2.2 Equilibrium and Net Force

Equilibrium means the vector sum of all forces on an object is zero. This does not mean each force is zero; it means their combined effect cancels. An object in equilibrium can be at rest or moving in a straight line at constant speed. What matters is zero acceleration, not whether velocity is zero.

In one dimension, this is often direct addition with sign. In two dimensions, treat each axis separately: ΣFx = 0 and ΣFy = 0 for equilibrium. The method scales to complex setups with ropes and inclines because component equations preserve vector logic without requiring geometric guesswork.

Axis choice is a strategic decision. On an incline, one axis parallel to the slope and one perpendicular usually keeps equations shortest. Along the slope, weight contributes mg sinθ; perpendicular to slope it contributes mg cosθ. This often turns difficult-looking diagrams into two clean equations with one unknown each.

When solving, delay arithmetic until symbolic structure is complete. Writing component equations first reduces sign errors and reveals whether your result is physically plausible. If a normal force solves negative, your assumed contact state is inconsistent and the model must be reconsidered.

On an incline, choose +x up the plane and +y normal to the plane. For a resting block, resolve weight into mg sinθ down the plane and mg cosθ into the plane. The normal equation gives R = mg cosθ. The parallel equation gives f_s = mg sinθ, where static friction points up the plane because it opposes impending downward motion.

At the largest angle that still allows rest, friction is at its limit: f_s,max = μsR. Substituting R = mg cosθ and f_s = mg sinθ gives μs = tanθ_max. This gives a practical interpretation of μs: it encodes the steepest no-slip incline. If the surface were frictionless, the same setup would have net force mg sinθ down the plane and acceleration g sinθ.

Interpretation target: with m = 8 kg and μ = 0.40, the static ceiling is about μmg ≈ 31.4 N. At 31 N the model should remain stuck with near-zero net force; a tiny increase above that creates a clear acceleration. This is a direct numerical visualization of the static-friction inequality.