Free-body diagram

A free-body diagram (FBD) is a diagram in which one chosen object or system is isolated from its surroundings and every relevant external force acting on that object or system is represented explicitly. The purpose of an FBD is analytical rather than artistic: it separates the object of interest from distracting geometric detail and provides the force model from which Newton's equations are written. A simple example is given in Figure 1.

Figure 1. A hanging mass with upward tension and downward weight.

For a particle or translating rigid body of constant mass,

$\displaystyle \sum \mathbf F_{\mathrm{ext}}=m\mathbf a.$

For a rigid body the same diagram also supplies the forces and moment arms used in rotational dynamics,

$\displaystyle \sum \boldsymbol\tau_O=\frac{d\mathbf L_O}{dt},$

or, in an appropriate fixed-axis special case,

$\displaystyle \sum\tau=I\alpha.$

The most important word in the definition is chosen. Before drawing force arrows, one must decide what object or collection of objects is the system. Whether a force is external or internal depends on this system boundary.

System boundary

A system boundary separates the chosen system from everything else. Forces exerted across the boundary by the surroundings are external and belong on the free-body diagram. Forces between parts of a multi-body system are internal and normally do not appear on a free-body diagram of the whole system.

For example, consider two blocks tied together. If only one block is selected, the string tension must appear on that block's FBD. If both blocks are selected together, the tension is internal to the two-block system and disappears from the system-level FBD. Choosing a useful system boundary is therefore an important problem-solving strategy.

A seven-step construction procedure

A reliable free-body diagram can be constructed by the following sequence.

Choose the system. State whether the system is one particle, one rigid body, or several bodies treated together.
Sketch the isolated body. For particle translation, a point or simple box is usually sufficient.
Identify every interaction crossing the system boundary.
Replace each interaction by a force vector acting on the chosen system.
Label forces by physical origin, for example $m\mathbf g$ , $\mathbf N$ , $\mathbf T$ , or $\mathbf f$ .
Choose coordinate axes that simplify the component equations.
Write Newton's equations only after the diagram is complete.

A useful final question is: “For every force arrow, what body in the surroundings exerts this force on the chosen system?” If that question cannot be answered, the arrow may not represent a real interaction.

Forces commonly appearing on free-body diagrams

Weight

Near the surface of Earth,

$\displaystyle \mathbf W=m\mathbf g,$

directed approximately toward the center of Earth. Weight should not be replaced by components until coordinates have been selected. On an incline, $mg\sin\theta$ and $mg\cos\theta$ are components of one gravitational force, not additional forces.

Normal force

A normal force is a contact force perpendicular to a surface. Its magnitude is not automatically $mg$ . It must be determined from the equations of motion and the geometry of contact.

Tension

An ideal flexible string or cable pulls along its own direction. In the common ideal model of a massless string over frictionless massless pulleys, the tension magnitude is the same throughout a continuous string.

Friction

Friction acts tangentially to a contact surface and opposes relative sliding or the tendency to slide. Static friction satisfies

$\displaystyle \vert f_s\vert\leq \mu_sN,$

whereas a common kinetic-friction model is

$\displaystyle f_k=\mu_kN.$

The equality $f_s=\mu_sN$ applies only at impending slip.

Spring force

For an ideal linear spring,

$\displaystyle \mathbf F_s=-k\mathbf x,$

where the minus sign indicates a restoring force.

Drag

Common approximations are

$\displaystyle \mathbf F_d=-b\mathbf v$

for linear drag and

$\displaystyle \mathbf F_d=-c\vert\mathbf v\vert\mathbf v$

for quadratic drag.

Forces are not their components

Suppose an applied force of magnitude $F$ makes angle $\theta$ with the positive $x$ -axis. One may write

$\displaystyle \mathbf F=F\cos\theta\,\hat{\mathbf i}+F\sin\theta\,\hat{\mathbf j}.$

The component terms are a representation of the same force; they are not additional interactions. The same warning applies to the components of weight on an inclined plane.

Newton's third law and FBDs

Newton's third law states

$\displaystyle \mathbf F_{A\to B}=-\mathbf F_{B\to A}.$

The two forces act on different bodies and therefore do not normally appear together on a single body's free-body diagram. For a book on a table, the table's normal force on the book and the book's weight are not a third-law pair because both forces act on the book.

Coordinate choices

Good axes simplify the equations.

For horizontal surfaces, horizontal and vertical axes are usually natural.
For inclined planes, axes parallel and perpendicular to the plane are usually best.
For circular motion, radial and tangential directions are often most useful.

No separate “centripetal force” should be added in an inertial frame. Centripetal force means the radial component of the net real force,

$\displaystyle \sum F_r=m\frac{v^2}{r}.$

Diagram gallery

Hanging mass

Figure 1. A hanging mass with upward tension and downward weight.

For a hanging mass attached to a Light string,

$\displaystyle T-mg=ma_y.$

Block on a horizontal surface

Figure 2. A rough horizontal surface with applied force, friction, normal force, and weight.

If vertical acceleration is zero,

$\displaystyle N-mg=0,$

and if the block slides,

$\displaystyle F_a-f_k=ma, \qquad f_k=\mu_kN.$

Block on an inclined plane

Figure 3. A block on an inclined plane.

