Units of Measurement
Physics relies on a consistent vocabulary of units — the SI base units, derived units, and dimensional analysis to convert between them.
Physics relies heavily on the language of mathematics to describe and explain the world around us — but mathematical expressions are meaningless without units. Every measurement must be expressed in units that clearly convey what's being measured, so before diving into the study of motion, it's essential to establish a consistent vocabulary of units.
Key Takeaways
The SI system defines seven base units: meter, kilogram, second, Kelvin, ampere, mole, and candela.
Derived units (like cubic meters for volume) are built by combining base units.
Metric prefixes (giga- through nano- and beyond) rescale units to make very large or very small values easier to express.
Dimensional analysis converts between units in three steps: identify the target unit, start with the given quantity, and apply conversion factors to cancel unwanted units.
Why Units Matter in Physics
Physics involves experimentation and measurement, and communicating those measurements accurately requires appropriate units. Every measurement has two parts — a number and a unit — and neither has meaning without the other. A given measurement can be expressed using different units, which is why a consistent, agreed-upon system is essential for scientific communication.
The SI System of Base Units
The agreed-upon unit system among scientists is the SI system (Système International), which is based on the metric system. It defines seven base units, each standardized and universally accepted as the foundation for all other measurements:
Quantity | Unit Name | Symbol |
|---|---|---|
Length | Meter | m |
Mass | Kilogram | kg |
Time | Second | s |
Temperature | Kelvin | K |
Electric current | Ampere | A |
Amount of substance | Mole | mol |
Luminous intensity | Candela | cd |
Every other unit you might be familiar with is a derived unit — obtained by combining these seven base units.
Derived Units
Derived units are created by combining base units. A key example is volume, which measures the amount of space an object occupies. Volume is expressed in cubic meters (m³) in the SI system. You calculate the volume of a cube, for example, using the formula:
V = length × width × height
Since all three dimensions are measured in meters, multiplying them gives a unit of cubic meters — a derived unit built directly from the base unit of length.
Metric Prefixes
Units are frequently modified using metric prefixes, which make it easier to refer to different magnitudes of a quantity without writing out long strings of zeros. For example, instead of saying "a human cell is anywhere between 0.00001 and 0.0001 meters," it's far more convenient to say "10 to 100 micrometers."
Prefix | Symbol | Power of Ten |
|---|---|---|
Giga- | G | 10⁹ |
Mega- | M | 10⁶ |
Kilo- | k | 10³ |
(base unit) | — | 10⁰ |
Centi- | c | 10⁻² |
Milli- | m | 10⁻³ |
Micro- | μ | 10⁻⁶ |
Nano- | n | 10⁻⁹ |
These prefixes let scientists express very large values (like giga-, 10⁹) or very small values (like nano-, 10⁻⁹) far more conveniently than writing out the full decimal.
Dimensional Analysis
Once you understand units, the next step is knowing how to use them effectively in problem-solving. Dimensional analysis provides a straightforward way to interconvert between units, and it's extremely useful throughout science and engineering.
Conversion factors are used to cancel unwanted units, and they can be developed from any equality (for example, 1 mile = 1.609 kilometers).
The Three-Step Method
Dimensional analysis follows three simple steps:
Write down what you want to know — identify the final unit(s) your answer should be in.
Start with the given measured quantity — begin with the value and unit you were given.
Apply conversion factors in order to cancel unwanted units — multiply by conversion factors (arranged so unwanted units cancel) until only the desired unit remains.
Worked Example
Suppose a car travels 90 miles using 3 gallons of gasoline, and you want to know its fuel efficiency in miles per gallon.
What you want to know: miles per gallon (mi/gal)
Given quantity: 90 miles and 3 gallons
Apply the conversion: 90 mi ÷ 3 gal = 30 mi/gal
The same three-step logic applies to any unit conversion — whether converting between metric prefixes, between imperial and metric units, or between compound units like meters per second and kilometers per hour: identify the target unit, start with what's given, and multiply by conversion factors until the unwanted units cancel out.
Common MCAT Mistakes
Mixing base and derived units mid-calculation. Plugging a value still in centimeters into a formula that expects meters (or vice versa) throws off the answer by a power of ten — convert everything to consistent units before calculating, not after.
Forgetting a conversion factor has two valid orientations. A conversion factor like (1 km / 1000 m) can be flipped to (1000 m / 1 km) — pick whichever orientation cancels the unit you're trying to remove, not whichever one is written first in a table.
Misreading metric prefixes as steps of ten instead of the actual power. Kilo- is 10³ and milli- is 10⁻³, so moving from kilometers to millimeters is a factor of 10⁶, not 10³ — always check the actual exponent, don't just count prefix "steps."
Treating dimensional analysis as optional for "obvious" conversions. Even simple-looking conversions (like mi/gal above) benefit from writing out the three-step method explicitly — skipping it is where unit-cancellation errors creep in under time pressure.
MCAT-Style Concept Check
Question: A student measures a bacterial cell width as 3 micrometers. Using the SI metric prefix system, this value is equivalent to how many meters?
A) 3 × 10³ m
B) 3 × 10⁻³ m
C) 3 × 10⁻⁶ m
D) 3 × 10⁻⁹ m
Answer: C
Explanation: The prefix "micro-" corresponds to a power of ten of 10⁻⁶. So 3 micrometers equals 3 × 10⁻⁶ m. Option A confuses micro- with a positive exponent, option B mistakes it for milli- (10⁻³), and option D mistakes it for nano- (10⁻⁹).
FAQ
What are the seven SI base units?
The seven SI base units are the meter (length), kilogram (mass), second (time), Kelvin (temperature), ampere (electric current), mole (amount of substance), and candela (luminous intensity). Every other unit used in physics is derived from combinations of these seven.
What's the difference between a base unit and a derived unit?
A base unit is one of the seven standardized units that form the foundation of the SI system. A derived unit is created by combining base units — for example, cubic meters (m³) for volume is derived by multiplying three length measurements together.
Why are metric prefixes used in physics?
Metric prefixes rescale a unit by a power of ten so very large or very small quantities can be expressed conveniently, without writing out long strings of zeros. For example, "10 to 100 micrometers" is far easier to work with than "0.00001 to 0.0001 meters."
What is dimensional analysis used for?
Dimensional analysis is a three-step method for converting a measurement from one unit to another: identify the target unit, start with the given quantity, and multiply by conversion factors arranged so unwanted units cancel out until only the desired unit remains.
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