The mechanics of things that do not move

An arrow with magnitude and direction, Fx = F cosθ, Fy = F sinθ, composition and unit vector

Tip-to-tail and parallelogram, ΣFx·ΣFy give R = √(ΣFx² + ΣFy²), decomposition is the inverse

Forces at a point sum to zero, closed polygon, ΣFx=0 and ΣFy=0, equilibrant and cable tension

M = F d, the moment arm (perpendicular distance), sign ccw+/cw-, ΣM and the seesaw

Zero resultant but a leftover moment, M = F d (spacing), free moment, the force-couple system

Both translation and rotation zero, three planar equations ΣFx, ΣFy, ΣM = 0, reactions, determinacy

Isolate the body, support=reaction, pin/roller/fixed reaction counts, complete FBD, the start of solving

Two-force members, tension/compression sign, joint=particle ΣFx·ΣFy=0, solving a triangular truss, zero-force members

Cut a section to expose bars, one side as a rigid-body FBD, the ΣM moment-center trick, member-force solve, sections vs joints

Multi-force members, dismembering at pins, the pin action-reaction pair, a beam-brace frame solve, a machine's mechanical advantage

Distributed-load intensity w, resultant=area at the centroid, uniform W=wL at mid, triangular W=1/2 w0L at 2/3, beam reactions

Centroid = balance point, the area average, on axes of symmetry, x̄ = ΣAx/ΣA weighted average, composite split, standard-shape centroids

I = ∫y²dA weights by distance squared, rectangle bh³/12, parallel-axis I_c + Ad², orientation effect, the I-beam

Friction is self-adjusting resistance F ≤ μs N, verge μs N, kinetic μk N, ramp slips at tan θ = μs, friction angle φ

Cutting exposes internal N, V, M, read from one-side FBD, the shear diagram, the moment diagram, dM/dx = V so M peaks where V=0