Physics for ME · Module 1 of 16
Physical Quantities, Units, Dimensions, and Scaling
Physics describes the world in measured quantities. Dimensions are the grammar; an equation that breaks them is wrong before any arithmetic.
Readiness check
From Math Module 1. Tick only what you can do closed-notes.
- Rearrange formulas with units carried through every step.
- Use scientific and engineering notation fluently.
- Round to 3 significant figures at the end, not the middle.
- Convert between SI prefixes (mm, m, km; g, kg; W, kW, MW).
- Estimate order of magnitude before computing.
The core idea
Every physical equation must balance in dimensions before it balances in numbers.
[F] = M·L/T² = kg·m/s² = NMass, length, time (plus temperature and current later) build every mechanical quantity. Checking dimensions catches wrong formulas for free, and scaling arguments (what happens if I double the size?) answer design questions before any detailed analysis.
The skills, taught in order
Seven small habits turn raw numbers into trustworthy engineering quantities. Each is quick on its own; together they are the error detector you carry into every later course.
1.1 Base and derived quantities
Three mechanical base dimensions build everything in this course: mass M, length L, and time T. (Temperature Θ and electric current I join in the thermal and electrical modules.) Every other mechanical quantity is a product of powers of these three, and its SI unit follows the same pattern.
| Quantity | Dimensions | SI unit |
|---|---|---|
| Length, position | L | m |
| Velocity | L/T | m/s |
| Acceleration | L/T² | m/s² |
| Force | M·L/T² | N = kg·m/s² |
| Energy, work | M·L²/T² | J = N·m |
| Power | M·L²/T³ | W = J/s |
| Pressure, stress | M/(L·T²) | Pa = N/m² |
1.2 SI units and prefixes
One coherent system, scaled by powers of ten. Keep one prefix per quantity, and convert the rest away before any arithmetic.
| Prefix | Symbol | Factor |
|---|---|---|
| giga | G | 109 |
| mega | M | 106 |
| kilo | k | 103 |
| (base) | 100 | |
| milli | m | 10−3 |
| micro | μ | 10−6 |
| nano | n | 10−9 |
Micro-example. 5000 N = 5 kN; 0.0025 s = 2.5 ms; 2.5 GPa = 2.5 × 109 Pa. The number moves, the physical size does not.
1.3 Dimensional analysis: check it, then derive it
Dimensions do two jobs. First they check: every term you add must share one dimension, and the two sides of an equation must match. Second they derive: if you know which quantities a result depends on, dimensions alone often fix its form.
Check. Is ½mv² an energy? [M]·[L/T]² = M·L²/T², the dimensions of the joule. The ½ is dimensionless, so it cannot change the answer's units.
Derive (Rayleigh's method). Suppose a pendulum's period T depends on length L, gravity g, and bob mass m: write T = k·Lagbmc. The right side must reduce to a time. Since only g carries time (g has dimension L/T²), matching powers forces c = 0, b = −½, a = ½, giving T = k√(L/g). Dimensions predict that mass does not matter, before any experiment.
1.4 Significant figures and rounding
Precision is a claim about how well you know a number. Carry full precision through the working, then round the final answer to about three significant figures, matched to your weakest input. A calculator's eight digits are not measurements.
Micro-example. A 12.0 N force over 7 mm of travel gives 1.71 N/mm, not 1.714286: the two-figure 7 limits the result.
1.5 Unit conversion
Multiply by 1 written as a ratio of two equal quantities, and cancel units like algebra until the answer's units appear.
Micro-example. 90 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 25 m/s. Each factor equals 1, so the value is unchanged; only the units are rewritten.
1.6 Engineering estimation
A Fermi estimate breaks a hard quantity into factors you can bound, takes round numbers, multiplies, and trusts the order of magnitude rather than the digits. It tells you within a factor of a few whether a design, a budget, or an answer is even plausible.
Micro-example. Energy to climb three storeys (about 10 m) with a 75 kg body: mgh ≈ 75 × 10 × 10 = 7500 J, roughly 2 food Calories. Small, which is why one flight of stairs is poor exercise.
1.7 Scaling laws
For geometrically similar objects, area grows as L² while volume (so mass, weight, and heat produced) grows as L³. A surprising number of design limits live in the gap between the two. Worked Example 2 turns this gap into a cooling problem.
Micro-example. Double every dimension: 8× the mass rests on 4× the cross-section, so the stress from self-weight doubles. Bigger is not simply more; it is proportionally heavier on its own supports.
Engineering connection: every course. Direct twin of Statics Module 1 and Math Module 1.
Worked example 1: estimating highway drag
Estimate the aerodynamic drag on a car at 30 m/s (108 km/h) and the engine power spent fighting it. Take ρ = 1.2 kg/m³, drag coefficient Cd = 0.30, frontal area A = 2.2 m².
- ProblemFind the drag force and the power it consumes for the car in Figure 1.
- Given / findρ = 1.2 kg/m³, Cd = 0.30, A = 2.2 m², v = 30 m/s. Find F and P.
- AssumptionsStill air, steady speed, standard drag model.
- ModelF = ½ρCdAv² and P = Fv. Check the dimensions first (concept figure): they collapse to N and W.
