Free-Body Diagrams

11.1 Why Free-Body Diagrams Matter

A free-body diagram (FBD) is a sketch that shows a single object in isolation, with every external force acting on it represented as a labeled arrow. The diagram makes the force analysis concrete, visual, and systematic, preventing the most common errors in Newton’s second law problems: missing a force, double-counting a force, or assigning the wrong direction.

The discipline of drawing FBDs before writing equations is one of the clearest markers that separates expert problem-solvers from novices. Students who draw FBDs make fewer errors, catch mistakes earlier, and solve more complex problems correctly. The FBD is not a preliminary sketch you do quickly before the real work; it is the real work.

11.2 Isolating a System or Object

The first step in drawing an FBD is choosing the system: the single object, or group of objects, you want to analyze with Newton’s second law. This choice determines which forces are relevant.

To isolate the object, imagine removing it from its environment by cutting every physical contact and every field interaction. Whatever was attached to the object at those cut points exerts a force on the object that must be represented in the FBD. What the object exerts on others is not shown; only forces on the chosen object appear.

11.3 Internal vs. External Forces

External forces act on the chosen object from outside the system boundary. These are the only forces that affect the acceleration of the object or system, and they are the only ones that appear in the FBD.

Internal forces are forces between parts within a system boundary. When you treat a system of two or more objects as a single unit, forces between those objects are internal and cancel in pairs. They do not appear in the system’s FBD and do not affect the system’s acceleration.

11.4 Drawing Objects as Simplified Bodies

In an FBD, the object itself is drawn as a simple shape, almost always a dot, square, or rectangle. Detail is irrelevant. What matters is the point of application of each force and its direction.

• Use a filled circle for a point mass when rotation can be neglected.
• Use a square or rectangle for an extended object when shape affects force distribution.
• Draw the object in a clear central position with space on all sides for force arrows.
• Do not draw the environment in the FBD; draw only the isolated object and its force vectors.

11.5 Representing Forces with Vectors

Each force in an FBD is represented as an arrow with direction, magnitude, and label. Direction shows the physical direction of the force. Arrow length represents the force’s strength. The label identifies the force type and, when known, its magnitude and unit.

• Arrow direction: Gravity points down; normal force is perpendicular to the surface; tension pulls along the rope toward the attachment.
• Arrow length: Longer arrows represent larger forces. Use a consistent scale within one diagram.
• Label: Write the force symbol next to the arrow, such as W, N, T, f, or F_app.

11.6 Common Forces in Free-Body Diagrams

Most mechanics problems involve a small set of recurring force types. Recognizing each type, its direction, and what determines its magnitude is the core skill of FBD construction.

11.7 Weight Force

Weight is the gravitational force Earth, or another body, exerts on an object. It always points straight downward, toward Earth’s center, regardless of the object’s shape, orientation, or motion. Its magnitude is W = mg, where g = 9.81 m/s² near Earth’s surface.

Weight acts at the object’s center of mass. For a uniform object, this is the geometric center. For FBD purposes, it is drawn from the center of the body representation, pointing straight down.

11.8 Normal Force

The normal force is the contact force that a surface exerts on an object perpendicular to, and away from, the surface. “Normal” means perpendicular. The normal force is a surface’s mechanical response to compression; it prevents objects from passing through each other.

The normal force is not always equal to the object’s weight. It equals the weight only on a horizontal surface with no vertical acceleration and no additional vertical forces. On an incline, in an elevator, or with an applied force at an angle, the normal force differs from the weight.

11.9 Tension Force

Tension is the pulling force transmitted through a rope, string, cable, or chain. Tension always pulls; it cannot push. The force on the object at one end of a rope is directed along the rope, away from the object, toward the other end.

For an ideal massless, inextensible rope, the tension magnitude is the same at every point along the rope. When a rope passes over an ideal massless, frictionless pulley, the tension magnitude is unchanged; only the direction changes.

11.10 Friction Force

Friction is a contact force parallel to the surface, opposing relative motion or the tendency for relative motion. Kinetic friction opposes sliding motion. Static friction opposes the tendency to slide.

A common error is placing friction in the wrong direction. Ask which way the surface would slide if friction disappeared. Friction opposes that direction.

11.11 Applied Forces and Contact Forces

Applied forces are forces exerted directly by an agent, such as a hand, motor, or person. Their direction is given in the problem. If an applied force is not along a coordinate axis, it must be resolved into components.

When an applied force is at an angle θ above the horizontal, its horizontal component F cosθ and vertical component F sinθ both appear in the force analysis. The vertical component affects the normal force; the horizontal component drives horizontal acceleration.

11.12 Net Force from a Free-Body Diagram

Once all forces are drawn and labeled in the FBD, apply Newton’s second law to find the net force and acceleration. Resolve all forces into x and y components, then sum each component separately.

11.13 Free-Body Diagrams on Inclines

Inclined surface problems require a rotated coordinate system: one axis along the slope and one axis perpendicular to the slope. This makes the normal force entirely perpendicular to the surface and gravity partially along the slope and partially into the surface.

Gravity has components in the rotated system: along the slope, mg sinθ; perpendicular to the slope, mg cosθ. For a frictionless incline, the acceleration down the slope is a = g sinθ.

11.14 Multi-Object Free-Body Analysis

When two or more objects are connected by a rope, rod, or direct contact, draw a separate FBD for each object. Then apply Newton’s second law to each FBD. Connected objects often share the same acceleration magnitude, which acts as a constraint.

