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Science Grade XI Physics

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Chapter 1: Units and Measurements

Grade 11 Science | Chapter 1

Units and Measurements

All of physics rests on careful measurement. This chapter sets out the SI system, dimensions, significant figures and the treatment of errors that every later calculation depends on.

Core Concepts

Key Principles

Worked Examples

Practice Sets

Contents

Introduction: Why We Measure

Physical Quantities and SI Units

Dimensions and Dimensional Formulae

Dimensional Analysis

Significant Figures and Rounding

Errors in Measurement

Key Reasoning (Principles)

Worked Examples (10)

Practice Sets A to D

10.

Summary and Exam Quick-Check

1. Introduction: Why We Measure

Physics describes the world with numbers, and a number is only meaningful when it carries a unit. To measure is to compare a quantity with an agreed standard. A length of 5 means nothing until we say 5 metres. This chapter sets up the shared system of units used worldwide, the idea of dimensions that lets us check equations, and the honest reporting of precision through significant figures and errors.

These tools are not a side topic; they run through the whole of physics. A dimensional check can catch a wrong formula in seconds, and a clear grasp of errors tells us how far to trust a result.

Core idea

A measurement is a number together with a unit. The SI system fixes the standards, dimensions let us check equations, and significant figures and errors report how precise a result really is.

2. Physical Quantities and SI Units

A physical quantity is anything that can be measured. Fundamental (or base) quantities are chosen as independent, and all others are derived from them. The SI system defines seven base quantities and their units; for example speed, a derived quantity, has the unit metre per second built from length and time.

Diagram 1 – The Seven SI Base Quantities

Seven SI base quantities around a central SI hub: length, mass, time, current, temperature, amount, luminous intensity

Fig 1. The seven base quantities and their SI units, from which every other unit in physics is built.

3. Dimensions and Dimensional Formulae

The dimensions of a quantity show how it is built from the base quantities mass (M), length (L) and time (T). Speed is length per time, so its dimensional formula is written [L T^-1]. Force, being mass times acceleration, has dimensions [M L T^-2]. Two quantities can be added only if they share the same dimensions.

Quantity

Formula

Dimensions

Speed

distance ÷ time

[L T^-1]

Acceleration

speed ÷ time

[L T^-2]

Force

mass × acceleration

[M L T^-2]

Work or energy

force × distance

[M L² T^-2]

4. Dimensional Analysis

Dimensional analysis uses the principle that both sides of a correct physical equation have the same dimensions. This lets us check whether an equation is possible, convert between systems of units, and even guess the form of a relationship. It cannot, however, find pure numbers or tell apart quantities with the same dimensions.

Diagram 2 – Thinking in Powers of Ten

A scale in powers of ten of metres from the nucleus to the universe

Fig 2. Physical lengths span an enormous range, from the nucleus of an atom to the size of the observable universe, shown here in powers of ten.

5. Significant Figures and Rounding

The significant figures in a measurement are the digits known reliably plus one estimated digit; they show how precisely a quantity was measured. All non-zero digits count, zeros between them count, leading zeros do not, and trailing zeros after a decimal point do. When rounding, a final digit of 5 or more rounds the previous digit up.

Watch out

In a calculation, the result cannot be more precise than the least precise measurement used. Keep the answer to the smallest number of significant figures among the data.

6. Errors in Measurement

No measurement is exact. The absolute error is the difference between a reading and the true value; the relative error is the absolute error divided by the true value; and the percentage error is the relative error expressed as a percentage. When quantities are multiplied or divided, their relative errors add, which is why a result can be less precise than any single measurement.

7. Key Reasoning (Principles)

Principle 1: Dimensional homogeneity

In any correct physical equation, every term has the same dimensions. A term that does not match signals an error, so a dimensional check is a fast first test of any formula.

Principle 2: Significant figures reflect precision

The number of significant figures in a result must not exceed that of the least precise measurement used, so we do not claim more precision than we actually have.

Principle 3: Relative errors add in products

When quantities are multiplied or divided, their percentage errors add. So measuring two quantities each to 1% gives a product known only to about 2%.

8. Worked Examples

Example 1

Q: Convert 72 km/h into metres per second.

▶ Show Solution

72 × 1000 ÷ 3600.

= 20 m/s.

Answer: 20 m/s.

Example 2

Q: Express 1 m² in cm².

▶ Show Solution

1 m = 100 cm, so 1 m² = 100 × 100.

= 10⁴ cm².

Answer: 10⁴ cm².

Example 3

Q: Express 1 m³ in cm³.

▶ Show Solution

1 m³ = 100 × 100 × 100.

= 10⁶ cm³.

Answer: 10⁶ cm³.

Example 4

Q: Write the dimensional formula of force.

▶ Show Solution

Force = mass × acceleration.

= [M] × [L T^-2] = [M L T^-2].

Answer: [M L T^-2].

Example 5

Q: How many significant figures are in 0.00203?

▶ Show Solution

Leading zeros do not count; 2, 0, 3 do.

So there are 3 significant figures.

Answer: 3 significant figures.

Example 6

Q: Round 3.14159 to 4 significant figures.

▶ Show Solution

The first four figures are 3, 1, 4, 1; the next is 5, so round up.

3.14159 becomes 3.142.

Answer: 3.142.

Example 7

Q: A length measured as 2.05 m has a true value of 2.00 m. Find the percentage error.

▶ Show Solution

Absolute error = 2.05 minus 2.00 = 0.05 m.

Percentage error = (0.05 ÷ 2.00) × 100 = 2.5%.

