SI units and angles
Shops use kilo, litre and metre separately; scientists give everyone one common language (SI). Link to earlier classes : the measurements and units of classes 9-10 now extend to accuracy, errors and dimensional analysis.
SI (International System of Units) is one measuring system for the whole world; the radian and steradian measure angles.
In the figure : When the arc equals the radius, θ = 1 rad.
Rule 1: SI base units — 7 base quantities : length (m), mass (kg), time (s), electric current (A), temperature (K), amount of substance (mol), luminous intensity (cd). All others are derived units, e.g. N = kg m s⁻², J = N m.
- Easy : The unit of force, newton, is derived, not a base unit.
- Medium : 1 J = 1 N × 1 m = 1 kg m² s⁻².
- Exam (CET) : Which is a base unit? (1) kelvin (2) newton (3) joule (4) watt → answer 1) Trick : K, A, mol, cd are base units.
Rule 2: Plane angle (radian) — θ = s ÷ r (s = arc, r = radius); unit radian (rad), dimensionless. 2π rad = 360°.
- Easy : 30° = 30 × π/180 = 0.5236 rad.
- Medium : r = 2 m, θ = 0.5 rad ⇒ s = θ r = 1.0 m.
- Exam (CET) : 1 rad ≈ ? (1) 57.3° (2) 180° (3) 360° (4) 3.14° → answer 1)
Rule 3: Solid angle (steradian) — Ω = A ÷ r² (A = area on the sphere); unit sr. A full sphere = 4π sr.
- Easy : A = 0.5 m², r = 2 m ⇒ Ω = 0.5 ÷ 4 = 0.125 sr.
- Medium : Sun : diameter 1.39 × 10⁹ m, distance 1.5 × 10¹¹ m ⇒ θ = 0.0093 rad ≈ 0.53°.
- Exam (CET) : Solid angle of a full sphere? (1) 4π sr (2) 2π sr (3) π sr (4) 360 sr → answer 1)
Derivation : 1 radian = 57.3°
- Step 1 : The arc of a full circle is its circumference 2π r.
- Step 2 : θ = s ÷ r = 2π r ÷ r = 2π rad.
- Step 3 : 2π rad = 360° ⇒ π rad = 180°.
- Step 4 : 1 rad = 180° ÷ π ≈ 57.3°.
θ = s/r; Ω = A/r²; π rad = 180°; 4π sr = full sphere.
rad and sr are supplementary units; angles are dimensionless; use π/180 correctly for degree ↔ radian.
Memory clue : "K A mol cd + s m kg" — the 7 base units.
Spot the difference
| Property | Plane angle | Solid angle |
|---|---|---|
| Formula | s/r | A/r² |
| Unit | rad | sr |
| Full value | 2π rad | 4π sr |
| Dimension | none | none |
Practice questions
- 1List the 7 SI base units.
- 2Formula for a radian?
- 345° in radians?
- 4s = 3 m, r = 1.5 m : θ?
- 5θ = 0.2 rad, r = 5 m : s?
- 6A = 2 m², r = 4 m : Ω?
- 7Moon : diameter 3.5 × 10⁶ m, distance 3.8 × 10⁸ m : θ in degrees?
- 8Arc for 2 rad at r = 7 m? Area for 1 sr at r = 7 m?
Show answers (try them yourself first)
- 1m, kg, s, A, K, mol, cd
- 2s ÷ r
- 3π/4 = 0.785 rad
- 4θ = 2 rad
- 5s = 1 m
- 6Ω = 0.125 sr
- 7θ = 0.53°
- 8s = 14 m; A = 49 m²
Measurement of length, mass and time
A near distance is read with a ruler, a far star's distance with a triangle — the method changes with the object.
Parallax is the apparent shift in an object's position when seen from two places; it reveals large distances.
In the figure : D = b ÷ θ; θ must be small.
Rule 1: Parallax method — For a distant object, measure the baseline b between two observation points and the parallax angle θ. D = b ÷ θ (θ in radians, small).
- Easy : b = 10 m, θ = 0.05 rad ⇒ D = 200 m.
- Medium : Star : b = 3 × 10¹¹ m, θ = 2 × 10⁻⁵ rad ⇒ D = 1.5 × 10¹⁶ m ≈ 1.6 light-years.
- Exam (CET) : D = ? (1) b/θ (2) bθ (3) θ/b (4) b² → answer 1)
Rule 2: Vernier and screw gauge — Vernier LC = 1 MSD − 1 VSD; screw gauge LC = pitch ÷ divisions. Reading = main + (coinciding × LC) ± zero error.