11.15 Common Free-Body Diagram Mistakes

1. Including forces the object exerts on others: An FBD shows only forces acting on the chosen object.
2. Drawing the net force as a force arrow: Net force is computed from the diagram; it is not an additional force.
3. Wrong direction for the normal force: Normal force is perpendicular to the contact surface and points away from it.
4. Assuming N = mg on inclines or in elevators: Normal force must be derived from the FBD equations.
5. Friction in the wrong direction: Kinetic friction opposes sliding; static friction opposes the tendency to slide.
6. Forgetting a force: Scan for gravity, contact, friction, tension, applied forces, and other given forces.

Chapter Summary

A free-body diagram isolates a single object and shows every external force acting on it as a labeled vector arrow. It is the essential first step for applying Newton’s second law to any mechanics problem.

• An FBD shows one object with all external forces as labeled arrows.
• Forces the object exerts on others are not shown.
• External forces appear in the FBD; internal forces within a chosen system do not.
• Common forces include weight, normal force, tension, friction, and applied force.
• Normal force is not always equal to weight; derive it from the FBD equations.
• On an incline, N = mg cosθ and the gravity component along the slope is mg sinθ.
• For connected objects, draw one FBD per object and use the shared acceleration as a constraint.
• The net force is computed from the FBD; it is never drawn as an additional force arrow.

Key Terms

free-body diagram (FBD)

A sketch showing a single isolated object with every external force represented as a labeled arrow.

external force

A force acting on the chosen object from outside the system boundary.

internal force

A force between parts within the chosen system boundary.

normal force

The contact force perpendicular to a surface and away from it.

tension

The pulling force transmitted through a rope, string, or cable.

system

The object or group of objects chosen for analysis.

incline problem

A mechanics problem involving a surface tilted at an angle.

constraint

A condition that restricts motion of connected objects.

Concept Review Questions

1. What does a free-body diagram show? What does it not show?
2. Why is it important to draw a separate FBD for each object in a multi-object system?
3. In what direction does the normal force always point? Give an example where it does not equal the object’s weight.
4. In what direction does tension act on an object attached to a rope?
5. Explain how to determine the direction of static friction for an object that has not yet started sliding.
6. Why should you never draw an arrow labeled F_net in a free-body diagram?
7. A box sits on a ramp at angle θ. What are the magnitudes of the normal force and the component of gravity along the ramp?
8. Two books are stacked. If you analyze only the bottom book, what forces act on it?

Diagram Construction Exercises

Draw the complete FBD for each object described. Label every force with its symbol and direction. If the magnitude is known, include it.

9. A 5.0 kg block rests on a horizontal frictionless table.
10. A 3.0 kg ball hangs from a single rope attached to the ceiling. The ball is at rest.
11. A 10.0 kg box slides to the right on a rough horizontal floor.
12. A 2.0 kg book rests against a vertical wall; a horizontal hand pushes the book into the wall with 15 N.
13. A 6.0 kg sled is pulled up a frictionless 20° ramp by a rope parallel to the ramp.
14. A 4.0 kg block hangs from a rope. A second rope below the block pulls it down with 10 N. Draw the FBD of the block.
15. A 50.0 kg person stands in an elevator that accelerates upward at 2.0 m/s².
16. A 12.0 kg box rests on a rough 35° incline and is in equilibrium.

Force Identification Practice

For each situation, list every force acting on the specified object, its direction, and what exerts it.

17. A car driving at constant velocity on a flat road.
18. A skydiver falling at terminal velocity.
19. A package resting on a conveyor belt that moves at constant speed.
20. A pendulum bob at the lowest point of its swing.
21. A satellite in circular orbit above Earth.

Inclined Plane Diagrams

For each problem, draw the FBD with the coordinate system rotated to align with the slope. Find the normal force and acceleration.

22. A 8.0 kg box slides down a frictionless 25° incline. Find N and a.
23. A 5.0 kg block on a 40° incline has μ_k = 0.30 and is sliding down. Find N, f_k, and a.
24. A 3.0 kg block is pushed up a frictionless 30° ramp by a force of 30.0 N parallel to the ramp. Find N and a.
25. A 10.0 kg block rests on a 20° incline. It does not slide. Find the minimum static friction force needed. Is f_s acting up or down the slope?
26. A 4.0 kg block on a 50° incline is held in place by a rope parallel to the slope with no friction. Find the tension in the rope.

Applied Newton’s Law Problems

27. Two blocks are stacked: Block A, 5.0 kg, on top of Block B, 8.0 kg, on a frictionless floor. A 26.0 N force pushes B to the right. The surfaces between A and B are rough with μ_s = 0.40. Do A and B move together or does A slide? Find the acceleration and the friction force on A.
28. Block A, 4.0 kg, and Block B, 6.0 kg, hang from either side of a pulley. Find the acceleration of the system and the tension in the rope. Assume a massless, frictionless pulley and massless rope.
29. A 2.0 kg block and a 3.0 kg block are connected by a rope on a horizontal frictionless surface. The 3.0 kg block is pulled by a 15.0 N force. Find the acceleration and the tension in the rope.
30. A 70.0 kg person stands on a scale in an elevator. The scale reads 840 N. Is the elevator accelerating up or down? What is the magnitude of the acceleration?
31. A 6.0 kg block on a 30° frictionless incline is connected by a string over a pulley to a hanging 4.0 kg block. Find the acceleration and tension.

Chapter 11 Checkpoint