How to Build a Net-Force Equation from a Free-Body Diagram

Start with the system boundary

A force equation is reliable only when every term refers to the same chosen system. Start by naming the body or collection of bodies whose motion you want to predict. Forces between parts inside that boundary are internal; forces exerted by objects outside it are external.

Draw forces, not motion

Represent the chosen body as a point or simple shape. Draw each external force that acts on it: weight, normal force, tension, friction, spring force, or an applied push or pull. Do not add velocity, acceleration, “motion force,” or a separate net-force arrow as though it were another interaction.

Choose axes after identifying the geometry. Resolve only the forces that are not already aligned with those axes. Then write one component equation per direction:

$$\sum F_x=ma_x, \qquad \sum F_y=ma_y.$$

Check a new example

Suppose an $11\,\mathrm{kg}$ sled is pulled horizontally with $30\,\mathrm{N}$ while kinetic friction is $8\,\mathrm{N}$ in the opposite direction. Vertical acceleration is zero, so the normal force balances weight. Horizontally,

$$\sum F_x=30-8=22\,\mathrm{N},$$

and

$$a_x=\frac{22}{11}=2.0\,\mathrm{m/s^2}.$$

The subtraction follows from the chosen positive direction, not from a rule that friction is always negative. If the axis were reversed, both the force signs and the acceleration sign would reverse while the physical motion stayed the same.

Use three closing checks

1. Every force has a named agent outside the system.
2. Units reduce to newtons on the force side and $\mathrm{m/s^2}$ after division by mass.
3. The calculated acceleration points in the same direction as the net-force vector.

Separate free-body diagrams are usually clearer for connected objects. Combining objects into one system can remove their mutual tension from the external-force sum, but it also changes which acceleration and mass belong in the equation.