under the given forces.

The importance of a free-body diagram in this situation cannot be overstated. It provides a clear and concise representation of the forces at play, thus allowing for an easier and more accurate calculation of the acceleration of the system. Without this tool, it would be much more difficult to understand and analyze the behavior of the blocks and the forces acting on them.