With axes parallel and perpendicular to the plane,

$\displaystyle W_{\parallel}=mg\sin\theta, \qquad W_{\perp}=mg\cos\theta.$

Rough-surface block pulled at an angle

Image free_body_diagram_of_a_rough_surface_block

Figure 4. A rough surface block pulled by an oblique force.

The vertical component of the pull changes $N$ and therefore changes the friction magnitude.

Block against a vertical wall

Image free_body_diagram_of_block_against_wall

Figure 5. A block held against a vertical wall.

If the block tends to slide downward, static friction acts upward.

Mass-spring system

Figure 6. A horizontal mass-spring system.

For an ideal spring,

$\displaystyle F_s=-kx.$

Pendulum bob

Figure 7. A pendulum bob with tension and weight.

Radial and tangential directions are especially convenient for pendulum dynamics.

Banked curve

Figure 8. A vehicle represented on a banked curve.

The normal force need not be vertical; its horizontal component can contribute to centripetal acceleration.

Loop-the-loop at the top

Figure 9. At the top of an inside loop, both N and mg point toward the center.

The radial equation is

$\displaystyle N+mg=m\frac{v^2}{r}.$

Atwood machine

Figure 10. An Atwood-machine system sketch and separate FBDs for the two masses.

For $m_2>m_1$ and an ideal string and pulley,

$\displaystyle T-m_1g=m_1a,$

$\displaystyle m_2g-T=m_2a,$

$\displaystyle a=\frac{(m_2-m_1)g}{m_1+m_2}.$

Worked examples

Example 1: horizontal pull

A $5.0\,\mathrm{kg}$ block is pulled horizontally by a $20\,\mathrm{N}$ force on a frictionless horizontal surface. The FBD gives

$\displaystyle 20=5.0a, \qquad N-5.0g=0.$

Hence

$\displaystyle a=4.0\,\mathrm{m/s^2}, \qquad N=49\,\mathrm{N}.$

Example 2: pull at an angle

A $10\,\mathrm{kg}$ block is pulled with force $50\,\mathrm{N}$ at $30^\circ$ above horizontal. The vertical equation is

$\displaystyle N+50\sin30^\circ-mg=0,$

$\displaystyle N=mg-25\,\mathrm{N}.$

If kinetic friction is present, its magnitude is $\mu_kN$ , showing why the vertical component of an oblique pull changes the horizontal friction force.

Example 3: frictionless incline

For a block on a frictionless incline of angle $\theta$ ,

$\displaystyle mg\sin\theta=ma, \qquad N=mg\cos\theta,$

$\displaystyle a=g\sin\theta.$

Example 4: rough incline

If a block slides down a rough incline,

$\displaystyle mg\sin\theta-f_k=ma, \qquad f_k=\mu_kmg\cos\theta,$

therefore

$\displaystyle a=g(\sin\theta-\mu_k\cos\theta).$

Example 5: terminal speed with linear drag

For downward positive direction,

$\displaystyle mg-bv=m\frac{dv}{dt}.$

At terminal speed, $dv/dt=0$

, hence

$\displaystyle v_t=\frac{mg}{b}.$

Example 6: car at the top of a loop

At the top of an inside loop, inward is downward. The real inward forces are $N$ and $mg$ :

$\displaystyle N+mg=m\frac{v^2}{r}.$

The minimum speed for contact occurs when $N=0$

$\displaystyle v_{\min}=\sqrt{gr}.$

Common mistakes

Omitting an external interaction.
Adding a force that acts on another body rather than the chosen body.
Drawing both members of a Newton's-third-law pair on one body's FBD.
Drawing both a force and its components as separate forces.
Treating “centripetal force” as an extra force.
Assuming without checking the vertical or normal equation.
Assuming static friction always equals $\mu_sN$ .
Choosing axes that make the geometry unnecessarily complicated.

Non-inertial frames

The diagrams above assume an inertial frame. If Newton's second law is written directly in an accelerating or rotating frame, inertial-force terms such as centrifugal and Coriolis forces may be introduced. They should be labeled explicitly as frame-dependent inertial terms rather than being confused with physical interactions such as gravity or contact forces.

Bridge to analytical mechanics

Free-body diagrams are central to Newtonian mechanics because they make individual forces explicit. In Lagrangian mechanics, ideal constraint forces can often be eliminated by choosing generalized coordinates adapted to the constraints. The conceptual progression is therefore

identify forces and constraints $\displaystyle \longrightarrow$ choose coordinates adapted to the constraints $\displaystyle \longrightarrow$ derive equations from $\displaystyle L=T-U.$

Free-body-diagram reasoning remains valuable even when the final equations are obtained from a variational principle because it clarifies the physical interactions and constraint assumptions.

Bibliography

1: J. Moore and contributors, Mechanics Map, Engineering LibreTexts. Creative Commons Attribution–ShareAlike 4.0. Mechanics Map
2: T. Weideman, UCD Physics 9A – Classical Mechanics, Physics LibreTexts. Creative Commons Attribution–ShareAlike 4.0. UCD Physics 9A

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