Units and Measurements lays the foundation of physics by defining how every physical quantity is measured against an agreed standard. It covers the seven SI base quantities and their derived units, dimensional formulae and the principle of homogeneity, and how to handle accuracy, errors and significant figures. Mastering it is essential for NEET because dimensional analysis and error propagation appear as direct one-mark questions and underpin numerical work across the whole syllabus.
Table of Contents
- Key Concepts — physical quantities, SI units, dimensions, significant figures, errors, accuracy
- Weightage in Board & Entrance Exams
- Important Definitions
- Solved Examples
- Important Questions for Board Exams
- Quick Revision Points
Key Concepts
1. Physical Quantities and Units
A physical quantity is any quantity that can be measured, such as length, mass, time, or temperature. Every measurement is written as a number followed by a unit — the chosen standard of that quantity (for example, 5 metres means 5 times the standard length called a metre).
Physical quantities are of two kinds. Fundamental (base) quantities are independent and cannot be expressed in terms of others — length, mass, and time are examples. Derived quantities are built from base quantities, such as speed (length/time) or force (mass × acceleration).
Key idea: The magnitude of a physical quantity = numerical value × unit. The bigger the unit, the smaller the numerical value (n₁u₁ = n₂u₂).
2. The SI System of Units
The SI system (Système Internationale d’Unités) is the internationally agreed set of units, built on seven base units. It is a coherent, rationalised, and metric system used worldwide in science.
The Seven Fundamental SI Units
| Base Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
There are also two supplementary units: the radian (rad) for plane angle and the steradian (sr) for solid angle.
Derived units are combinations of base units — for example the newton (N = kg·m/s²) for force and the joule (J = kg·m²/s²) for energy.
3. Dimensions and Dimensional Formulae
The dimensions of a physical quantity are the powers to which the base quantities must be raised to represent it. The base quantities are written as M (mass), L (length), T (time), A (current), K (temperature), mol, and cd.
The dimensional formula is the expression showing these powers. For example, the dimensional formula of force is [M¹L¹T⁻²], and the corresponding dimensional equation is [F] = [MLT⁻²].
Dimensional Formulae of Common Quantities
| Quantity | Formula | Dimensional Formula |
|---|---|---|
| Area | length × breadth | [L²] |
| Velocity | displacement/time | [M⁰L¹T⁻¹] |
| Acceleration | velocity/time | [M⁰L¹T⁻²] |
| Force | mass × acceleration | [M¹L¹T⁻²] |
| Work / Energy | force × displacement | [M¹L²T⁻²] |
| Power | work/time | [M¹L²T⁻³] |
| Pressure | force/area | [M¹L⁻¹T⁻²] |
| Momentum / Impulse | mass × velocity | [M¹L¹T⁻¹] |
Dimensionless quantities (like strain, angle, refractive index, and pure numbers) have all powers zero, i.e. [M⁰L⁰T⁰].
4. Principle of Homogeneity of Dimensions
An equation is dimensionally correct only if every term on both sides has the same dimensions. You can only add or equate quantities of the same dimensions — you cannot add a length to a time.
This principle of homogeneity is the basis of dimensional analysis. For example, in v = u + at, every term has the dimension of velocity [LT⁻¹], so the equation is dimensionally consistent.
Key idea: Dimensional correctness is necessary but not sufficient — an equation can be dimensionally correct yet still wrong by a numerical factor.
5. Applications of Dimensional Analysis
Dimensional analysis is a powerful checking and deriving tool. It has three main uses.
- Checking the correctness of an equation using the principle of homogeneity.
- Converting a unit from one system to another using n₁[M₁ᵃL₁ᵇT₁ᶜ] = n₂[M₂ᵃL₂ᵇT₂ᶜ].
- Deriving a relation between physical quantities when the dependence is known (for example, deriving T = 2π√(l/g) for a pendulum, up to the constant).
6. Limitations of Dimensional Analysis
Dimensional analysis cannot do everything. Knowing its limits is a favourite board question.
- It cannot find the value of dimensionless constants (like 2π or ½).
- It cannot be used if a quantity depends on more than three other quantities.
