1.2 Units, dimensions, and systems of measurement

Units, dimensions, and measurement systems are the building blocks of mechanics. They let you quantify physical quantities and perform calculations that actually mean something. Getting comfortable with these concepts now will save you from frustrating errors throughout the rest of the course.

This section covers fundamental units, derived units, unit conversions, and dimensional analysis, which is your go-to tool for checking whether an equation makes physical sense.

Fundamental Units in Mechanics

Base Units and Their Relationships

Every quantity in mechanics can be traced back to three base dimensions: length, mass, and time. All other units you'll encounter are built from combinations of these three.

The two unit systems you need to know:

• SI (metric) system:
  • Meter (m) for length
  • Kilogram (kg) for mass
  • Second (s) for time
• US customary system:
  • Foot (ft) for length
  • Slug for mass
  • Second (s) for time

A note on the US customary system: the slug is the standard unit of mass in this system when working with Newton's second law. You might also see pound-mass (lbm), but in statics and dynamics courses, slug is what keeps $F = ma$ consistent with pounds-force. One slug is defined so that a 1-slug mass accelerated at $1 \, ft/s^2$ experiences a force of 1 lbf.

Derived Units

Derived units are formed by combining base units through multiplication or division. The most important one for this course is force.

• Newton (N) in SI: $1 \, N = 1 \, kg \cdot m/s^2$
• Pound-force (lbf) in US customary: $1 \, lbf = 1 \, slug \cdot ft/s^2$

Other derived units you'll see:

Notice how each derived unit traces back to the three base units. A Pascal, for instance, expands to $kg/(m \cdot s^2)$.

Unit System Conversion

Conversion Factors

Conversion factors let you translate a quantity from one unit system to another without changing its physical meaning. You multiply by a ratio that equals 1 (since the numerator and denominator represent the same quantity).

Common conversion factors to know:

• $1 \, m = 3.281 \, ft$
• $1 \, kg = 0.06852 \, slug$
• $1 \, N = 0.2248 \, lbf$
• $1 \, Pa = 1.450 \times 10^{-4} \, psi$

Example: Convert 50 N to pound-force.

$50 \, N \times \frac{0.2248 \, lbf}{1 \, N} = 11.24 \, lbf$

The "N" cancels, leaving you with lbf. Always write out the units and cancel them explicitly; this is the single best way to catch conversion mistakes.

Consistent Units in Equations

Before plugging values into any equation, check that every term uses the same unit system. Mixing systems is one of the most common errors in mechanics problems.

For example, if you're using $F = ma$ and your mass is in kg but your acceleration is in $ft/s^2$, you need to convert the acceleration to $m/s^2$ first (divide by 3.281), or convert the mass to slugs. Pick one system and stick with it for the entire problem.

Dimensions and Dimensional Homogeneity

Physical Nature of Quantities

Dimensions describe what type of physical quantity something represents, independent of which unit system you use. We denote the basic dimensions with brackets:

• [L] for length
• [M] for mass
• [T] for time

Any derived quantity has dimensions built from these. For example:

• Velocity: $[L][T]^{-1}$
• Acceleration: $[L][T]^{-2}$
• Force: $[M][L][T]^{-2}$
• Pressure: $[M][L]^{-1}[T]^{-2}$

Think of dimensions as the "type" of a quantity. Meters and feet are different units, but they share the same dimension: length [L].

Dimensional Homogeneity

An equation is dimensionally homogeneous if every term has the same dimensions. This is a requirement for any physically valid equation.

Take $v = v_0 + at$. Check each term:

• $v$: $[L][T]^{-1}$
• $v_0$: $[L][T]^{-1}$
• $at$: $[L][T]^{-2} \times [T] = [L][T]^{-1}$

All three terms have dimensions of $[L][T]^{-1}$, so the equation is dimensionally homogeneous.

Dimensionless quantities like angles (in radians), ratios, and coefficients of friction have no dimensions, represented as [1]. You can't add a dimensionless quantity to one that has dimensions.

Dimensional Analysis for Equation Consistency

Verifying Equation Consistency

Dimensional analysis is a quick way to check whether an equation could be correct. Here's the process:

1. Replace each variable in the equation with its dimensions.
2. Simplify the dimensions on each side using algebra.
3. Compare the left-hand side to the right-hand side. If the dimensions match, the equation is dimensionally consistent. If they don't, something is wrong.

Example: Verify that $s = v_0 t + \frac{1}{2}at^2$ is dimensionally consistent.

• Left side: $s$ has dimensions $[L]$
• Right side, first term: $v_0 t = [L][T]^{-1} \times [T] = [L]$
• Right side, second term: $\frac{1}{2}at^2 = [L][T]^{-2} \times [T]^2 = [L]$

Both terms on the right reduce to [L], matching the left side. The equation checks out.

Limitations and Applications

Dimensional analysis can catch errors, but it has limits. An equation can be dimensionally correct and still be wrong. For instance, $s = 2v_0 t + 3at^2$ passes a dimensional check, but the numerical coefficients are wrong. Dimensional analysis catches structural mistakes (like adding a force to a velocity), not numerical ones.

The Buckingham Pi theorem takes dimensional analysis further: any physically meaningful equation with $n$ variables and $k$ independent dimensions can be rewritten using $p = n - k$ dimensionless groups. This is especially useful in fluid mechanics and heat transfer for scaling experiments, but you likely won't need it much in an introductory statics course. Just know it exists as a powerful extension of the dimensional reasoning you're learning here.