A runner’s speed can look larger in kilometres per hour than in metres per second without the runner speeding up. The number and the unit work as a pair. We will change that pair while keeping the physical quantity fixed.
Unit conversion is often treated as a preliminary chore. In physics it is also a diagnostic tool. Following the units through an equation reveals a missing square, an inverted rate or a pressure mistaken for a force before an answer reaches the final line. We will build a method that works without memorising separate rules for every pair of units.
What you will learn to do
- Build conversion factors that equal one physical ratio.
- Convert powers and compound units without losing their exponents.
- Check whether an equation is dimensionally possible.
- Distinguish temperature values from temperature intervals.
- Preserve measurement information through an exact conversion.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number Learning goals: - Build conversion factors that equal one physical ratio. - Convert powers and compound units without losing their exponents. - Check whether an equation is dimensionally possible. - Distinguish temperature values from temperature intervals. - Preserve measurement information through an exact conversion. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Why is multiplying by 100 cm / 1 m allowed in a conversion?
Compare your reasoning
The numerator and denominator represent equal lengths, so the ratio is one. Choose its orientation so the unwanted unit cancels. The physical value is unchanged.
Revisit the explanation →01 / Units: keep the quantity, change the number
The number and unit belong together
Write a length as a numerical value multiplied by a unit: L = 2.4 m. A unit is an agreed reference quantity of the same kind. Choosing centimetres means comparing the length with a reference one hundred times smaller, so the numerical value becomes one hundred times larger. The product still describes the same length.
A dimension identifies the physical kind of quantity: length, time or mass, for example. A unit is a particular standard for that dimension. Metres and centimetres have the same dimension. Metres and seconds do not. You may add compatible lengths after expressing them consistently; you cannot add a length directly to a time.
Speed is a derived quantity: length divided by time. Force, energy and pressure have different combinations of base dimensions. A named unit is a convenient abbreviation for that combination, not a new mathematical exception.
| Quantity | Named SI unit | In base units |
|---|---|---|
| Force | newton, N | kg·m/s² |
| Energy or work | joule, J | kg·m²/s² |
| Power | watt, W | kg·m²/s³ |
| Pressure | pascal, Pa | kg/(m·s²) |
The physical quantity Q stays fixed. Changing the chosen unit u changes the number n inversely.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number
Focus: The number and unit belong together
Learning goals:
- Build conversion factors that equal one physical ratio.
- Convert powers and compound units without losing their exponents.
- Check whether an equation is dimensionally possible.
- Distinguish temperature values from temperature intervals.
- Preserve measurement information through an exact conversion.
Equations in this section (LaTeX):
Q=nu=n^{\prime}u^{\prime}
Meaning: The physical quantity Q stays fixed. Changing the chosen unit u changes the number n inversely.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Units: keep the quantity, change the number
Make the unwanted unit cancel
The equality 1 km = 1000 m supplies two useful conversion ratios. Both ratios equal one when their numbers and units are kept together. Choose the orientation that cancels the unit you already have. This is more reliable than deciding whether a decimal point should move left or right from memory.
Write every factor on the page until the method becomes automatic. Treat numerator and denominator units algebraically. A remaining kilometre in an answer requested in metres is a visible sign that a factor was inverted. Multiplying by an exact conversion factor cannot change the underlying physical quantity.
WORKED EXAMPLE
A speed in two different unit systems
A sensor records a constant speed of 54.0 km/h. Express it in m/s and find the distance travelled in 8.0 s, using ordinary measurement reporting rules.
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STOP & THINK
Convert 90 km/h to m/s. Treat this as a unit conversion of an ideal model value.
Answer in m/s.
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number Focus: Make the unwanted unit cancel Learning goals: - Build conversion factors that equal one physical ratio. - Convert powers and compound units without losing their exponents. - Check whether an equation is dimensionally possible. - Distinguish temperature values from temperature intervals. - Preserve measurement information through an exact conversion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Units: keep the quantity, change the number
The exponent belongs to the conversion factor too
An area unit is the product of two length units. A square with side 1 cm has side 0.01 m, so its area is 0.0001 m². Converting cm² to m² requires squaring the complete length conversion factor. A volume similarly requires cubing it.