- It fails for equations involving trigonometric, exponential, or logarithmic functions.
- It cannot tell whether a quantity is a scalar or a vector.
- It cannot derive relations that contain a sum of terms, such as s = ut + ½at².
7. Significant Figures
Significant figures are the digits in a measurement that are known reliably plus the first uncertain digit. They tell you the precision of a measurement — 2.50 m is more precise than 2.5 m.
Rules for Counting Significant Figures
- All non-zero digits are significant (234 has 3).
- Zeros between non-zero digits are significant (2.004 has 4).
- Leading zeros are not significant (0.0025 has 2).
- Trailing zeros after a decimal point are significant (2.300 has 4).
- Trailing zeros in a number without a decimal are ambiguous (best written in scientific notation).
Rules for Rounding Off and Calculations
- Addition/subtraction: the result keeps as many decimal places as the term with the fewest decimal places.
- Multiplication/division: the result keeps as many significant figures as the term with the fewest significant figures.
- If the digit to be dropped is greater than 5, round up; if less than 5, round down; if exactly 5, round to the nearest even digit.
8. Errors in Measurement
No measurement is perfectly exact — every measurement has an error, the difference between the measured value and the true value. Understanding errors lets you state results honestly.
Types of Errors
| Type | Description |
|---|---|
| Systematic error | Consistent, one-directional error from faulty instruments, technique, or surroundings; can be minimised |
| Random error | Irregular error varying in size and sign; reduced by taking many readings and averaging |
| Gross error | Human mistake in reading or recording observations |
Calculating Errors
If a quantity is measured n times giving readings a₁, a₂, … aₙ, the best value is the arithmetic mean.
- Mean value: aₘₑₐₙ = (a₁ + a₂ + … + aₙ)/n
- Absolute error of each reading: Δaᵢ = |aₘₑₐₙ − aᵢ|
- Mean absolute error: Δaₘₑₐₙ = (|Δa₁| + |Δa₂| + … + |Δaₙ|)/n
- Relative (fractional) error: Δaₘₑₐₙ/aₘₑₐₙ
- Percentage error: (Δaₘₑₐₙ/aₘₑₐₙ) × 100%
9. Combination (Propagation) of Errors
When a result is calculated from several measured quantities, their errors combine. The rules below tell you the maximum possible error.
- Sum or difference (Z = A ± B): the absolute errors add — ΔZ = ΔA + ΔB.
- Product or quotient (Z = AB or A/B): the relative errors add — ΔZ/Z = ΔA/A + ΔB/B.
- Power (Z = A^p·B^q/C^r): ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C).
Key idea: In a product/quotient, a quantity raised to a higher power contributes more to the total error, so it must be measured most carefully.
10. Accuracy and Precision
These two words sound similar but mean different things, and the difference is a classic 1-mark question.
- Accuracy is how close a measured value is to the true value. High accuracy means small systematic error.
- Precision is how close repeated measurements are to each other — the resolution or fineness of the instrument. High precision means small random error.
A measurement can be precise but not accurate (consistently wrong) or accurate but not precise. The best measurements are both.
11. Measurement of Large Distances — Parallax Method
Parallax is the apparent shift in the position of an object when viewed from two different points. Hold a pen at arm’s length and look at it with each eye in turn — its position shifts against the background. That shift is parallax.
To measure the distance D of a far object (like a planet or star), it is observed from two points separated by a known basis b. If θ is the parallax angle subtended (in radians), then:
D = b/θ
The same idea (angular diameter) gives the size of a planet: d = Dα, where α is the angle the planet subtends at the observer.
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Weightage in Board & Entrance Exams
| Exam | Typical Weightage | Most-Tested Areas |
|---|---|---|
| CBSE Board (Class 11) | 4–5 marks | Dimensional formulae, error analysis, significant figures |
| JEE Main / Advanced | 1–2 questions | Dimensional analysis, combination of errors, unit conversion |
| NEET | 1–2 questions | Dimensional formulae, significant figures, percentage error |
[TABLE: Question-type split — VSA (1 mark): SI units, accuracy vs precision, significant figures; SA (2–3 marks): dimensional formulae, error calculations; LA (5 marks): checking/deriving equations by dimensions, combination of errors.]
