Statics Day 1

Distances between particles in a table (or any body) do not change: the table is a rigid body. You cannot model a rigid body as a single particle in a free body diagram (FBD) for all problems.
In this course, there will be a dedicated discussion of body diagrams (FBDs) to account for how extended bodies behave. Not all free body diagrams will be point masses.
Internal forces vs external forces: when analyzing a system of connected bodies, you must separate the bodies and draw internal forces explicitly on each body’s FBD. These internal forces will cancel appropriately when you analyze the whole system or a closed set of bodies.
FBDs are used to set up equations of equilibrium and dynamics for the bodies involved.

A concentrated force is an idealization where the effect of loading is assumed to act at a single point as a single resultant force (a normal force, for example).
In reality, a contact force like a normal force is a pressure distributed over an area, but it can be modeled as a single force acting at a point for simplicity.
Other common concentrated-force examples discussed: gravity (weight) and its role in FBDs.
Gravity is an acceleration field; the related force is weight: $W = m g$
- Gravity is an acceleration; the force associated with gravity on a body of mass m is the weight W.
- Weight is often treated as acting at the center of mass (center of gravity) for simplicity, but in reality the weight distribution occurs throughout the body; center of gravity is the point where the total weight effectively acts.
Important nuance: weight is not a single point force in the real body, but for many problems we model it as a concentrated force at the center of gravity.

Newton’s First Law: An object at rest stays at rest, and an object in motion stays in motion unless acted upon by an external force.
Newton’s Second Law: When a body is accelerated by external forces, the relation is
$\boldsymbol{F} = m \boldsymbol{a}$
Newton’s Third Law: For every action, there is an equal and opposite reaction. This will come up when considering normal forces, friction, and contact pairs between two bodies.
In this course, the second law is the one most frequently used to set up equations of motion for free-body diagrams.

To analyze a system, you may need to draw the FBD for each body separately.
Internal forces appear on the FBDs of the individual bodies (and they come in action–reaction pairs between bodies).
When you sum forces to apply Newton’s laws, you typically include all external forces on each body and account for internal forces only as interactions between the bodies being considered.
FBDs are used to derive equations of equilibrium (for static or quasi-static problems) or dynamics (for motion).

A famous motivating example: Mars Climate Orbiter mission failure due to a units mismatch (feet vs meters).
- The orbiter cost around $0.3$ billion in 1998; value cited as about $0.6$ billion by 2025.
- Ground control used one unit system (feet) while the orbiter used meters, leading to a catastrophic error when commands were executed with inconsistent units.
- This illustrates that missing or inconsistent units can render a calculation meaningless, even if the numerical values seem reasonable.
Lesson: Always keep track of units; a final numeric answer without units is often ambiguous or incorrect.

SI (International System of Units)
- Four base quantities: length, time, mass, and force (the latter is derived via Newton’s second law).
- Derived unit for force in SI: the Newton,
  $1 ext{ N} = 1~ ext{kg}\frac{ ext{m}}{ ext{s}^2}.$
US customary (sometimes called English engineering units)
- Mass unit commonly used: the slug (in some contexts, pounds mass “lbm” or pounds-force “lbf” are used depending on convention).
- In this course, you’ll see a conversion table between SI and US customary units; you won’t be required to memorize all conversion factors, but you should be able to use the conversion table correctly.
Always be able to switch between these systems using explicit conversion factors; do not mix units without converting.

Example: converting a force value from pounds to newtons
- Given a force of $F = 47~ ext{lb}$ (pounds-force), convert to newtons.
- Use the conversion factor: $1~ ext{lb} = 4.48~ ext{N}.$
- Set up the conversion with proper cancellation of units:
  $F = 47~ ext{lb} imes \frac{4.48~ ext{N}}{1~ ext{lb}} = 210.56~ ext{N}.$
- The pounds cancel, leaving newtons as the unit.
Two quick example conversions to illustrate the format:
- Convert $50~ ext{kN}$ to newtons:
  $50~ ext{kN} = 50 imes 10^{3}~ ext{N}.$
- Convert $60~ ext{nm}$ to meters:
  $60~ ext{nm} = 60 imes 10^{-9}~ ext{m}.$
Approach you can adopt for solutions:
- Write the quantity you want to convert.
- Multiply by the appropriate conversion factor so that units cancel correctly.
- If needed, combine numbers and exponents separately: multiply the numerical factors and add the exponent terms.
When performing unit algebra, keep prefixes consistent and ensure you apply prefixes to the entire unit (e.g., mm to m, mm^2 to m^2, etc.).

No plurals for SI units (e.g., use kg, not kgs).
Use a dot to indicate multiplication between units when writing composite units (e.g., $ext{m} \cdot ext{s}$ instead of $ext{ms}$ for meter-second).
Distinguish between milliseconds (ms) and meter-seconds (m·s).
Most unit symbols are lowercase, with some exceptions:
- Newton: $ext{N}$ (uppercase).
- Pascal: $ext{Pa}$ (uppercase P, lowercase a).
Prefixes (multipliers on powers of ten):
- Kilo: $ext{k}$ (10^3), Mega: $ext{M}$ (10^6), Giga: $ext{G}$ (10^9).
- Milli: $ext{m}$ (10^{-3}), Micro: $ext{μ}$ (10^{-6}), Nano: $ext{n}$ (10^{-9}).
- Important: prefixes apply to the entire unit (e.g., mm = 10^{-3} m, mm^2 = (10^{-3} m)^2 = 10^{-6} m^2).
Prefix rules and dictionary-like tables are typically provided in the textbook and lecture materials; you will be given a conversion table when needed.

Example conversions (summarized):
- $1~ ext{lb} = 4.48~ ext{N}$ and hence $$47~ ext{lb} = 47 imes 4.48~ ext{N} \