The notation cm² means (cm)². The prefix is inside the square. It does not mean a hundredth of a square metre. This is why a correct length conversion can still produce a pressure or density wrong by a factor of one hundred or one thousand if its exponent is lost.
A negative power reverses the scale: 1 cm⁻¹ = 100 m⁻¹. Always apply the exponent to the entire conversion.
WORKED EXAMPLE
A small area makes a substantial pressure
An ideal uniform pressure is produced by a 24 N normal force spread over 6.0 cm². Calculate the pressure in pascals.
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WORKED EXAMPLE
Both the numerator and denominator change
A material has a density of 1.25 g/cm³. Express it in kg/m³ without changing its significant-figure information.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number
Focus: The exponent belongs to the conversion factor too
Learning goals:
- Build conversion factors that equal one physical ratio.
- Convert powers and compound units without losing their exponents.
- Check whether an equation is dimensionally possible.
- Distinguish temperature values from temperature intervals.
- Preserve measurement information through an exact conversion.
Equations in this section (LaTeX):
1\ \mathrm{cm^2}=(10^{-2}\ \mathrm m)^2=10^{-4}\ \mathrm{m^2};\qquad 1\ \mathrm{cm^3}=10^{-6}\ \mathrm{m^3}
Meaning: A negative power reverses the scale: 1 cm⁻¹ = 100 m⁻¹. Always apply the exponent to the entire conversion.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Units: keep the quantity, change the number
Prefixes describe scale, not precision
A prefix is a multiplier attached to a unit. Milli means 10⁻³ and micro means 10⁻⁶; kilo means 10³ and mega means 10⁶. Prefix symbols are case-sensitive. An m before a unit symbol is not interchangeable with an M.
Choose a working unit that keeps the calculation clear. Converting everything to SI base units is a reliable default when using SI constants, but a ratio of two lengths can be evaluated in centimetres if both use centimetres. Consistency matters more than mechanically changing units that already cancel.
Be especially careful with graph headings such as “current / mA.” A tick labelled 3 means 3 mA, which is 0.003 A. A fitted slope read numerically from such axes carries the displayed vertical unit divided by the displayed horizontal unit. The number alone is not a resistance, acceleration or spring constant until those axis units are included.
| Prefix | Symbol | Multiplier |
|---|---|---|
| nano | n | 10⁻⁹ |
| micro | μ | 10⁻⁶ |
| milli | m | 10⁻³ |
| centi | c | 10⁻² |
| kilo | k | 10³ |
| mega | M | 10⁶ |
STOP & THINK
A voltage–current graph uses volts vertically and milliamperes horizontally. Its numerical slope is 2.0 V/mA. What resistance does this represent?
- 2.0 Ω
- 0.0020 Ω
- 2000 Ω
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number Focus: Prefixes describe scale, not precision Learning goals: - Build conversion factors that equal one physical ratio. - Convert powers and compound units without losing their exponents. - Check whether an equation is dimensionally possible. - Distinguish temperature values from temperature intervals. - Preserve measurement information through an exact conversion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Units: keep the quantity, change the number
Temperature values and intervals are different conversions
Most familiar unit conversions in mechanics multiply by a factor. Temperature scales can also have different zero points. A Celsius temperature value becomes a kelvin value by adding 273.15. A temperature rise of 5 °C, however, is a rise of 5 K: the offset cancels when two values are subtracted.
This distinction matters when an equation describes an absolute temperature, as in a gas law, versus a temperature change, as in heat capacity. Adding 273.15 to a temperature difference creates an unphysical extra change. Conversely, inserting 20 instead of 293.15 into an ideal-gas calculation confuses Celsius with the thermodynamic temperature.
