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2.2.1.4. Free-Body Diagrams

Table of Contents

Isolating the Object

A free body diagram is a simple drawing that shows all the external forces acting on one chosen object. The idea is to isolate the object from everything around it and represent each interaction by a force arrow. This helps turn a physical situation into a clear physics problem.

The phrase "free body" means the object is considered by itself, as if it were separated from its surroundings. You do not draw the entire scene in detail. You draw only the object of interest and the forces acting on it.

For example, if a book rests on a table, the book interacts with Earth and with the table. In the free body diagram for the book, you draw the book as a dot or box, then draw the weight downward and the normal force upward. You do not draw forces that the book exerts on other objects, because the diagram is only about forces acting on the chosen object.

A free body diagram includes only forces acting on the chosen object, not forces the object exerts on something else.

Why Free-Body Diagrams Matter

In mechanics, the motion of an object depends on the net force acting on it. A free body diagram lets you identify those forces before writing equations. Without this step, it is easy to miss a force, add a force that does not exist, or confuse an action-reaction pair.

A good free body diagram answers three questions clearly. What is the object? What forces act on it? In what directions do those forces act?

Once the forces are identified, Newton's second law can be applied along chosen axes.

After drawing a free body diagram, the main next step is to apply Newton's second law to the forces on that object:
$$\sum \vec{F} = m\vec{a}$$

How to Draw a Free-Body Diagram

The method is systematic. First choose the object you want to analyze. Then replace the object by a simple shape, often a box or a dot. Next identify every external interaction. Each interaction gives one force on the object. Draw each force as an arrow starting on the object and pointing in the direction of that force. Label each arrow clearly.

If the object is on a surface, there may be a normal force, and possibly friction. If the object is attached to a rope, there may be tension. If the object is near Earth, there is usually weight. If someone pushes or pulls it, include that applied force.

The length of arrows does not need to be perfectly exact unless the problem gives enough information. The most important features are the correct force names and directions.

Common Forces Seen in Free-Body Diagrams

Many beginner problems use a small set of forces. The table below shows the most common ones.

ForceSymbol often usedTypical directionCause
Weight$W$ or $mg$Downward, toward EarthGravity
Normal force$N$Perpendicular to surfaceContact with surface
Tension$T$Along rope or stringPull of rope or string
Friction$f$Along surface, opposes relative motion or tendency of motionContact between surfaces
Applied force$F_{\text{app}}$Depends on push or pullExternal agent
Drag or air resistance$F_d$Opposes motion through fluidFluid interaction

The normal force is not always upward. It is always perpendicular to the contact surface. On a slope, it points perpendicular to the slope, not vertically.

Friction is not always opposite velocity. More precisely, it opposes relative motion or the tendency for relative motion between surfaces.

Tension acts along the rope or string, pulling away from the object.

What Not to Include

A free body diagram becomes confusing if extra things are added. Do not include objects that are not part of the chosen body, except perhaps as a light sketch for orientation if really needed. Do not draw velocity or acceleration arrows as if they were forces. Velocity and acceleration may be useful separately, but they are not forces.

Also do not include "internal forces" if the chosen system contains multiple parts and those parts pull or push on each other. A free body diagram for a single object usually includes only external forces on that object.

Velocity, acceleration, and motion direction are not forces. Do not place them on a free body diagram as force arrows.

Example, Book on a Table

Consider a book at rest on a horizontal table. The book has two main external forces. Its weight acts downward. The table exerts a normal force upward.

If the book is at rest and not accelerating vertically, then the upward and downward forces balance.

$$\sum F_y = N - mg = 0$$

So in this case,

$$N = mg$$

This equality is true here because the acceleration is zero in the vertical direction. It is not a universal rule for every contact situation.

Free-body diagram of a book on a table

Example, Box Pulled Across a Floor

Now consider a box on a rough floor pulled to the right by a rope or a person. The forces may be weight downward, normal force upward, applied force to the right, and friction to the left.

If the box accelerates horizontally, Newton's second law in the horizontal direction becomes

$$\sum F_x = F_{\text{app}} - f = ma_x$$

In the vertical direction, if there is no vertical acceleration,

$$\sum F_y = N - mg = 0$$

This shows why choosing axes is useful. Each direction can be treated separately.

Free-body diagram of a box pulled on a rough surface

Example, Object on an Inclined Plane

An inclined plane is one of the most important uses of free body diagrams. If a block rests or moves on a slope, its weight still acts vertically downward. The normal force acts perpendicular to the surface. Friction, if present, acts along the surface.

In these problems, it is often helpful to choose axes parallel and perpendicular to the incline. Then the forces can be analyzed more easily. The weight may be resolved into components along those axes, while the normal force stays entirely perpendicular to the surface.

You should remember that the free body diagram must show the real forces, not the components and the original force in a mixed and confusing way. If you choose to show components of weight, do so clearly and consistently.

Free-body diagram of a block on an incline

Choosing the Correct Object

The most important decision is often the first one, choosing what object the diagram is for. If a problem involves two blocks, you may need one free body diagram for each block. If a problem treats both blocks together as one system, then you draw one diagram for the combined system.

This choice affects which forces appear. A tension force between two parts of a chosen system may disappear from the external force list if both parts are included together. So the diagram depends on the object or system being analyzed.

Relation to Equilibrium

A free body diagram is especially useful when an object is at rest or moving with constant velocity. In such cases the acceleration is zero, so the net force is zero.

$$\sum \vec{F} = \vec{0}$$

That does not mean no forces act. It means the forces balance.

For example, a hanging lamp is not force-free. It has weight downward and tension upward. The diagram shows both forces, and equilibrium means they cancel.

Equilibrium does not mean absence of forces. It means the vector sum of all forces is zero:
$$\sum \vec{F} = \vec{0}$$

Common Mistakes

One common mistake is drawing both members of an action-reaction pair on the same free body diagram for one object. For instance, for a book on a table, the force of the table on the book belongs on the book's diagram, but the force of the book on the table does not.

Another common mistake is assuming that $N = mg$ always. This is only true in some special cases, such as a book at rest on a horizontal table with no additional vertical forces.

A third mistake is forgetting friction when surfaces are rough, or adding friction when the surface is explicitly frictionless.

A fourth mistake is drawing force arrows in directions that follow motion rather than interaction. Forces come from interactions, not simply from the way the object moves.

A Simple Checklist

When you finish a free body diagram, check it by asking whether the object is isolated, whether every arrow is a real external force, whether each force has a clear label, and whether the directions make physical sense. Then choose axes and write Newton's law component by component.

Free-body diagram checklist:
$$\text{Choose object} \rightarrow \text{Identify external forces} \rightarrow \text{Draw and label arrows} \rightarrow \text{Apply } \sum \vec{F} = m\vec{a}$$

Final Picture

A free body diagram is one of the most powerful tools in classical mechanics because it turns words and situations into force equations. For beginners, the main skill is not artistic drawing, but careful thinking. Identify the object, identify the interactions, and represent only the forces acting on that object. Once that is done, the physics becomes much easier to organize and solve.

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