Statics: Forces, Moments, Centre of Gravity, Stress & Buoyancy - Quick Reference

Forces

• Force: anything that tends to cause motion, change of motion, or prevent motion.

• Work: product of a force and the distance moved; $W = F \cdot s$.

• Unit: Newton (N); $1\ \text{N} = 1\ \text{kg} \cdot \text{m}/\text{s}^2$.

• Newton’s laws (summary):

  • 1st: A body at rest stays at rest unless acted on by external force.

  • 2nd: $F = m a$.

  • 3rd: For every action, there is an equal and opposite reaction.

• Forces on a body: often multiple forces; e.g., weight and normal reaction.

Scalars & Vectors

• Scalar quantity: magnitude only (no direction).

• Vector quantity: magnitude and direction.

• Examples:

  • Scalars: length, time, mass, temperature, density, volume, etc.

  • Vectors: force, velocity, displacement, acceleration, momentum, etc.

• Vector representation: symbol often with arrow; components along axes.

Vector Addition & Resultants

• Displacements/forces add to a resultant vector: C is the vector sum of A and B.

• If collinear: simply add/subtract magnitudes along the line.

• If not collinear: use parallelogram law.

• For a path A (\rightarrow) B, the same resultant as a single displacement C: vector addition property.

• 2D components: A = ($A<em>x$, $A</em>y$) with

  • $A<em>x = A \cos\theta, \quad A</em>y = A \sin\theta$

  • $\Vert A\Vert = \sqrt{A<em>x^2 + A</em>y^2}$

Equilibrium & Equilibrants

• Equilibrium: all forces and moments cancel; net effect is zero.

• For rotational equilibrium, sum of clockwise moments equals sum of anticlockwise moments about any pivot.

• Equilibrant: a single force equal in magnitude and opposite in sense to the resultant.

Moments & Couples

• Moment of a force: rotation produced by a force. Magnitude: $M = F \cdot d$ where $d$ is the perpendicular distance from the pivot (lever arm).

• Sign: clockwise vs anticlockwise (negative/positive convention).

• If force line of action passes through the pivot, no moment.

• Couple: two parallel, equal and opposite forces; no net translation, only rotation.

• Resultant force = 0; resultant moment = torque of the couple.

Centre of Gravity (CG)

• CG is the point where all weight can be considered to act.

• For a system: $W = \sum w<em>i, \quad \text{cg} = \frac{\sum (w</em>i \cdot d<em>i)}{W}$ where $d</em>i$ are distances from a reference line.

• In flight, aircraft/rockets rotate about CG.

• Stability: lower CG and wider base increase stability; CG must lie within specified limits for safe flight.

Elements of Theories: Stress, Strain, Elasticity

• Stress: internal force per area; $\sigma = \dfrac{F}{A}$; units: Pa (N/m^2).

• Strain: relative deformation; $\varepsilon = \dfrac{\Delta L}{L_0}$; no units.

• Hooke’s law (elastic region): $\sigma = E \varepsilon$; E is Young’s modulus.

• Types of stress: tension, compression, torsion, bending, shear.

• Elastic limit: linear (elastic) region; beyond it, plastic deformation occurs.

• Materials tend to return to original shape if below elastic limit.

Pressure & Buoyancy in Liquids (Barometers)

• Pressure: $P = \dfrac{F}{A}$; internal resistance to external force.

• Pascal’s law: pressure acts equally in all directions within a fluid.

• Gauge vs Absolute Pressure:

  • Gauge pressure is relative to ambient (atmospheric) pressure.

  • Absolute pressure: $P<em>{abs} = P</em>{gauge} + P_{atm}$

• Atmospheric pressure: standard values measured in inHg, mb, psi, etc.

• Barometer: measures atmospheric pressure.

• Buoyancy (Archimedes’ principle):

  • Buoyant force: $F_B = \rho g V$

  • If FB > weight of object (\rightarrow) floats; if FB < weight (\rightarrow) sinks; if equal, neutral buoyancy.

• Practical note: ships float due to large displaced