Free-body diagrams and applying Newton's second law

Why free-body diagrams matter

A free-body diagram is a simplified drawing that shows all external forces acting on one chosen object or system. It is one of the most important problem-solving tools in mechanics. Many errors in Newton's law problems come from missing a force, adding a force that does not belong, or confusing forces on different objects.

The word free means the object is isolated from its surroundings in the diagram. The surroundings are replaced by force arrows.

Step 1: choose the object

Before drawing forces, decide what object or system you are analyzing. Is it a single block? Two connected blocks? A person in an elevator? A cart plus hanging mass? The forces depend on this choice.

If you analyze a single block, tension from a rope may be an external force. If you analyze the block and rope together, that same interaction may become internal or require a different model.

Step 2: identify real forces

Only real forces belong on a free-body diagram. Common forces include weight, normal force, tension, friction, spring force, drag, thrust, and applied pushes or pulls.

Weight near Earth is

ec{W}=mec{g}

The normal force is perpendicular to a contact surface. Tension acts along a rope or string, pulling away from the object. Friction acts along a surface, opposing actual or impending relative motion.

Do not draw velocity or acceleration as forces. They are motion quantities, not interactions.

Step 3: choose axes

Choose coordinate axes to simplify the equations. If an object moves on a horizontal surface, horizontal and vertical axes are natural. For an incline, it is often best to choose one axis parallel to the plane and one perpendicular to it.

Newton's second law is then written component by component:

$sum F_x=ma_x$

$sum F_y=ma_y$

Components allow vector equations to become scalar equations.

Example: block on a horizontal surface

A block of mass $m$ is pulled horizontally by force $F$ on a frictionless table. Forces are weight $mg$ downward, normal force $N$ upward, and applied force $F$ horizontally.

Vertical acceleration is zero, so

$N-mg=0$

Horizontal motion gives

$F=ma$

Thus

a=rac{F}{m}

Example: block on an incline

For a block on a frictionless incline at angle $heta$, choose axes parallel and perpendicular to the surface. Weight has components:

$mgsin heta$

parallel down the incline, and

$mgcos heta$

perpendicular into the surface.

The normal force is

$N=mgcos heta$

The acceleration down the incline is

$a=gsin heta$

This result shows that steeper inclines produce larger acceleration.

Constraint equations

Many systems contain constraints. If two blocks are connected by a taut, massless string over an ideal pulley, they often have accelerations of equal magnitude. If an object is constrained to remain on a surface, acceleration perpendicular to that surface may be zero.

Constraint equations are not force laws, but they are necessary for solving systems.

Common mistakes

A common mistake is assuming $N=mg$ in every problem. That is true only in certain cases. On an incline, in an accelerating elevator, or with extra vertical forces, the normal force differs from $mg$.

Another mistake is adding a vague force called motion force. Motion does not require a force in its direction. Forces cause acceleration, not velocity.

The big idea

Free-body diagrams translate physical situations into mathematical equations. The method is systematic: choose the object, draw only real external forces, choose useful axes, write Newton's second law by components, include constraints, and solve. This process is the backbone of classical dynamics.