- Easy : 1 MSD = 1 mm, 10 VSD = 9 MSD ⇒ LC = 0.1 mm.
- Medium : Pitch 1 mm, 100 divisions ⇒ LC = 0.01 mm; main 4 mm, circular 37 ⇒ reading = 4.37 mm.
- Exam (CET) : Screw gauge LC = ? (1) pitch/divisions (2) pitch × divisions (3) divisions/pitch (4) 1 mm → answer 1)
Rule 3: SI definitions — Metre : distance light covers in 1/299 792 458 s; second : 9 192 631 770 periods of Cs-133 radiation; kilogram : fixed through h = 6.62607015 × 10⁻³⁴ J s.
- Easy : Light travels 299 792 458 m in 1 s.
- Medium : 1 u = 1.66 × 10⁻²⁷ kg (atomic mass unit).
- Exam (CET) : The second is based on? (1) Cs-133 periods (2) Earth's rotation (3) a pendulum (4) the Moon → answer 1)
Derivation : D = b ÷ θ
- Step 1 : The two points and the object form a long, narrow triangle.
- Step 2 : D acts as the radius; b acts as the arc (θ small).
- Step 3 : θ = arc ÷ radius = b ÷ D.
- Step 4 : D = b ÷ θ.
D = b/θ; vernier LC = 1 MSD − 1 VSD; screw gauge LC = pitch/divisions.
Do not use θ in degrees in D = b/θ; use radians. Never forget to correct the zero error.
Memory clue : "LC = least division gap".
Spot the difference
| Instrument | LC formula | LC (e.g.) |
|---|---|---|
| Metre scale | smallest division | 1 mm |
| Vernier | 1 MSD − 1 VSD | 0.1 mm |
| Screw gauge | pitch ÷ divisions | 0.01 mm |
Practice questions
- 1What is parallax?
- 2What is least count?
- 3What is a screw gauge used for?
- 4b = 6 m, θ = 0.03 rad : D?
- 5Pitch 0.5 mm, 50 divisions : LC?
- 6Main 2.5 mm, circular 18, LC 0.01 mm : reading?
- 7Zero error +0.05 mm, reading 3.27 mm : true reading?
- 8D = 2 × 10⁶ m, b = 4 m : θ?
Show answers (try them yourself first)
- 1Apparent shift in position seen from two points
- 2Smallest value an instrument can measure
- 3Measuring small thickness/diameter
- 4D = 200 m
- 5LC = 0.01 mm
- 62.68 mm
- 73.22 mm
- 8θ = 2 × 10⁻⁶ rad
Dimensions and dimensional analysis
You cannot add apples and oranges; you cannot add metres and seconds either — dimensions must match.
The dimension of a quantity shows its powers of the base quantities (M, L, T).
In the figure : Use dimensions on T = k l^a g^b to find a and b.
Rule 1: Dimensional formulas — Base dimensions : M, L, T, (I, K, mol, cd). Velocity [L T⁻¹], acceleration [L T⁻²], force [M L T⁻²], work/energy [M L² T⁻²], power [M L² T⁻³], pressure [M L⁻¹ T⁻²], momentum [M L T⁻¹].
- Easy : Force = m a ⇒ [M] [L T⁻²] = [M L T⁻²].
- Medium : Pressure = force ÷ area = [M L T⁻²] ÷ [L²] = [M L⁻¹ T⁻²].
- Exam (CET) : Dimension of energy? (1) M L² T⁻² (2) M L T⁻² (3) M L² T⁻³ (4) M L T⁻¹ → answer 1) Trick : energy = force × length.
Rule 2: Principle of homogeneity — Every term of an equation must have the same dimension. A correct equation may pass; a wrong one is surely rejected.
- Easy : v = u + a t : [L T⁻¹] = [L T⁻¹] + [L T⁻²][T] ✓.
- Medium : s = u t + a t : [L] vs [L T⁻²][T] = [L T⁻¹] ✗ (wrong).
- Exam (CET) : Which is homogeneous? (1) v² = u² + 2as (2) v = u + a (3) s = u + at (4) v = at² → answer 1)
Rule 3: Conversions and deriving formulas — Converting between systems : n₁ u₁ = n₂ u₂. 1 N = 10⁵ dyne; 1 J = 10⁷ erg. Limits : constants (2π) are not found; fails for sin, e^x; sums are awkward.
- Easy : 5 N = 5 × 10⁵ dyne.
