Sketch the object → choose axes → resolve forces → add components → use ΣF = ma → check units and signs.
| Situation | Relationships |
|---|
| Vectors (angle from right) | Fₓ = F cos θ; Fᵧ = F sin θ
|ΣF| = √[(ΣFₓ)² + (ΣFᵧ)²]
aₓ = ΣFₓ/m; aᵧ = ΣFᵧ/m |
| Surface with a pull | θ: ramp angle; α: applied-force angle measured from up the ramp toward the outward normal.
N = mg cos θ − F sin α, while contact holds and a⊥ = 0.
ΣF∥ = F cos α − mg sin θ + f
On a flat floor θ = 0°. |
| Friction | |fₛ| ≤ μₛN; maximum fₛ = μₛN.
Static friction matches what is needed for equilibrium, up to its limit.
|fₖ| = μₖN, opposite sliding. It is not always opposite acceleration. |
| Inclines | Weight acts vertically down. Its components are mg sin θ down the ramp and mg cos θ into it. These components replace the weight in a component sum; do not count them as extra forces. |
| Equilibrium | ΣF = 0 means a = 0, not necessarily v = 0. Normal force is not always mg. Zero force has no direction. |
| Torque | τ = rF sin φ; counterclockwise positive.
Static balance requires ΣF = 0 and Στ = 0.
Include a uniform rod’s weight at its midpoint. |
Use degree mode. In surface questions, right/up the ramp is positive. A negative acceleration may describe an object slowing while moving in the positive direction.