Units and Measurements Class 11 Notes | CBSE Physics Chapter 1

Units and Measurements is Chapter 1 of CBSE Class 11 Physics — the chapter that quietly decides how many easy marks you collect all year. Every numerical you ever solve depends on getting units, dimensions, significant figures, and errors right, so this is the toolkit chapter for the whole of physics. Get it solid now and you stop losing silly marks in JEE, NEET, and boards.

By the end of these notes you will be able to write the dimensional formula of any quantity, check whether an equation is correct using dimensional analysis, round off answers to the right significant figures, and calculate absolute, relative, and percentage errors confidently. This is a scoring chapter carrying roughly 4–5 marks in boards, and the foundation for every numerical you will attempt across mechanics, thermodynamics, and electricity.

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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

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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).

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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.

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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].

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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.

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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).

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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.

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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.

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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%

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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.

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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.

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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.]

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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

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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.

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Important Questions for Board Exams

1-Mark Questions (VSA)

1. Name the seven fundamental SI units.
2. What is the difference between accuracy and precision?
3. How many significant figures are there in 0.00230?
4. Write the dimensional formula of work.
5. Are all dimensionally correct equations physically correct? Justify.

2–3-Mark Questions (SA)

1. State the principle of homogeneity and use it to check v = u + at.
2. Define absolute error, relative error, and percentage error.
3. State any three limitations of dimensional analysis.
4. Explain how the parallax method is used to measure the distance of a far-off planet.

5-Mark Questions (LA)

1. What is dimensional analysis? Explain its three applications with one example each.
2. Derive the rule for the combination of errors in a product Z = AB, and hence in Z = A^p·B^q.
3. State the rules for counting significant figures and for rounding off, with one example of each.

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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

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Next Chapter: Chapter 2 — Motion in a Straight Line