This MCQ module is based on: Units and Si System
TOPIC 1 OF 33
Units and Si System
🎓 Class 11
Physics
CBSE
Theory
Ch 1 – Units and Measurements
⏱ ~14 min
🌐 Language: [gtranslate]
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Units and Si System
1.1 Introduction: What is Measurement?
Physics is a quantitative science built on measurement. Every observation in physics involves measuring something — the length of a table, the mass of a ball, or the time taken for a pendulum swing. But what exactly does it mean to measure?
To measure any physical quantity, we compare it with a well-defined, internationally accepted reference standard called a unit. The result of any measurement is expressed as:
\[\text{Physical Quantity} = \text{Numerical Value} \times \text{Unit}\]
For instance, the length of a classroom might be 12 metres. Here, 12 is the numerical value and metre (m) is the unit.
Fundamental vs Derived Quantities
Physical quantities are broadly classified into two categories:
| Type | Description | Examples |
|---|---|---|
| Fundamental (Base) Quantities | Quantities that are independent of other quantities; they cannot be expressed in terms of simpler quantities. | Length, Mass, Time, Temperature, Electric Current, Amount of Substance, Luminous Intensity |
| Derived Quantities | Quantities obtained by combining base quantities through multiplication, division, or both. | Velocity = Length/Time, Force = Mass × Acceleration, Energy = Force × Length |
Key Idea: All physical quantities in the universe can ultimately be expressed in terms of just 7 base quantities. This is the foundation of the SI system.
1.2 The International System of Units (SI)
Historically, scientists in different countries used different systems of units. The three most common were:
| System | Length | Mass | Time |
|---|---|---|---|
| CGS (Gaussian) | centimetre (cm) | gram (g) | second (s) |
| FPS (British) | foot (ft) | pound (lb) | second (s) |
| MKS | metre (m) | kilogram (kg) | second (s) |
In 1971, the General Conference on Weights and Measures (CGPM) adopted the SI system (Systeme International d'Unites) as a universal, rational, and coherent system for all measurements in science and technology.
The 7 SI Base Units
Supplementary Units: Radian and Steradian
Besides the 7 base units, the SI system originally defined two supplementary units for measuring angles:
| Quantity | Unit | Symbol | Definition |
|---|---|---|---|
| Plane angle | radian | rad | Angle subtended at the centre of a circle by an arc equal in length to the radius |
| Solid angle | steradian | sr | Solid angle subtended at the centre of a sphere by a surface area equal to the square of its radius |
Common Derived Units
All other physical quantities are derived from the 7 base quantities. Some important derived SI units include:
| Derived Quantity | Expression | SI Unit | Special Name |
|---|---|---|---|
| Velocity | Length / Time | m/s | — |
| Acceleration | Velocity / Time | m/s² | — |
| Force | Mass × Acceleration | kg·m/s² | newton (N) |
| Work / Energy | Force × Distance | kg·m²/s² | joule (J) |
| Power | Energy / Time | kg·m²/s³ | watt (W) |
| Pressure | Force / Area | kg/(m·s²) | pascal (Pa) |
| Density | Mass / Volume | kg/m³ | — |
1.3 Significant Figures
Every measurement has a degree of uncertainty associated with it. The concept of significant figures tells us how precise a measurement is. The significant figures in a measured value include all digits that are known reliably plus one digit that is uncertain (estimated).
Rules for Determining Significant Figures
1All non-zero digits are significant.
Example: 2456 has 4 significant figures; 78.3 has 3.
2Zeros between non-zero digits are significant.
Example: 3007 has 4 significant figures; 1.05 has 3.
3Leading zeros (before the first non-zero digit) are NOT significant.
Example: 0.0045 has 2 significant figures (only 4 and 5 count).
4Trailing zeros in a number WITH a decimal point are significant.
Example: 4.700 has 4 significant figures; 0.0500 has 3.
5Trailing zeros in a number WITHOUT a decimal point may or may not be significant (ambiguous).
Example: 123000 may have 3, 4, 5, or 6 significant figures. Use scientific notation to remove ambiguity.
6Exact numbers (counting numbers, defined values) have infinite significant figures.
Example: 12 eggs, π (exact), 1 inch = 2.54 cm (exact definition).
Scientific Notation and Order of Magnitude
In scientific notation, any number is expressed as:
\[a \times 10^b \quad \text{where } 1 \le a < 10 \text{ and } b \text{ is an integer}\]
For example: \(3.08 \times 10^{-3}\) has 3 significant figures. The power of 10 is irrelevant for counting sig figs — only the coefficient \(a\) matters.