Important Definitions
| Term | Definition |
|---|---|
| Physical quantity | A measurable property, expressed as numerical value × unit |
| Fundamental unit | An independent base unit not derived from others (m, kg, s, etc.) |
| Derived unit | A unit built from base units (e.g. N = kg·m/s²) |
| Dimensional formula | Expression of a quantity in powers of M, L, T (e.g. force = [MLT⁻²]) |
| Principle of homogeneity | Every term in a correct equation has the same dimensions |
| Significant figures | Reliably known digits plus the first uncertain digit |
| Absolute error | Magnitude of the difference between mean and measured value |
| Relative error | Ratio of mean absolute error to mean value |
| Accuracy | Closeness of a measurement to the true value |
| Precision | Closeness of repeated measurements to one another |
| Parallax | Apparent shift of an object viewed from two different points |
Solved Examples
Example 1
Find the dimensional formula of pressure.
Answer: Pressure = force/area = [MLT⁻²]/[L²] = [M¹L⁻¹T⁻²].
Example 2
Check whether the equation v² = u² + 2as is dimensionally correct.
Answer: [v²] = [L²T⁻²]; [u²] = [L²T⁻²]; [2as] = [LT⁻²][L] = [L²T⁻²]. All terms have dimension [L²T⁻²], so the equation is dimensionally correct.
Example 3
The radius of a sphere is measured as 2.1 cm. Express its volume to the correct significant figures.
Answer: V = (4/3)πr³ = (4/3)(3.14)(2.1)³ = 38.79 cm³. Since the radius has 2 significant figures, V = 39 cm³.
Example 4
The length of a rod is measured as 25.4 cm with an absolute error of 0.1 cm. Find the relative and percentage error.
Answer: Relative error = 0.1/25.4 = 0.0039. Percentage error = 0.0039 × 100 = 0.39%.
Example 5
A physical quantity Z = A²B/C. If the percentage errors in A, B, and C are 1%, 2%, and 3%, find the maximum percentage error in Z.
Answer: ΔZ/Z = 2(ΔA/A) + (ΔB/B) + (ΔC/C) = 2(1) + 2 + 3 = 7%.
Example 6
Convert a force of 1 newton into dyne using dimensional analysis. (1 N = 1 kg·m/s², 1 dyne = 1 g·cm/s²)
Answer: 1 N = (1000 g)(100 cm)/s² = 1000 × 100 g·cm/s² = 10⁵ dyne.
Important Questions for Board Exams
1-Mark Questions (VSA)
- Name the seven fundamental SI units.
- What is the difference between accuracy and precision?
- How many significant figures are there in 0.00230?
- Write the dimensional formula of work.
- Are all dimensionally correct equations physically correct? Justify.
2–3-Mark Questions (SA)
- State the principle of homogeneity and use it to check v = u + at.
- Define absolute error, relative error, and percentage error.
- State any three limitations of dimensional analysis.
- Explain how the parallax method is used to measure the distance of a far-off planet.
5-Mark Questions (LA)
- What is dimensional analysis? Explain its three applications with one example each.
- Derive the rule for the combination of errors in a product Z = AB, and hence in Z = A^p·B^q.
- State the rules for counting significant figures and for rounding off, with one example of each.
Quick Revision Points
- Physical quantity = numerical value × unit; n₁u₁ = n₂u₂
- Seven SI base units: m, kg, s, A, K, mol, cd; plus radian and steradian
- Dimensional formula: powers of M, L, T (force = [MLT⁻²])
- Principle of homogeneity: all terms must have the same dimensions
- Dimensional analysis: check equations, convert units, derive relations
- Limits: no dimensionless constants, no trig/exp/log, no sums of terms
- Significant figures show precision; follow add/subtract and multiply/divide rules
- Errors: absolute Δa, relative Δa/a, percentage (Δa/a) × 100
- Sum/difference: absolute errors add; product/quotient: relative errors add
- Power rule: ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C)
- Accuracy = closeness to true value; precision = closeness of repeats
- Parallax: D = b/θ; angular size: d = Dα
Next Chapter: Chapter 2 — Motion in a Straight Line
Class 11 Physics · Chapter 2 – swipe through all 10 cards to understand the whole chapter.