- Medium : 20 J = 20 × 10⁷ erg = 2 × 10⁸ erg.
- Exam (CET) : 1 J = ? (1) 10⁷ erg (2) 10⁵ erg (3) 10³ erg (4) 10⁻⁷ erg → answer 1)
Derivation : T = k √(l ÷ g) (by dimensions)
- Step 1 : Assume T = k l^a g^b (k dimensionless).
- Step 2 : Dimensions : [T] = [L]^a [L T⁻²]^b = L^(a+b) T^(−2b).
- Step 3 : T : −2b = 1 ⇒ b = −1/2; L : a + b = 0 ⇒ a = 1/2.
- Step 4 : T = k √(l ÷ g); k = 2π is found by experiment.
[F] = M L T⁻²; [E] = M L² T⁻²; [P] = M L² T⁻³; check homogeneity.
Matching dimensions do not make a formula certainly right (2π, ½ unknown); mismatched ones are certainly wrong.
Memory clue : "Force = MLT⁻²; energy = force × L".
Spot the difference
| Quantity | Dimension | Unit |
|---|---|---|
| Velocity | L T⁻¹ | m s⁻¹ |
| Force | M L T⁻² | N |
| Energy | M L² T⁻² | J |
| Power | M L² T⁻³ | W |
| Pressure | M L⁻¹ T⁻² | Pa |
Practice questions
- 1Dimension of force?
- 2State the principle of homogeneity.
- 31 N = ? dyne
- 4Dimension of power?
- 5Check F = m v²/r.
- 63 J = ? erg
- 7In T = 2π√(m/k) find the dimension of k.
- 8Compare the dimensions of impulse and momentum.
Show answers (try them yourself first)
- 1M L T⁻²
- 2Every term has the same dimension
- 310⁵ dyne
- 4M L² T⁻³
- 5[M][L²T⁻²]/[L] = [M L T⁻²] ✓
- 63 × 10⁷ erg
- 7k = m/T² : M T⁻²
- 8Both M L T⁻¹ — equal
Accuracy, precision and errors
Two kids shoot arrows : one hits the centre, the other's all land together but off to one side.
Error is the difference between the measured and true values.
In the figure : Green : accurate and precise; orange : precise but not accurate.
Rule 1: Accuracy and precision — Accuracy = closeness to the true value; precision = closeness of repeated readings to each other. Both are independent.
- Easy : All readings near 9.8 m/s² (true 9.8) ⇒ accurate.
- Medium : Readings only 9.30–9.31, true 9.8 ⇒ precise, not accurate.
- Exam (CET) : Precision means? (1) closeness of readings (2) near the true value (3) zero error (4) large LC → answer 1)
Rule 2: Types of errors — Systematic : fault of instrument/observer/method, one direction. Random : irregular. Gross : carelessness, misreading.
- Easy : An instrument with zero error ⇒ systematic.
- Medium : Reading fluctuating from air currents ⇒ random.
- Exam (CET) : Zero error is? (1) systematic (2) random (3) gross (4) both → answer 1)
Rule 3: Absolute, relative and percentage error — Δa_i = |a_mean − a_i|; Δa_mean = (ΣΔa_i) ÷ n; relative = Δa_mean ÷ a_mean; percentage = relative × 100. Result : a = a_mean ± Δa_mean.
- Easy : T : 4.52, 4.48, 4.55, 4.50, 4.45 s ⇒ a_mean = 4.50 s.
- Medium : Δa_i = 0.02, 0.02, 0.05, 0.00, 0.05 ⇒ Δa_mean = 0.028 ≈ 0.03 s; percentage = 0.62%.
- Exam (CET) : Unit of relative error? (1) none (2) m (3) s (4) % only → answer 1)
Derivation : Mean absolute error
- Step 1 : n readings a₁, a₂, … aₙ ⇒ a_mean = Σa ÷ n.
- Step 2 : Absolute error of each Δa_i = |a_mean − a_i|.
- Step 3 : Mean Δa_mean = (Δa₁ + … + Δaₙ) ÷ n.
- Step 4 : Result a = a_mean ± Δa_mean.
Δa_i = |a_mean − a_i|; relative = Δa_mean ÷ a_mean; percentage × 100.
Accuracy and precision differ; relative error has no unit; Δa is always positive.
Memory clue : "Accurate = true; precise = same".
Spot the difference
| Property | Absolute error | Relative error | Percentage | ||
|---|---|---|---|---|---|
| Formula | a_m − a_i | Δa/a_m | Δa/a_m × 100 | ||
| Unit | of the quantity | none | % |
Practice questions
- 1What is accuracy?