The order of magnitude of a quantity is the power of 10 closest to it. For instance:
- \(5 = 0.5 \times 10^1\), so order of magnitude = \(10^1\)
- \(0.007 = 7 \times 10^{-3}\), so order of magnitude = \(10^{-2}\) (since 7 > 3.16)
Worked Examples — Significant Figures
Example 1: Counting Significant Figures
Determine the number of significant figures in each: (a) 0.00308 (b) 4.700 (c) 123000 (d) \(6.022 \times 10^{23}\)
Solution:
(a) 0.00308: Leading zeros are not significant. Digits 3, 0, 8 are significant (zero between 3 and 8 counts). 3 significant figures.
(b) 4.700: Trailing zeros after decimal point are significant. Digits 4, 7, 0, 0 all count. 4 significant figures.
(c) 123000: Ambiguous. Without a decimal point, trailing zeros may or may not be significant. If written as \(1.23 \times 10^5\), it has 3 sig figs. If \(1.2300 \times 10^5\), it has 5. Ambiguous — typically assumed 3.
(d) \(6.022 \times 10^{23}\): In scientific notation, count digits in the coefficient: 6, 0, 2, 2. 4 significant figures.
(a) 0.00308: Leading zeros are not significant. Digits 3, 0, 8 are significant (zero between 3 and 8 counts). 3 significant figures.
(b) 4.700: Trailing zeros after decimal point are significant. Digits 4, 7, 0, 0 all count. 4 significant figures.
(c) 123000: Ambiguous. Without a decimal point, trailing zeros may or may not be significant. If written as \(1.23 \times 10^5\), it has 3 sig figs. If \(1.2300 \times 10^5\), it has 5. Ambiguous — typically assumed 3.
(d) \(6.022 \times 10^{23}\): In scientific notation, count digits in the coefficient: 6, 0, 2, 2. 4 significant figures.
Example 2: Arithmetic with Significant Figures — Addition
Add 436.32 g, 227.2 g, and 0.301 g with proper significant figures.
Solution:
Rule for addition/subtraction: The result should have the same number of decimal places as the quantity with the fewest decimal places.
\[\begin{aligned} &436.32 \quad &(\text{2 decimal places})\\ +\;&227.2\phantom{0} \quad &(\text{1 decimal place — fewest})\\ +\;&\phantom{00}0.301 \quad &(\text{3 decimal places})\\ \hline &663.821 \quad &(\text{raw sum}) \end{aligned}\] The least number of decimal places among the addends is 1 (in 227.2).
Round the result to 1 decimal place: \(\boxed{663.8 \text{ g}}\)
Rule for addition/subtraction: The result should have the same number of decimal places as the quantity with the fewest decimal places.
\[\begin{aligned} &436.32 \quad &(\text{2 decimal places})\\ +\;&227.2\phantom{0} \quad &(\text{1 decimal place — fewest})\\ +\;&\phantom{00}0.301 \quad &(\text{3 decimal places})\\ \hline &663.821 \quad &(\text{raw sum}) \end{aligned}\] The least number of decimal places among the addends is 1 (in 227.2).
Round the result to 1 decimal place: \(\boxed{663.8 \text{ g}}\)
Example 3: Arithmetic with Significant Figures — Multiplication
A rectangular sheet has length \(l = 16.2\) cm and breadth \(b = 10.1\) cm. Find the area with correct significant figures.
Solution:
Rule for multiplication/division: The result should have the same number of significant figures as the quantity with the fewest significant figures.
\[\text{Area} = l \times b = 16.2 \times 10.1 = 163.62 \text{ cm}^2\] Here, \(l = 16.2\) has 3 sig figs and \(b = 10.1\) has 3 sig figs. So the result should have 3 significant figures.
Round 163.62 to 3 sig figs: \(\boxed{164 \text{ cm}^2}\)
Rule for multiplication/division: The result should have the same number of significant figures as the quantity with the fewest significant figures.
\[\text{Area} = l \times b = 16.2 \times 10.1 = 163.62 \text{ cm}^2\] Here, \(l = 16.2\) has 3 sig figs and \(b = 10.1\) has 3 sig figs. So the result should have 3 significant figures.
Round 163.62 to 3 sig figs: \(\boxed{164 \text{ cm}^2}\)
Example 4: Rounding Off
Round the following to 3 significant figures: (a) 2.745 (b) 2.735 (c) 0.08356
Solution:
Rounding rules:
• If the digit to be dropped is less than 5, leave the preceding digit unchanged.
• If the digit to be dropped is more than 5, increase the preceding digit by 1.
• If the digit to be dropped is exactly 5, round to the nearest even number (banker's rounding).
(a) 2.745 → 2.74 (digit dropped is 5; preceding digit 4 is even, so leave unchanged) = \(\boxed{2.74}\)
(b) 2.735 → 2.74 (digit dropped is 5; preceding digit 3 is odd, so round up) = \(\boxed{2.74}\)
(c) 0.08356 → We need 3 sig figs. The significant digits are 8, 3, 5, 6. Keep first three (8, 3, 5), drop 6 (which is > 5, so round up): = \(\boxed{0.0836}\)
Rounding rules:
• If the digit to be dropped is less than 5, leave the preceding digit unchanged.
