Free-body diagrams Knowledge-Based Study

Study notes

1. What Is a Free-Body Diagram?

   A free-body diagram (FBD) is a simplified graphical representation used to visualize the forces acting upon a specific isolated body or system. In physics, isolating the object helps determine the net external force and analyze the resulting motion using Newton's laws.

   Key principles of an FBD include:

   • Isolating the object: Represent the body as a simplified point particle or a single rigid block, removing all surrounding surfaces, ropes, and supports.
   • External forces only: Only draw forces exerted on the chosen object by its environment. Never include internal forces or forces that the object exerts on other bodies.
   • Vector representation: Each force is drawn as an arrow starting at the object (or its center of mass) and pointing strictly in the direction of the force. Arrow lengths should qualitatively reflect the magnitude of each force.

   Identifying External Forces

   When constructing a free-body diagram, systematic identification of forces prevents common errors such as missing contact forces or inventing non-existent ones.

   • Gravitational Force ($F_g$ or $W$): Always acts vertically downward toward the center of the Earth, regardless of the incline or state of motion. Its magnitude is given by $F_g = mg$.
   • Normal Force ($F_N$ or $N$): A contact force exerted perpendicular to the surface of contact. It pushes away from the surface, preventing objects from passing through each other.
   • Frictional Force ($f_s$ or $f_k$): A contact force acting parallel to the interface between two surfaces, resisting relative motion or tendency toward motion.
   • Tension ($T$): A contact pulling force exerted by a taut string, cord, cable, or chain, directed strictly along the length of the connector away from the object.
   • Applied Force ($F_{app}$): An external push or pull directly exerted by an external agent.

   Step-by-Step Construction

   1. Define the system: Clearly identify the single object being analyzed.
   2. Set up a coordinate system: Choose perpendicular coordinate axes ($x$ and $y$). For an object on an inclined plane, aligning one axis parallel to the surface and the other perpendicular to it simplifies the equations.
   3. Draw and label all forces: Sketch force vectors originating from the center. Do not include velocity, acceleration, or momentum vectors on the diagram itself, as they are kinematic quantities, not forces.
   4. Resolve components: Any force not aligned with the chosen axes is broken into components:$$F_x = F \cos\theta$$$$F_y = F \sin\theta$$

   Connecting FBDs to Equations of Motion

   Once the free-body diagram is complete, Newton's second law is applied independently along each orthogonal axis:

   $$\sum F_x = m a_x$$$$\sum F_y = m a_y$$

   Consider an object of mass $m$ sliding down a frictionless incline tilted at an angle $\theta$ above the horizontal:

   • Forces acting: The downward gravitational force $F_g = mg$ and the perpendicular normal force $F_N$.
   • Component along the incline: The component of gravity parallel to the surface is $F_{g,\parallel} = mg \sin\theta$. Thus, $\sum F_x = mg \sin\theta = m a_x$, leading to $a_x = g \sin\theta$.
   • Component perpendicular to the incline: The normal force balances the perpendicular component of gravity: $\sum F_y = F_N - mg \cos\theta = 0$, so $F_N = mg \cos\theta$.