- 2Example of a systematic error?
- 3What is relative error?
- 4Readings 2.1, 2.3, 2.2 : mean?
- 5Mean absolute error?
- 6a = 50, Δa = 0.5 : percentage?
- 7LC 0.1 cm, reading 4.6 cm : percentage error?
- 8How is random error reduced?
Show answers (try them yourself first)
- 1Closeness to the true value
- 2Zero error
- 3Δa_mean ÷ a_mean
- 42.2
- 50.067 ≈ 0.07
- 61%
- 72.17%
- 8By taking many readings and averaging
Combination of errors
When you add two bills each slightly off, the small mistakes pile up in the total.
Combination : finding the error in a result calculated from two or more measured quantities.
In the figure : Blue A ± ΔA, green B ± ΔB, purple the sum.
Rule 1: Sum and difference — Z = A ± B ⇒ ΔZ = ΔA + ΔB (absolute errors add in both cases).
- Easy : (12.5 ± 0.1) + (7.2 ± 0.1) = 19.7 ± 0.2.
- Medium : (12.5 ± 0.1) − (7.2 ± 0.1) = 5.3 ± 0.2 (add even for difference).
- Exam (CET) : Z = A − B, ΔA = 0.2, ΔB = 0.1 ⇒ ΔZ? (1) 0.3 (2) 0.1 (3) 0.2 (4) 0.05 → answer 1)
Rule 2: Product and quotient — Z = A B or A/B ⇒ ΔZ/Z = ΔA/A + ΔB/B (relative errors add).
- Easy : V = 10 ± 0.2 V, I = 2 ± 0.1 A ⇒ R = V/I : ΔR/R = 0.02 + 0.05 = 0.07 ⇒ 7%.
- Medium : R = 5 Ω ⇒ ΔR = 0.07 × 5 = 0.35 Ω.
- Exam (CET) : In A/B the relative errors? (1) add (2) subtract (3) multiply (4) divide → answer 1)
Rule 3: Powers — Z = Aᵖ Bᵠ ÷ Cʳ ⇒ ΔZ/Z = p ΔA/A + q ΔB/B + r ΔC/C; the quantity with the larger power contributes most.
- Easy : g = 4π² l / T² : l error 1%, T error 2% ⇒ g error = 1 + 2 × 2 = 5%.
- Medium : Cube side with 2% error ⇒ volume 3 × 2 = 6%.
- Exam (CET) : Z = A²B ⇒ % error in Z? (1) 2a + b (2) a + b (3) a + 2b (4) 2ab → answer 1) (a, b = % errors of A, B).
Derivation : Relative error in a product
- Step 1 : Z = A B; Z ± ΔZ = (A ± ΔA)(B ± ΔB).
- Step 2 : = A B ± A ΔB ± B ΔA ± ΔA ΔB; ΔA ΔB is tiny ⇒ neglect it.
- Step 3 : ± ΔZ = ± A ΔB ± B ΔA (maximum).
- Step 4 : Divide by A B : ΔZ/Z = ΔA/A + ΔB/B.
± : ΔA + ΔB; × ÷ : ΔA/A + ΔB/B; Aⁿ : n ΔA/A.
Errors add even in a subtraction; multiply by the exponent for powers; add relative errors in %.
Memory clue : "No subtracting errors — always add".
Spot the difference
| Operation | Rule | What adds |
|---|---|---|
| A ± B | ΔA + ΔB | absolute |
| A × B, A ÷ B | ΔA/A + ΔB/B | relative |
| Aⁿ | n ΔA/A | relative × n |
Practice questions
- 1Error in a sum?
- 2Error in a quotient?
- 3Relative error of Z = A³?
- 4(8.0 ± 0.1) + (3.0 ± 0.2) : ?
- 5V = 5 ± 0.1, I = 2 ± 0.05 : % error in R?
- 61% error in radius : error in sphere volume?
- 7m = 5 ± 0.15 kg, ρ = 5 ± 0.2 : % error in V = m/ρ?
- 8l = 100 ± 0.1 cm, T = 2 ± 0.01 s : % error in g?
Show answers (try them yourself first)
- 1Absolute errors add
- 2Relative errors add
- 33 ΔA/A
- 411.0 ± 0.3
- 54.5%
- 63%
- 77%
- 81.1%
Significant figures and order of magnitude
A shopkeeper says 'about two and a half kilos'; science states how many digits are reliable.
