SI units, dimensional checks and estimation
Key idea: A complete H2 Physics lesson on SI quantities, estimation, errors, uncertainty and coplanar vectors.
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The core idea
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Big question: How do units and estimates expose an impossible physics result?
Begin with the six required SI base quantities and their units, then express prefixes as powers of ten before calculating. Derived units can be reduced to base units to test dimensional homogeneity. A sound estimate states reasonable assumptions, calculates from them and checks the order of magnitude; it is reasoned approximation, not a guess.
Build every unit from a small reliable core
The six required SI base quantities are mass, length, time, electric current, thermodynamic temperature and amount of substance, with units kg, m, s, A, K and mol. Derived units are products or quotients of these base units: for example, N = kg m s⁻² and J = kg m² s⁻².
Replace prefixes by powers of ten before calculating. The prefix belongs to the whole unit, so 1 mm² = (10⁻³ m)² = 10⁻⁶ m². Letter case matters: m is milli while M is mega.
Check your understanding: Express one pascal in SI base units.
Pa = N m⁻² = kg m⁻¹ s⁻².
Use dimensions as an error detector
Terms that are added or subtracted must have the same dimensions, and both sides of a physical equation must be dimensionally homogeneous. Replace each quantity by powers of M, L, T and any other required base dimensions, then compare.
Passing this test is necessary but not sufficient. It cannot establish a numerical constant, choose between two dimensionally identical models or prove that either model describes nature.
Check your understanding: Why does dimensional agreement not prove s = ut + at²?
Both terms have dimension L, but dimensional agreement cannot determine the missing numerical factor 1/2 or prove the constant-acceleration model.
Make an estimate from stated physical assumptions
A physical estimate is a short model, not an unsupported guess. Choose plausible dimensions, rates or material properties, state them clearly, calculate and then round the result to a defensible order of magnitude.
Different assumptions can give different numerical answers and still be sound. What matters is whether the assumptions suit the situation and whether the final scale is plausible.
Check your understanding: Why is a ten-digit answer unsuitable for a classroom-volume estimate?
The assumed room dimensions are approximate, so many digits would imply precision the model does not have.
Key ideas to keep
- Case matters in prefixes: m means 10⁻³ while M means 10⁶.
- Dimensional homogeneity can reject an equation but cannot prove its numerical factor or physical model.
- An estimate earns credibility from stated dimensions, rates or material properties.
See the reasoning
Worked example
Estimate the mass of water in a bathtub
Question: Estimate the mass of water in a domestic bathtub and give an order of magnitude.
Step 1: Choose a simple shape and dimensions
Why: An estimate needs explicit assumptions that another student could challenge or improve.
Working: Model the filled part as 1.5 m × 0.55 m × 0.25 m, allowing for rounded ends.
Step 2: Find the approximate volume
Why: Mass follows from density times volume.
Working: V ≈ 1.5(0.55)(0.25) = 0.206 m³.
Step 3: Use water density and round sensibly
Why: The assumed dimensions do not justify fine precision.
Working: m ≈ 1000(0.206) ≈ 2 × 10² kg, so the order of magnitude is 10² kg.
Answer: A defensible estimate is about 200 kg, of order 10² kg.
Check: This is comparable with a few adults and much less than a tonne, so the scale is plausible.
Another worked model
Model 1
Show that power has SI base units kg m² s⁻³, starting from P = E/t.
Check the worked solution
Energy has unit J = kg m² s⁻². Dividing by time gives kg m² s⁻²/s = kg m² s⁻³.
Model 2
Estimate the energy transferred by a 2.0 kW kettle operating for 3 minutes.
Check the worked solution
Use E = Pt with 2.0 kW = 2.0 × 10³ W and 3 min = 180 s. E = 3.6 × 10⁵ J, sensibly of order 10⁵ J.
Use a hint if needed
Practise with support
Question 1
Convert 7.2 GW to watts and write the pascal in SI base units.
Hint: Replace giga by 10⁹; use pressure = force/area.
Check your answer
7.2 GW = 7.2 × 10⁹ W. Pa = N m⁻² = kg m s⁻² m⁻² = kg m⁻¹ s⁻².
Question 2
Check the homogeneity of p = ρgh using SI base dimensions.
Hint: Write density as mass per volume.
Check your answer
[ρgh] = (M L⁻³)(L T⁻²)(L) = M L⁻¹ T⁻², the same dimensions as pressure.
Now work without the hint
Practise independently
Question 1
Convert 450 nm to metres and 0.75 MJ to joules. Derive the SI base unit of electric charge from Q = It.
Check your answer
450 nm = 4.50 × 10⁻⁷ m; 0.75 MJ = 7.5 × 10⁵ J; charge has base unit A s.
Question 2
A proposed pendulum relation is T = 2π√(l/g). Test its dimensional homogeneity and explain what that test cannot establish.
Check your answer
[l/g] = L/(L T⁻²) = T², so its square root has dimension T. The relation is homogeneous, but this does not prove the factor 2π or that the model applies.
Avoid these traps
Common mistakes
Common mistake
A prefix changes only the written symbol, not the numerical scale.
What is wrong with this reasoning?
Show better thinking
Replace every prefix by its power of ten before calculating. Case matters: m is 10⁻³ while M is 10⁶.
Common mistake
Matching dimensions proves a proposed physical equation is correct.
What is wrong with this reasoning?
Show better thinking
Homogeneity is necessary, not sufficient. It can reject an equation with mismatched dimensions but cannot prove numerical factors or the physical model.
Common mistake
A physical estimate is a guess that should copy an exact reference value.
What is wrong with this reasoning?
Show better thinking
A defensible estimate states reasonable dimensions, rates or material properties, calculates from them and checks the resulting order of magnitude.
Write for the examiner
Exam guidance
Show every prefix conversion and base-unit step, then state why your final scale is physically plausible.
Exam-style practice [7 marks]
The time T for a pendulum is proposed to be T = C√(l/g). Test the equation dimensionally and state what the test cannot determine. Then estimate the mass of air in a room 8 m by 6 m by 3 m using air density 1.2 kg m⁻³, giving an order of magnitude.
Plan before you answer
- Reduce l/g to base dimensions.
- Separate dimensional consistency from proof.
- State the room model before estimating.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
[l/g] = L/(LT⁻²) = T², so √(l/g) has dimension T and C is dimensionless. This does not determine C or prove that the pendulum follows the model. Treating the room as a cuboid, V ≈ 8(6)(3) = 144 m³ and m ≈ 1.2(144) = 173 kg, so the mass is of order 10² kg.
Come back in three days
Check what stayed with you
Recall question 1
Name the six required SI base units.
Check the answer
kg, m, s, A, K and mol.
Recall question 2
What can a failed homogeneity test establish?
Check the answer
The proposed equation is wrong as written.
Recall question 3
What makes a physical estimate defensible?
Check the answer
Plausible stated assumptions, a calculation and a sensible order-of-magnitude check.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes.
- GCE A-Level H2 PhysicsTopic 1(a) / Topic 1(b) / Topic 1(c) / Topic 1(d) / Topic 1(e) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027