Answer: 2.5%.

Example 8

Q: Two quantities are each measured to 1%. Find the percentage error in their product.

▶ Show Solution

Relative errors add for a product.

1% + 1% = 2%.

Answer: 2%.

Example 9

Q: Convert a density of 1 g/cm³ into kg/m³.

▶ Show Solution

1 g = 0.001 kg and 1 cm³ = 10^-6 m³.

1 g/cm³ = 0.001 ÷ 10^-6 = 1000 kg/m³.

Answer: 1000 kg/m³.

Example 10

Q: Light travels at 3 × 10⁸ m/s. Estimate the distance light covers in one year (a light year).

▶ Show Solution

Seconds in a year ≈ 365.25 × 24 × 3600 ≈ 3.156 × 10⁷ s.

Distance = 3 × 10⁸ × 3.156 × 10⁷ ≈ 9.46 × 10¹⁵ m.

Answer: About 9.46 × 10¹⁵ m.

9. Practice Sets A to D

Set A – Multiple Choice (Basic)

1. The SI unit of amount of substance is the: (a) kilogram (b) mole (c) candela (d) kelvin

2. The dimensional formula of speed is: (a) [L T] (b) [L T^-1] (c) [L² T] (d) [M L T^-1]

3. How many significant figures are in 100.0? (a) 1 (b) 2 (c) 3 (d) 4

4. 1 m³ equals: (a) 10² cm³ (b) 10⁴ cm³ (c) 10⁶ cm³ (d) 10³ cm³

5. Dimensional analysis cannot find: (a) units (b) pure numbers (c) errors in dimensions (d) consistency

▶ Reveal Answers

1. (b) mole.

2. (b) [L T^-1].

3. (d) 4.

4. (c) 10⁶ cm³.

5. (b) pure numbers.

Set B – Short Answer (Understanding)

1. State the difference between a fundamental and a derived quantity.

2. What is meant by the dimensions of a physical quantity?

3. State the principle of dimensional homogeneity.

4. Define percentage error.

5. Why can a result not be more precise than the least precise measurement used?

▶ Reveal Answers

1. A fundamental quantity is chosen as independent; a derived quantity is built from fundamental ones.

2. They show how the quantity is built from the base quantities mass, length and time.

3. Every term in a correct physical equation has the same dimensions.

4. The relative error expressed as a percentage of the true value.

5. Because the uncertainty of the least precise value limits how precisely the result can be known.

Set C – Application and Reasoning

1. Convert 90 km/h into metres per second.

2. Write the dimensional formula of work.

3. How many significant figures are in 0.0450?

4. A measurement of 5.10 m has a true value of 5.00 m. Find the percentage error.

5. Two lengths measured to 2% each are multiplied. Find the percentage error of the product.

▶ Reveal Answers

1. 90 × 1000 ÷ 3600 = 25 m/s.

2. Work = force × distance = [M L T^-2] × [L] = [M L² T^-2].

3. 3 significant figures.

4. (0.10 ÷ 5.00) × 100 = 2%.

5. 2% + 2% = 4%.

Set D – Higher Order (Challenge)

1. Use dimensions to check whether v = u + at is dimensionally correct.

2. The period of a pendulum is thought to depend on length L and g. Use dimensions to find how T depends on L and g.

3. A cube has side measured as 2.0 cm to 2 significant figures. State its volume to the correct number of significant figures.

4. A density is found from mass (1% error) and volume (2% error). Find the percentage error in the density.

5. Explain why dimensional analysis cannot find a numerical constant such as 2 or pi.

▶ Reveal Answers

1. [v] = [L T^-1]; [u] = [L T^-1]; [at] = [L T^-2][T] = [L T^-1]. All terms match, so it is dimensionally correct.

2. Assuming T depends on L and g, matching dimensions gives T proportional to the square root of (L ÷ g).

3. Volume = 2.0³ = 8.0 cm³, kept to 2 significant figures.

4. Density = mass ÷ volume, so relative errors add: 1% + 2% = 3%.

5. Dimensional analysis only balances dimensions; pure numbers have no dimensions, so they cannot be found this way.

Chapter Summary

Measurement and Units

A measurement is a number with a unit; the SI system fixes seven base units.

Fundamental and Derived

Base quantities are independent; all others are derived from them.

Dimensions

Show how a quantity is built from M, L and T; used to check equations.

Dimensional Analysis

Checks equations and converts units, but cannot find pure numbers.

Significant Figures

Reflect precision; a result keeps the least precise data’s figures.

Errors

Absolute, relative and percentage; relative errors add in products.

Quantity

Unit

Symbol

Length

metre

Mass

kilogram

Time

second

8-Point Exam Quick-Check

A measurement is a number with a unit; the SI system has seven base units.

Fundamental quantities are independent; derived quantities are built from them.

Dimensions show how a quantity is built from mass, length and time.

In a correct equation every term has the same dimensions (homogeneity).

Dimensional analysis checks equations and converts units but cannot find pure numbers.

Significant figures show precision; keep the least precise data’s count.

Percentage error = (absolute error divided by true value) times 100.

When quantities multiply or divide, their relative errors add.

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Class 11 Physics Chapter 1: Units and Measurements, Complete Notes and Practice

This revision guide follows the current NCERT Class 11 Physics syllabus and covers measurement and the SI system, fundamental and derived quantities, dimensions and dimensional formulae, dimensional analysis, significant figures and rounding, and errors in measurement, with two diagrams, ten worked examples and graded practice. Visit SchoolRevise.com to revise, practise and excel.