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In dynamics, ΣF = ma is applied component-by-component in the chosen coordinate system to find acceleration given forces, or to find required forces given a desired motion. In Cartesian form: ΣFx = max, ΣFy = may. In normal-tangential form: ΣFt = maₜ = m(dv/dt), ΣFn = maₙ = mv²/ρ. In polar form: ΣFr = m(r̈ − rθ̇²), ΣFθ = m(rθ̈ + 2ṙθ̇). The FBD shows only real forces; ma is kept on the equation's right side as the kinetic resultant.
Draw the FBD and a separate kinetic diagram (showing the ma vector) side by side. Choose the coordinate system consistent with the kinematics. For circular motion, identify centripetal acceleration direction explicitly to avoid sign errors.
Newton's second law ΣF = ma looks simple, but applying it to particle dynamics requires two things you did not need in statics: a careful choice of coordinate system and a clean separation between the free-body diagram (real forces) and the kinetic diagram (the ma resultant). In statics ΣF = 0, so the coordinate system barely matters. In dynamics, the choice of coordinates determines how messy your algebra gets.
For Cartesian coordinates the equations are ΣFx = max and ΣFy = may, with constant unit vectors i and j. This works well for straight-line or projectile-type problems. For motion along a curved path, normal-tangential (n-t) coordinates are often cleaner: ΣFt = maₜ = m(dv/dt) gives the rate at which speed changes, while ΣFn = maₙ = mv²/ρ relates the net inward force to the centripetal requirement. Notice that even at constant speed (aₜ = 0), a net inward force is still needed to keep the particle curving — this is a common point of confusion. For problems phrased in terms of an angle from a fixed point, polar coordinates give ΣFr = m(r̈ − rθ̇²) and ΣFθ = m(rθ̈ + 2ṙθ̇), where the −rθ̇² and 2ṙθ̇ terms encode centripetal and Coriolis effects respectively.
The most persistent error in dynamics problems is treating ma as a force on the FBD — sometimes called a "D'Alembert inertia force." This leads students to write ΣF = 0 (equilibrium!) by moving ma to the left side, which is mathematically equivalent but pedagogically dangerous because it blurs the distinction between real forces and acceleration. Draw the FBD with real forces only, then set their sum equal to the mass times acceleration. Keeping these diagrams separate is the professional engineering practice for a reason: it prevents sign errors and makes the physics transparent.
When setting up any dynamics problem, the workflow is: (1) draw the FBD with all real forces, (2) choose the coordinate system that aligns naturally with the motion (circular path → n-t, angle-defined path → polar), (3) write the scalar equations component by component, and (4) bring in kinematic relationships from curvilinear kinematics to express acceleration in terms of position, speed, or time. Problems that combine a force law with a kinematic constraint — like a bead on a rotating rod — require all three coordinate-system forms to work together.