Statics and Dynamics (Rigid-Body Mechanics)
Rigid-body mechanics rewards free-body discipline: if the diagram is right, the equations are usually straightforward.
Core rules
- Equilibrium: ΣF = 0 and ΣM = 0; a 2D problem gives three independent equations. A two-force member carries force along the line between its pins.
- Trusses: method of joints for the whole truss, method of sections for one member. Zero-force members occur at unloaded joints with two non-collinear members.
- Friction: F ≤ μ_s N at rest, F = μ_k N once sliding begins.
- Centroids and second moments use the parallel-axis theorem, I = I_c + Ad².
- Dynamics: F = ma per axis; v = v₀ + at; s = s₀ + v₀t + ½at²; normal acceleration a_n = v²/r.
- Work–energy: U = ΔT. Impulse–momentum: ∫F dt = Δ(mv). Angular momentum about a point is conserved when the net moment there is zero.
Worked example
A 10 kg block slides down a 30° incline with μ_k = 0.2. Normal force N = mg cos30° = 10(9.81)(0.866) = 84.9 N. Friction = 0.2(84.9) = 17.0 N. Gravity along the incline = 10(9.81)(0.5) = 49.1 N. Net force = 32.1 N, so a = 3.21 m/s². The block accelerates because 49.1 N exceeds the 17.0 N of friction.
Traps
- Using mg instead of mg cosθ for the normal force on an incline.
- Forgetting that friction reverses direction with impending motion.
- Using the wrong radius in rolling or circular-motion relations.
- Mixing static and kinetic coefficients within one calculation.