Drawing Free-Body Diagrams

A free-body diagram (FBD) is a stripped-back sketch that shows a single object in isolation, with every force acting on it represented as an arrow. The arrow's direction shows where the force points, its length suggests the force's relative magnitude, and it is drawn starting from (or pointing toward) the object's centre of mass. By removing all surrounding objects and replacing their influence with force vectors, an FBD transforms a complicated real-world scene into a clean mathematical picture that Newton's second law can be applied to directly. Every force you draw must be one that is physically real and currently acting on your chosen object. Common forces include gravitational weight (W = mg, always pointing straight down toward Earth's centre), the normal force (perpendicular to any surface the object rests on), friction (parallel to the surface, opposing relative motion), tension (along a rope or cable, pulling the object), and applied forces from pushes or contacts. A crucial rule: you never draw forces that the object exerts on other things — only forces exerted on the object itself. Free-body diagrams are the essential first step in solving virtually every dynamics problem. Once drawn correctly, you choose a coordinate system, resolve each force vector into components, and sum them to apply ΣF = ma. A carefully drawn FBD prevents sign errors and missing forces — the two most common reasons students lose marks in mechanics calculations.

Key takeaways
  • Identify the system object
  • List every force acting ON the object
  • Draw and label each force arrow
  • Check Newton's third-law pairs — then exclude them
  • Set up a coordinate system and resolve

Identify the system object

Decide which single object you are analysing. Draw a dot or simple box to represent it. Everything else in the problem — the table, the rope, the planet Earth — becomes an 'outside agent' whose influence you will represent as a force arrow, not as a drawn object.

List every force acting ON the object

Work through a mental checklist: (1) Is there gravity? Always yes, near Earth's surface — draw W = mg downward. (2) Is the object in contact with a surface? Draw a normal force perpendicular to that surface. (3) Is there a rope, spring or rod attached? Draw tension or compression along it. (4) Is there sliding or impending motion? Draw kinetic or static friction opposing that motion. (5) Any other applied forces, air resistance, or buoyancy? Add those too.

Draw and label each force arrow

Each force is a vector arrow originating from the object's centre of mass (for point-object FBDs). The arrow must point in the correct physical direction. Label each arrow with a clear symbol: W or F_g for weight, N or F_N for normal force, T for tension, f or F_f for friction, F_app for an applied force. Use subscripts if multiple forces of the same type act simultaneously.

Check Newton's third-law pairs — then exclude them

Newton's third law tells you that for every force your object experiences, it exerts an equal and opposite force back on the other object. For example, the table pushes the book upward (normal force on book) and the book pushes the table downward (normal force on table). The FBD for the book includes only the upward normal force — never the reaction force the book exerts on the table. This is the most common FBD mistake.

Set up a coordinate system and resolve

Once all forces are drawn, choose x and y axes that simplify the algebra — often aligning one axis with the direction of acceleration (e.g., along an incline). Resolve every force into its x and y components, then apply ΣF_x = ma_x and ΣF_y = ma_y separately. The FBD is complete when it contains all real forces, no fictitious ones, and every arrow is labelled.

The Key Equation

F⃗net=∑iF⃗i=ma⃗\vec{F}_{\text{net}} = \sum_{i} \vec{F}_i = m\vec{a}
  • \vec{F}_{\text{net}} — Net force (N (newtons)): The single resultant vector you obtain by adding all individual force vectors from the FBD head-to-tail (or by summing components).
  • \vec{F}_i — Individual force (N (newtons)): Each separate force arrow drawn on the FBD — weight, normal, friction, tension, applied force, etc.
  • m — Mass (kg (kilograms)): The scalar measure of the object's inertia; stays constant regardless of where the object is or how many forces act on it.
  • \vec{a} — Acceleration (m s⁻² (metres per second squared)): The resulting acceleration vector. If the FBD is drawn correctly and the net force is zero, this is zero (the object is in equilibrium or moving at constant velocity).

Worked Example

Problem. A 5.0 kg book rests on a horizontal table. A student pushes it horizontally with an applied force of 12 N. The coefficient of kinetic friction between the book and table is 0.20. Draw the free-body diagram, then find the book's acceleration.

a=Fapp−fkm=12−9.85.0=0.44 m s−2a = \frac{F_{\text{app}} - f_k}{m} = \frac{12 - 9.8}{5.0} = 0.44 \ \text{m s}^{-2}

Answer. The book accelerates at 0.44 m/s² in the direction of the applied force.

Figure
Diagram
Free-Body Diagram: Book on a TableBook5.0 kgTable surfaceN = 49 NW = 49 Nf_k9.8 NF_app12 N■ Normal (N)■ Weight (W)■ Applied (F_app)■ Friction (f_k)
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton formulated the three laws of motion published in his landmark 1687 work Philosophiæ Naturalis Principia Mathematica. His second law — that net force equals mass times acceleration — is the very equation that free-body diagrams are designed to apply. By systematically isolating objects and cataloguing every force acting upon them, physicists put Newton's framework into practice exactly as he envisioned. Newton reportedly developed much of his foundational work on motion and gravity during 1665–1666, when Cambridge University was closed due to the Great Plague. He referred to this as his 'year of wonders' (annus mirabilis) — arguably the most productive enforced study leave in scientific history.

  • Newton's Second Law — Free-body diagrams are drawn specifically to apply Newton's second law: once all force vectors are identified and summed, ΣF = ma gives the object's acceleration directly.
  • Newton's Third Law — Understanding action–reaction pairs is essential for drawing correct FBDs — you must identify which forces act on your chosen object and deliberately exclude the reaction forces it exerts on other objects.
  • Friction Forces — Friction is one of the most commonly included forces in FBDs; knowing whether to draw static or kinetic friction, and in which direction, is a key skill built on FBD practice.
  • Normal Force and Weight — Weight and normal force are almost always the first two arrows drawn on an FBD; understanding why they are equal only in specific equilibrium situations (and not always) deepens FBD accuracy.