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Engineering Mechanics

From transcript: 20CV210 Engineering Mechanics (B)

Cheat sheet

Printable study sheet: formulas → definitions → topic notes → traps.

Formulas

ΣF_x=0, ΣF_y=0, ΣM=0 (2D static equilibrium)
Moment M = r × F ; M = F·d (perpendicular lever arm)
Varignon: moment of resultant = sum of moments of components
Centroid x̄ = ∫x dA / A ; composite: Σ(A_i x_i)/ΣA_i
Parallel axis: I = I_cm + A d² ; polar J = I_x + I_y (plane)
Friction: F ≤ μ_s N (static); F = μ_k N (kinetic); impending at =
Belt friction: T1/T2 = e^{μθ} (flat belt, impending slip)
Particle: ΣF = m a ; work–energy: U_{1→2} = ΔT ; impulse ∫F dt = Δp
Rigid body planar: ΣF=m a_G ; ΣM_G = I_G α ; or ΣM_IC = I_IC α
Instantaneous center: v = ω · r_⊥ from IC
Relative motion: v_B = v_A + ω × r_{B/A} ; a_B = a_A + α×r − ω²r + 2ω×v_rel (Coriolis)
Gyroscopic couple: τ = I ω ω_p (spin × precession)
Truss method of joints: ΣF=0 at each pin; sections: cut ≤3 unknowns
ΣF = 0, ΣM = 0 (statics equilibrium)
F = ma ; M = Iα
Friction: F ≤ μN (static μs, kinetic μk)
Work–energy: U = ΔT + ΔV
Impulse–momentum: ∫F dt = Δp

Definitions

Free-body diagram (FBD)
Isolated body with all external forces/moments drawn
Statically determinate
Unknowns solvable from equilibrium alone
Couple
Pure moment; force sum zero; free vector in plane
Dry friction
Tangential force at contact opposing slip tendency
Centroid vs center of mass
Geometry vs mass-weighted; coincide if uniform density
Moment of inertia I
Second moment of area (bending) or mass (dynamics)
Impulse–momentum
Integrated force equals change of linear momentum
Work–energy
Work of all forces equals change of kinetic energy
IC of rotation
Point of zero velocity at an instant for planar rigid body
Degrees of freedom
Independent coordinates to fix configuration
Free-body diagram
Isolate body; draw all external forces/moments
Centroid / CG
Balance point of area / mass distribution

Topic-wise short notes

Study these first — one block per syllabus topic. Then read the deep notes below.

Statics

  • Support types: roller (1 force), pin (2), fixed (2+moment), cable (tension along cable).
  • Trusses: assume pin joints, loads at joints; zero-force members by inspection.
  • Frames/machines: multi-force members — cut and use FBDs of parts.
  • Shear & BMD: cut beam, ΣV and ΣM on remaining; V=dM/dx, w=−dV/dx.
  • Friction wedges/screws: self-locking if lead angle < friction angle.

Dynamics

  • Rectilinear vs curvilinear: a_t=dv/dt, a_n=v²/ρ.
  • Energy: conservative forces → potential V; T+V conserved if only conservative work.
  • Impact: e = relative separation / approach along line of impact; 0≤e≤1.
  • Rigid planar kinetics: mass moment I_G; parallel axis for other points.
  • Gyroscopes: effect direction by right-hand rule on angular momentum change.

Free-body diagrams & equilibrium

  • Draw known magnitudes with sense; unknowns as components or angled unknowns.
  • Check units N vs kN; moments N·m.
  • Internal forces: cut member, expose axial/shear/moment.

Friction & particle/rigid dynamics

  • Impending motion: use μ_s and equality; if not impending, friction is unknown ≤μ_s N.
  • Rolling without slip: a = α R kinematic link; friction may be static < μN.
  • Belt/brake problems: tight vs slack side — T_tight/T_slack = e^{μθ}.

Virtual work & energy methods

  • Σ δW = 0 for equilibrium of ideal systems.
  • Castigliano: ∂U/∂Pi = δi — deflection of linearly elastic structures.
  • Conservation laws when forces conservative / no dissipation.

Exam traps & quick notes

Comprehensive notes

Statics

Concurrent/non-concurrent force systems, trusses (method of joints/sections), beams with shear/moment diagrams from equilibrium.

External: MIT Eng Mechanics I ↗ Engineering Toolbox statics ↗

Dynamics

Kinematics (s-t, v-t) vs kinetics (Newton, energy, impulse). Relative motion and rigid-body rotation about fixed axis.

External: HyperPhysics Mechanics ↗

Free-body diagrams & equilibrium

Isolate the body; include reactions, friction, distributed loads as resultants. ΣF=0 and ΣM=0 in 2D statics. Method of joints/sections for trusses. Shear and bending moment diagrams from cutting sections — interview gold.

External: NPTEL ↗ Engineering Toolbox ↗

Friction & particle/rigid dynamics

Dry friction F≤μN; impending motion vs motion. Newton’s laws, work–energy, impulse–momentum. Instantaneous center for planar rigid bodies. Gyroscopic couple τ=Iωωp for spinning rotors on turning vehicles/aircraft.

External: NPTEL ↗ Engineering Toolbox ↗

Interview Q&A for this subject

Q: How do you draw an FBD for a ladder on rough floor/smooth wall?
A: Weight at CG, normal+friction at floor, normal at wall (no friction if smooth); take moments about floor contact to find wall reaction.

External: ASQ / quality ↗ MIT OCW ↗