PHSX 220 Physics I (w/ Calculus) · Unit 1: Kinematics & Newton’s laws · on Exam 1
Ch 1 Units, dimensions, and estimation
Study notes for Units, dimensions, and estimation in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- SI base units for mechanics: meter (m), kilogram (kg), second (s). Everything else is built from them: newton $\text{N} = \text{kg·m/s}^2$, joule $\text{J} = \text{N·m}$, watt $\text{W} = \text{J/s}$, pascal $\text{Pa} = \text{N/m}^2$.
- Convert units by multiplying by 1 written as a ratio: $72\ \text{km/h} \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{h}}{3600\ \text{s}} = 20\ \text{m/s}$. Squared or cubed units convert twice or three times: $1\ \text{cm}^3 = (10^{-2}\ \text{m})^3 = 10^{-6}\ \text{m}^3$.
- Dimensional analysis catches wrong equations: both sides must have the same dimensions, and you can only add quantities with the same units. Arguments of $\sin$, $\exp$, $\ln$ must be dimensionless.
- Report answers to about three significant figures (the precision of the data), with units, and a sign or direction when the quantity is a vector.
- Fermi estimates: round to powers of ten, multiply, and sanity-check the order of magnitude. Clicker questions love these.
Key formulas
Key conversions
$$1\ \text{km/h} = \tfrac{1}{3.6}\ \text{m/s},\quad 1\ \text{mi} = 1.609\ \text{km},\quad 1\ \text{in} = 2.54\ \text{cm},\quad 1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$$
Derived units
$$\text{N} = \text{kg·m/s}^2,\quad \text{J} = \text{N·m} = \text{kg·m}^2\text{/s}^2,\quad \text{W} = \text{J/s},\quad \text{Pa} = \text{N/m}^2$$
Worked example
Check whether $x = v t^2$ could describe a position.
Show the solution
Dimensions of the right side: $[v][t]^2 = \frac{\text{L}}{\text{T}}\cdot\text{T}^2 = \text{L·T}$. Position has dimension $\text{L}$, so the equation is wrong. ($x = \tfrac12 a t^2$ works: $\frac{\text{L}}{\text{T}^2}\cdot\text{T}^2 = \text{L}$.)
Common mistakes
- Forgetting to square the conversion factor for areas (cm² → m² is ×10⁻⁴, not ×10⁻²).
- Mixing km/h with seconds in a kinematics equation.
- Dropping units on the final answer: the graders and Expert TA both check them.
Exam tipBefore submitting any Expert TA answer, run the units through in your head. A velocity answer in m/s² is an instant flag that a formula was misapplied.
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