Significant figures = the certain digits plus the first doubtful digit.
In the figure : 2.6 certain, 5 doubtful = 3 significant figures.
Rule 1: Rules for significant figures — 1) All non-zero digits count. 2) Zeros between non-zero digits count. 3) Leading zeros (0.0014) do not. 4) Trailing zeros after a decimal point (1.500) count. 5) 12500 has only 3 (ambiguous) ⇒ write 1.25 × 10⁴.
- Easy : 178.43 : 5; 165.02 : 5.
- Medium : 0.001405 : 4; 1.500 : 4; 0.01500 : 4.
- Exam (CET) : Significant figures in 0.00250? (1) 3 (2) 5 (3) 2 (4) 6 → answer 1) Trick : drop leading zeros.
Rule 2: Rounding and arithmetic — Rounding : dropped digit > 5 ⇒ round up; < 5 ⇒ keep; = 5 ⇒ round up if the previous digit is odd, keep if even. Sum/difference : fewest decimal places. Product/quotient : fewest significant figures.
- Easy : 2.34 + 1.2 = 3.54 ⇒ 3.5.
- Medium : 2.5 × 3.14 = 7.85 ⇒ 7.9 (2 s.f.).
- Exam (CET) : 5.28 × 1.1 ⇒ answer? (1) 5.8 (2) 5.81 (3) 5.808 (4) 6 → answer 1)
Rule 3: Order of magnitude — Write the number as A × 10ⁿ with 0.5 ≤ A < 5; n is the order of magnitude (as per the textbook).
- Easy : Earth's radius 6400 km = 6.4 × 10⁶ m = 0.64 × 10⁷ m ⇒ order 7.
- Medium : e = 1.6 × 10⁻¹⁹ C ⇒ order −19.
- Exam (CET) : Order of magnitude of 3 × 10⁸? (1) 8 (2) 9 (3) 7 (4) 3 → answer 1)
Derivation : Why sums keep the fewest decimals
- Step 1 : In 2.34 the last digit 4 is doubtful (hundredths).
- Step 2 : In 1.2 the digit 2 is doubtful (tenths).
- Step 3 : Digits beyond the tenths place are meaningless in the sum.
- Step 4 : 3.54 → 3.5 (fewer places).
+/− : fewest decimals; × ÷ : fewest significant figures; N = A × 10ⁿ (0.5 ≤ A < 5).
In a product count significant figures, not decimals; use the textbook rule for order of magnitude.
Memory clue : "+ − = decimals; × ÷ = figures".
Spot the difference
| Number | Sig. figs | Reason |
|---|---|---|
| 0.0052 | 2 | leading zeros dropped |
| 3.050 | 4 | middle and trailing count |
| 12500 | 3 (ambiguous) | write 1.25 × 10⁴ |
| 6400 km | 2 | 0.64 × 10⁷ m |
Practice questions
- 1Sig. figs in 0.0450?
- 2Sig. figs in 10.20?
- 3What is order of magnitude?
- 412.3 + 0.456 : ?
- 54.5 × 2.34 : ?
- 6Round 2.65 to 2 s.f.
- 7Order of 1.5 × 10⁻¹⁹?
- 8Order of 7.6 × 10⁴?
Show answers (try them yourself first)
- 13
- 24
- 3n in A × 10ⁿ
- 412.8
- 511
- 62.6
- 7−19
- 85
Quick revision of the whole lesson
| Concept | Key formula / fact | Unit / dimensions |
|---|---|---|
| Plane angle | θ = s/r | rad |
| Solid angle | Ω = A/r² | sr |
| Parallax | D = b/θ | m |
| Vernier | LC = 1 MSD − 1 VSD | mm |
| Screw gauge | LC = pitch/divisions | mm |
| Dimensions | force MLT⁻²; energy ML²T⁻² | — |
| Percentage error | Δa/a × 100 | % |
| Product error | ΔZ/Z = ΔA/A + ΔB/B | — |
| Sig. figures | +− : decimals; ×÷ : figures | — |
🎯 Most important points for the exam
- 7 base units; π rad = 180°; 4π sr = sphere.
- D = b/θ (θ in rad).
- Dimensions : force MLT⁻², energy ML²T⁻², pressure ML⁻¹T⁻².
- Accuracy ≠ precision; zero error is systematic.
- Sum/difference : add Δ; product : add relative.
- Powers : n ΔA/A.
- Order of magnitude : A × 10ⁿ, 0.5 ≤ A < 5.
Check units → match dimensions → add errors → keep significant figures.