An exact scale conversion does not supply finer experimental knowledge. A thermometer reading to whole degrees does not suddenly justify hundredths simply because the conversion offset contains decimals. Follow the stated uncertainty or reporting rule.
WORKED EXAMPLE
Convert the endpoints before interpreting the difference
An idealised thermal model changes from 18 °C to 43 °C. Find both temperatures in kelvin and the temperature change in kelvin. Treat the stated values as exact model inputs.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number Focus: Temperature values and intervals are different conversions Learning goals: - Build conversion factors that equal one physical ratio. - Convert powers and compound units without losing their exponents. - Check whether an equation is dimensionally possible. - Distinguish temperature values from temperature intervals. - Preserve measurement information through an exact conversion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Units: keep the quantity, change the number
Use dimensions to reject an impossible equation
A physically meaningful sum needs terms with the same dimensions. In s = ut + ½at², the velocity–time product and acceleration–time-squared product both have dimension length. Replacing t² by t in the second term would attempt to add a length and a velocity.
Dimensional consistency is necessary, but not sufficient, for correctness. The expressions at² and ½at² have the same dimensions; a dimensional check cannot select their coefficient. Nor does it check whether acceleration was constant, whether a sign convention was followed, or whether a vector direction was ignored.
The argument of an exponential, logarithm or trigonometric function in a physical model must be dimensionless. A decay law can contain exp(−t/τ), because the times cancel. Writing exp(−t) without a coefficient carrying inverse-time units would make the prediction depend on whether time was measured in seconds or minutes. Keep the physical model invariant under a change of units.
Square brackets here mean dimensions, with M, L and T denoting mass, length and time. They do not mean a numerical value in a particular unit.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Units: keep the quantity, change the number
Focus: Use dimensions to reject an impossible equation
Learning goals:
- Build conversion factors that equal one physical ratio.
- Convert powers and compound units without losing their exponents.
- Check whether an equation is dimensionally possible.
- Distinguish temperature values from temperature intervals.
- Preserve measurement information through an exact conversion.
Equations in this section (LaTeX):
[p]=\frac{[F]}{[A]}=\frac{MLT^{-2}}{L^2}=ML^{-1}T^{-2}
Meaning: Square brackets here mean dimensions, with M, L and T denoting mass, length and time. They do not mean a numerical value in a particular unit.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.BRING THE IDEAS TOGETHER
What should stay with you
- Changing units preserves the physical quantity and changes its numerical representation.
- Apply powers to conversion factors as well as to unit symbols.
- A rate or density requires converting numerator and denominator consistently.
- A temperature interval does not include the Celsius–kelvin offset.
- Dimensional checks reject some errors but cannot prove a model or numerical coefficient.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
Convert 7.5 mm² to square metres. Give a decimal or scientific-notation value.
Answer in m².
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SELF CHECK 2
Water flows at 2.4 litres per minute. Using 1 litre = 10⁻³ m³, find the flow in m³/s.
Answer in m³/s.
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SELF CHECK 3
Two proposed free-fall distances are gt² and ½gt². What can a dimensional check establish?
- Only the second is dimensionally correct
- Both are dimensionally possible; more physics is needed
- Neither has length dimensions
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
A density in two unit systemsIntermediate · Free sample · Hints and guided reasoningA temperature interval has no zero offsetBeginner · Hints and guided reasoningA speedometer meets the SI systemBeginner · Hints and guided reasoningMake the exponent dimensionlessIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Explain why 1 cm² is not 0.01 m².
Compare your reasoning
One centimetre is 0.01 metre, and both sides of an area must be converted. The factor is squared, giving 0.0001 m². A volume conversion would cube it.
Revisit the explanation →A few days later: change the situation
A temperature rises from 20 °C to 30 °C. What is the rise in kelvin, and why do you not add 273.15 to the rise?
Compare your reasoning
The rise is 10 K. Celsius and kelvin have equally sized intervals; their fixed zero offset cancels in a difference. Absolute temperatures still require the offset.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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