Why units exist
A measurement is meaningless until a number is paired with a unit.
A bare number in a numerical answer loses marks.
- The unit is the agreed standard you compare against
- Same quantity, different units: 5 m/s = 18 km/h
- Always write the unit with the number
The 7 SI Base Quantities
Seven independent base quantities are the building blocks of all physics.
Mnemonic: Lovely Maths Teachers Are Truly Magical Companions.
- Length→metre, Mass→kilogram, Time→second
- Current→ampere, Temperature→kelvin
- Amount→mole, Luminous intensity→candela
Derived Units
Anything that is not one of the 7 base units is built from them.
Newton, joule, watt, pascal are derived — NOT base.
- Force = mass × acceleration → kg·m·s⁻2
- Energy = force × distance → kg·m2·s⁻2
- 1 Pa = 1 N/m2 = kg·m⁻1·s⁻2
Base vs Derived (NEET trap)
Classic MCQ: which is NOT a base unit?
Kelvin has no degree sign — never write °K.
- Mass base unit is the kilogram, not the gram
- Candela IS base; lumen and lux are derived
- Newton / joule / watt / pascal are all derived
Dimensional Formulae
Express any mechanical quantity in powers of M, L and T.
Angle, strain, refractive index are dimensionless: [M0L0T0].
- Acceleration = [L T⁻2], Area = [L2]
- Work / Energy = [M L2 T⁻2]
- Add current (A) as a 4th dimension for electrical quantities
Principle of Homogeneity
Every term in a valid physical equation must have the same dimensions.
Can prove an equation WRONG, but never fully right.
- Check correctness, convert units, guess a relation
- It misses dimensionless constants like ½ and 2π
- Work and torque share [M L2 T⁻2] but differ physically
Accuracy vs Precision
Two different things — closeness to truth vs consistency.
You can be precise but inaccurate (always off by the same amount).
- Accuracy = how close to the true value
- Precision = how finely / consistently you measure
- Least count sets the smallest error you can claim
Error Propagation
Errors always pile up — never cancel (worst-case rule).
Even a divided quantity contributes a PLUS sign.
- Sums / differences → add absolute errors
- Products / powers → add relative (%) errors
- A power becomes a multiplier of the % error
Power = Multiplier
When a quantity is raised to a power, its % error multiplies.
Surface area ∝ r2 would give 4%; constant 4⁄3 π adds nothing.
- Find the power, then multiply the % error by it
- Forgetting this multiplier is the top NEET slip
- Pure constants like π have zero error
Significant Figures
Sig figs are the reliably known digits in a measurement.
Multiply/divide: answer keeps sig figs of the least precise input.
- All non-zero digits, and zeros between them, count
- Leading zeros NEVER count (they only place the decimal)
- Trailing zeros after a decimal point DO count
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Frequently Asked Questions
A base unit measures one of the seven independent SI base quantities (metre, kilogram, second, ampere, kelvin, mole and candela). A derived unit is built by combining base units, for example the newton (kg and m per s squared) or the joule (kg and m squared per s squared).
It states that every term in a valid physical equation must have the same dimensions. It can be used to check whether an equation could be correct, convert between unit systems and guess the form of a relation, but it cannot prove an equation fully right because it misses dimensionless constants like one half or two pi.
For products and powers you add the relative (percentage) errors, and a power multiplies its term’s percentage error. For example, if radius has a 2 percent error, the volume (which is proportional to radius cubed) has a 3 times 2 equal to 6 percent error.
Yes. It is part of the Class 11 Physics NEET syllabus and usually contributes about one question. Dimensional analysis, error propagation and significant figures are favourite topics, and the concepts support numerical accuracy throughout the rest of physics.
Accuracy is how close a measurement is to the true value, while precision is how finely and consistently you can measure. A reading can be precise but inaccurate if it is consistently off by the same amount.