Physics_1_Vectors_and_Statics_Essentials.pptx

Physics_1_Vectors_and_Statics_Essentials.pptx

1. Vectors & Statics Essentials Fast‑track guide to vectors, moments, FBDs, and 2D equilibrium

2. Why Vectors Matter Forces and velocities have magnitude AND direction. Component form lets you add, resolve, and project. Statics = ΣF = 0 and ΣM = 0 using vector components.

3. Vector Notation Unit vectors: i, j, k point along x, y, z Example: F = ⟨Fx, Fy⟩ = Fx i + Fy j Magnitude: |F| = √(Fx² + Fy²)

4. Components & Resultants Resolve: Fx = F cos θ, Fy = F sin θ (choose θ carefully) Resultant: R = ΣF = ⟨ΣFx, ΣFy⟩; |R| = √((ΣFx)² + (ΣFy)²) Direction of R: tan⁻¹(ΣFy / ΣFx) (adjust quadrant) x y

5. Dot Product & Projection Dot product: a · b = |a||b|cosθ = axbx + ayby (+ azbz) Scalar projection of a onto b: comp_b(a) = (a·b)/|b| Vector projection: proj_b(a) = [(a·b)/|b|²] b

6. Moments & Couples Moment about O: M_O = r × F (2D scalar: M = r⊥F) Positive CCW, negative CW (choose and stick to convention) Couple = two equal, opposite forces separated by d ⇒ M = F·d

7. Varignon’s Theorem Moment of a resultant = sum of moments of components. Take moments with forces placed at their lines of action. Great for distributed forces or many forces.

8. Free‑Body Diagrams (FBDs) — Steps Isolate the body: draw only external forces/reactions. Show weight at the center of mass. Replace supports with reactions (unknown magnitudes/directions). Label all known magnitudes, angles, and dimensions. Tip: keep FBDs clean. Too many arrows = confusion.

9. Common Supports & Reactions (2D) Roller: 1 reaction ⟂ to surface Pin: 2 reactions (Ax, Ay) Fixed: 3 reactions (Ax, Ay, Mz)

10. 2D Equilibrium ΣFx = 0, ΣFy = 0, ΣM_O = 0 Pick a smart moment point to kill unknowns. Solve linear system ⇒ reactions and internal forces.

11. Mini Example (No Numbers) A 4 m beam with a pin at A and a roller at B carries a 6 kN point load at midspan. Unknowns: Ax, Ay, By. Take moments about A to solve for By; then ΣFy, ΣFx. Check: do reaction directions match final signs? x y Strategy: choose A for moments ⇒ Ax, Ay vanish.

12. Truss Method of Joints (Quick Peek) Assume all member forces are tension (positive). At each joint: ΣFx=0, ΣFy=0 to solve unknowns. Zero‑force members appear when loads/geometry meet rules.

13. Checklist & Pitfalls Always draw an FBD before writing equations. Keep sign convention consistent (CCW positive). Track units; convert kN↔N, m↔mm when needed. If an answer is negative, your assumed direction was opposite.