• If the digit to be dropped is more than 5, increase the preceding digit by 1.
• If the digit to be dropped is exactly 5, round to the nearest even number (banker's rounding).
(a) 2.745 → 2.74 (digit dropped is 5; preceding digit 4 is even, so leave unchanged) = \(\boxed{2.74}\)
(b) 2.735 → 2.74 (digit dropped is 5; preceding digit 3 is odd, so round up) = \(\boxed{2.74}\)
(c) 0.08356 → We need 3 sig figs. The significant digits are 8, 3, 5, 6. Keep first three (8, 3, 5), drop 6 (which is > 5, so round up): = \(\boxed{0.0836}\)
Activity — Exploring Measurement Precision
L3 Apply
Predict first: If you measure the length of your physics textbook with (a) a metre ruler marked in cm, and (b) a ruler marked in mm, which measurement will have more significant figures? Why?
- Take your physics textbook and measure its length using a ruler marked only in centimetres. Record the measurement.
- Now measure the same length using a ruler marked in millimetres. Record this measurement.
- Repeat each measurement three times and note the readings.
- Compare the number of significant figures in each set of readings.
Observation:
With the cm ruler, you might record: 26.4 cm (3 sig figs). With the mm ruler: 26.42 cm (4 sig figs).
Conclusion: A more precise instrument (smaller least count) gives measurements with more significant figures. The mm ruler has a least count of 0.1 cm compared to 1 cm for the cm ruler, allowing one additional significant digit to be estimated.
With the cm ruler, you might record: 26.4 cm (3 sig figs). With the mm ruler: 26.42 cm (4 sig figs).
Conclusion: A more precise instrument (smaller least count) gives measurements with more significant figures. The mm ruler has a least count of 0.1 cm compared to 1 cm for the cm ruler, allowing one additional significant digit to be estimated.
Interactive: Significant Figures Counter L3 Apply
Enter any number and check how many significant figures it contains:
Competency-Based Questions
A physics laboratory requires students to measure the thickness of a thin wire using a screw gauge with a least count of 0.01 mm. Three students measure the same wire and report their results as: Student A: 1.20 mm, Student B: 1.2 mm, Student C: 1.200 mm.
Q1. L1 Remember How many significant figures are in the measurement reported by Student C (1.200 mm)?
Answer: D. 4 significant figures. In 1.200, the trailing zeros after the decimal point are significant. The digits 1, 2, 0, 0 are all significant, giving 4 sig figs.
Q2. L2 Understand Explain why the measurements of Student A and Student B convey different levels of precision, even though 1.20 and 1.2 appear numerically equal. (2 marks)
Answer: Student A reports 1.20 mm (3 significant figures), which means the measurement is precise to the hundredths place (0.01 mm). Student B reports 1.2 mm (2 significant figures), precise only to the tenths place (0.1 mm). The trailing zero in 1.20 is significant because it indicates that the student actually measured and confirmed the digit in the hundredths place. Therefore, Student A's measurement conveys higher precision.
Q3. L3 Apply The mass of an object is measured as \(m = 4.237\) g and its volume as \(V = 2.51\) cm³. Calculate the density of the object with the correct number of significant figures. (3 marks)
Answer:
\[\rho = \frac{m}{V} = \frac{4.237}{2.51} = 1.6882...\text{ g/cm}^3\] \(m\) has 4 sig figs, \(V\) has 3 sig figs. For division, the result takes the fewest sig figs = 3.
Rounding to 3 sig figs: \(\boxed{\rho = 1.69 \text{ g/cm}^3}\)
\[\rho = \frac{m}{V} = \frac{4.237}{2.51} = 1.6882...\text{ g/cm}^3\] \(m\) has 4 sig figs, \(V\) has 3 sig figs. For division, the result takes the fewest sig figs = 3.
Rounding to 3 sig figs: \(\boxed{\rho = 1.69 \text{ g/cm}^3}\)
Q4. L4 Analyse A student measures the diameter of a sphere as 4.28 cm. They then calculate the volume as \(V = \frac{4}{3}\pi r^3\) and report \(V = 41.03396\) cm³. Identify the error in reporting and give the correct answer. (3 marks)
Answer: The diameter \(d = 4.28\) cm has 3 significant figures, so \(r = 2.14\) cm also has 3 sig figs.
\[V = \frac{4}{3}\pi (2.14)^3 = \frac{4}{3}\times 3.14159 \times 9.800344 = 41.03...\text{ cm}^3\] The student reported 7 sig figs (41.03396), which is incorrect. Since the measured quantity has only 3 sig figs, the result should also have 3 sig figs:
\(\boxed{V = 41.0 \text{ cm}^3}\)
The error is that the student reported far more digits than the precision of the measurement justified.
\[V = \frac{4}{3}\pi (2.14)^3 = \frac{4}{3}\times 3.14159 \times 9.800344 = 41.03...\text{ cm}^3\] The student reported 7 sig figs (41.03396), which is incorrect. Since the measured quantity has only 3 sig figs, the result should also have 3 sig figs:
\(\boxed{V = 41.0 \text{ cm}^3}\)
The error is that the student reported far more digits than the precision of the measurement justified.
