Statics: Quick Reference (Key Concepts and Rules)

What is Statics?

• A branch of mechanics; study of bodies when acted upon by forces
• Statics deals with systems in equilibrium: objects at rest or moving with a constant velocity

Branches of Mechanics

• Mechanics includes: Rigid Body, Deformable Body, Fluid
• Subfields: Statics, Dynamics
• Fluid classification: Compressible, Incompressible

Fundamental Units and Systems of Units

• Four fundamental quantities (dimensions): Length, Mass, Time, Force
• Base units: Length in meters (m), Mass in kilograms (kg), Time in seconds (s), Force in newtons (N)
• Relationships between units underpin all calculations

Base and Derived Units

• SI (International System of Units):
  • Length: $ext{m}$, Time: $ext{s}$, Mass: $ext{kg}$, Force: $ext{N}$
• FPS (US Customary):
  • Length: $ext{ft}$, Time: $ext{s}$, Mass: $ext{slug}$, Force: $ext{lb}$

Unit Conversion

• 1 lb ≈ 4.448 N
• 1 slug ≈ 14.59 kg
• 1 ft = 0.3048 m
• These relations convert between FPS and SI units
  $F<em>{ ext{N}} = 4.448 imes F</em>{ ext{lb}}$
  $m<em>{ ext{kg}} = 14.59 imes m</em>{ ext{slug}}$
  $L<em>{ ext{m}} = 0.3048 imes L</em>{ ext{ft}}$

Dimensional Homogeneity

• The terms of all equations must be dimensionally homogeneous
• Dimensions on the Left-Hand Side (LHS) must equal those on the Right-Hand Side (RHS)
• Example (motion): $v = v_0 + a t$
  • Dimensions:
  • $[v] = [L T^{-1}]$, $[v_0] = [L T^{-1}]$, $[a t] = [L T^{-1}]$

Models and Idealizations

• Models simplify reality to solve problems:
  • Particle: has mass but negligible size
  • Rigid Body: collection of particles fixed relative to one another under load
  • Concentrated Force: force applied over a relatively small area on a rigid body

Newton’s Laws of Motion

• 1st Law: A particle at rest or in motion will remain so unless acted upon by an unbalanced external force
• 2nd Law:
  \sum \mathbf{F} = m \mathbf{a}
  
• 3rd Law: For every action there is an equal and opposite reaction

Law of Gravitational Attraction

• Two bodies attract with a force given by:
  $F = G \frac{m<em>1 m</em>2}{r^2}$
• Gravitational constant: $G \approx 6.673 \times 10^{-11} \ \text{N m}^2\text{/kg}^2$

Weight of a Body

• Weight is the gravitational force on a body:
  $W = m g$
• On Earth, $g \approx 9.81 \ \text{m s}^{-2}$

Accuracy of Numerical Calculations

• Accuracy depends on:
  • Accuracy of the input data
  • Accuracy of the computations
• The solution cannot be more accurate than the least accurate data

Significant Figures

• Indicates the precision of a number
• Examples:
  • 4903 has $4\;\text{sig figs}$
  • 0.00356 has $3\;\text{sig figs}$
• Engineering notation (preferred):
  • $4.903 \times 10^3$ (4 sig figs)
  • $23.5 \times 10^3$ (3 sig figs)
  • $23.50 \times 10^3$ (4 sig figs)

Calculations and Rounding Rules

• Round only the final answer
• Show calculations with up to $4\;\text{sig figs}$; round final answer to $3\;\text{sig figs}$
• Keep units throughout calculations and in the final answer

Sample Problems (Units in Calculations)

• Example: A car at $55.0\ \text{mph}$: convert to
  • $55.0\ \text{mph} \to 88.5\ \text{km/h}$
  • $55.0\ \text{mph} \to 24.6\ \text{m/s}$
• Note: Use consistent significant figures and units

Unit Rules

• No plurals (e.g., use $5\ \text{kg}$, not $5\ \text{kgs}$)
• Separate units with a dot: $\text{m} \cdot \text{s}$, not $\text{ms}$
• Symbols are mostly lowercase (e.g., $m, \ kg, s$)
  • Exceptions: $N, Pa, M, G$
• Exponential power applies to units: $\text{cm}^2$
• Compound prefixes should not be used

Important Reminders

• Develop a sense for numbers and units
• Use engineering language and terminology
• Understand load paths (how loads travel between bodies)
• Use appropriate units and four significant digits in intermediate steps; round to three for answers
• Use engineering notation
• Marks may be deducted for not following basic rules