- EquationsF = ½ρCdAv² P = F·v
- SolveF = 0.5 × 1.2 × 0.30 × 2.2 × 900 = 356 N. P = 356 × 30 = 10 690 W ≈ 10.7 kW.
- CheckScale: 356 N is about the weight of 36 kg, plausible for highway drag. The v² scaling predicts 4× the force and 8× the power at 60 m/s: this is why fuel economy collapses at high speed.
- ConclusionAbout 10.7 kW (a third of a small engine's output) is spent on air alone at 108 km/h. The estimation pattern (model, dimensions, numbers, scaling story) is the everyday tool of design reviews.
Worked example 2: the square-cube law
A small brushless motor with characteristic size L = 40 mm sheds its waste heat passively and stays at a safe temperature. For a bigger machine it is scaled up geometrically to 80 mm, every dimension doubled. The waste heat a motor makes grows with its volume; the heat it can shed grows with its surface. Find how the heat output and the surface heat flux of the larger motor compare with the original, and say what that means for cooling.
- ProblemCompare the heat output and the surface heat flux of the 80 mm motor with the 40 mm original.
- Given / findLinear scale factor k = 80/40 = 2; heat output proportional to volume; heat shed across surface area. Find P2/P1 and the flux ratio q2/q1.
- AssumptionsGeometric similarity (every length ×2), same materials and magnetics, heat generated ∝ volume, heat removed ∝ exposed surface area.
- ModelVolume ∝ L³, area ∝ L². Surface heat flux is q = P/A.
- EquationsP2/P1 = k³ A2/A1 = k² q2/q1 = k³ / k² = k
- SolveP2/P1 = 2³ = 8, so an original 50 W of waste heat becomes 400 W. A2/A1 = 2² = 4. Flux ratio = 8 / 4 = 2: every square centimetre of the big motor's surface must pass twice the heat.
- CheckHeat flux has units W/m² = (M·L²/T³)/L² = M/T³; the ratio of two fluxes is dimensionless, as any ratio of like quantities must be. The general result q ∝ L says flux always climbs with size under geometric scaling.
- ConclusionScaling a design up is not free. Heat made (as L³) outruns the surface that loses it (as L²), so the large motor runs hotter on the same cooling scheme and needs fins, a fan, or liquid cooling that the small one did without. The square-cube law settled this before any heat-transfer calculation, which is exactly what scaling arguments are for.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| kg used as a force | Loads "in kilograms" inside force equations | "Is this matter or a push?" | Mass in kg; force in N; W = mg converts. The Statics habit starts here. |
| Adding quantities of different dimensions | Lengths added to areas, energies to powers | "Do all added terms share one unit?" | Only like adds to like. A dimension check on every sum. |
| Precision invented by the calculator | Eight digits reported from two-digit inputs | "How precise were my worst inputs?" | Result precision follows the weakest input. Three significant figures is the engineering default. |
| Scaling intuition from length alone | "Twice the size, twice the weight" | "Does this quantity follow L, L², or L³?" | Area goes as L², volume and mass as L³: double the size means 8× the weight on 4× the cross-section. |
Practice ladder
Convert 108 km/h to m/s, and 10.7 kW to horsepower (1 hp = 746 W).
Show answer
108/3.6 = 30 m/s. 10 690/746 = 14.3 hp.
Check dimensionally whether T = 2π√(L/g) can be a period formula, and whether T = 2π√(g/L) can.
Show answer
√(L/g) = √(m/(m/s²)) = √(s²) = s: valid. √(g/L) gives 1/s: a frequency, not a period. Dimensions alone reject the second form.
Fermi estimate: how many litres of fuel does the worked-example car burn per 100 km fighting drag alone? Assume engine efficiency 30% and fuel energy 34 MJ per litre.
Show answer
100 km at 30 m/s takes 3333 s; drag energy = 10 690 × 3333 ≈ 35.6 MJ. Fuel energy needed = 35.6/0.30 = 119 MJ, so about 3.5 litres per 100 km for drag alone. The right order of magnitude against real consumption, which is the point of a Fermi estimate.
Take any datasheet quantity (battery Wh, motor torque, pump flow) and run the full discipline: state its dimensions, convert it to base SI, scale it (what if the device were twice as large?), and sanity-check one derived number the datasheet implies.
What good work looks like
Dimensions written in M, L, T form, a clean conversion chain, an explicit L, L², or L³ scaling claim, and one recomputed datasheet number with agreement or a flagged discrepancy.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Write a one-page "Estimation Memo" on a question you care about (drag on your bike, heat from your PC, energy in your stairs climb): model, dimensional check, numbers, scaling statement, and a comparison against one measured or published value.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Express force, energy, and power in base dimensions and SI units.
Force: ML/T² = N. Energy: ML²/T² = J. Power: ML²/T³ = W.
2. What does dimensional homogeneity demand, and what does it buy you?
Every added term shares the same dimensions. It rejects wrong equations for free and exposes algebra slips.
3. State the significant-figures discipline.
Carry full precision through the work; round the final answer to about three significant figures, matching the weakest input.
4. How do area and volume scale with size, and why do engineers care?
Area as L², volume (and mass) as L³: strength-to-weight worsens as machines grow. Scaling decides what designs are possible.
5. What are the steps of a Fermi estimate?
Model the quantity, break it into factors you can bound, take round numbers, multiply, and trust the order of magnitude, not the digits.