Q5. L5 Evaluate Why is the SI system preferred over the CGS or FPS systems for scientific measurements? Give at least two reasons. (2 marks)
Answer:
(i) Universality: The SI system is internationally accepted and used by scientists worldwide, enabling easy communication and comparison of experimental results across countries.
(ii) Coherence: All derived units in SI are obtained by simple multiplication or division of base units without introducing any extra numerical factors (unlike CGS where 1 N = 10&sup5; dyne, which can cause confusion).
(iii) Rational definitions: SI base units are defined using fundamental constants of nature (speed of light, Planck's constant, etc.), making them reproducible anywhere in the universe.
(i) Universality: The SI system is internationally accepted and used by scientists worldwide, enabling easy communication and comparison of experimental results across countries.
(ii) Coherence: All derived units in SI are obtained by simple multiplication or division of base units without introducing any extra numerical factors (unlike CGS where 1 N = 10&sup5; dyne, which can cause confusion).
(iii) Rational definitions: SI base units are defined using fundamental constants of nature (speed of light, Planck's constant, etc.), making them reproducible anywhere in the universe.
Assertion-Reason Questions
Assertion (A): The number 0.0500 has three significant figures.
Reason (R): Trailing zeros after the decimal point are significant, but leading zeros are not.
Answer: A. 0.0500 has three significant figures (5, 0, 0). The two leading zeros merely fix the decimal position and are not significant. The trailing zeros (after 5) are significant because they appear after the decimal point. The Reason correctly explains why.
Assertion (A): The radian is a dimensionless quantity.
Reason (R): The radian is defined as the ratio of arc length to radius, both of which have the dimension of length.
Answer: A. Since \(\theta = \frac{\text{arc length}}{\text{radius}} = \frac{[L]}{[L]}\), the dimensions cancel out, making the radian dimensionless. Both statements are true, and R correctly explains A.
Assertion (A): When we add 7.21 and 12.3, the result should be reported as 19.5.
Reason (R): In addition, the result must have the same number of significant figures as the number with fewest significant figures.
Answer: C. The Assertion is true: 7.21 + 12.3 = 19.51, rounded to 1 decimal place gives 19.5. However, the Reason is false. For addition/subtraction, the rule is based on the fewest decimal places, not the fewest significant figures. The correct rule: "the result should have the same number of decimal places as the quantity with the fewest decimal places."
Frequently Asked Questions - Units and Si System
What is the main concept covered in Units and Si System?
In NCERT Class 11 Physics Chapter 1 (Units and Measurements), "Units and Si System" covers core principles and equations needed for board exam success. The MyAiSchool lesson explains the topic with definitions, derivations, worked examples, and interactive simulations. Key formulas and dimensional analysis are included to build conceptual depth and problem-solving skills aligned with the CBSE 2025-26 syllabus.
How is Units and Si System useful in real-life applications?
Real-life applications of Units and Si System from NCERT Class 11 Physics Chapter 1 include engineering design, satellite mechanics, sports biomechanics, transportation safety, and electrical/electronic devices. The MyAiSchool lesson links every concept to a tangible example so students see physics as a problem-solving framework for the physical world, not as abstract formulas.
What are the key formulas in Units and Si System?
Key formulas in Units and Si System (NCERT Class 11 Physics Chapter 1 Units and Measurements) are derived step-by-step in the MyAiSchool lesson. Students should memorize the final formula AND understand its derivation for full board marks. Each formula is listed with its dimensional formula, SI unit, applicability range, and common pitfalls. The Summary section at the end of each part includes a quick-reference formula card.
How does this part connect to other parts of Chapter 1?
NCERT Class 11 Physics Chapter 1 (Units and Measurements) is structured so each part builds on the previous one. Units and Si System connects directly to neighbouring parts via shared definitions, units, and methodology. The MyAiSchool lesson cross-references related concepts with internal links so students can navigate the whole chapter as one connected story rather than disconnected fragments.
What types of CBSE board questions come from Units and Si System?
CBSE board questions from Units and Si System typically include: (1) 1-mark MCQs on definitions and formulas, (2) 2-mark short-answer derivations or applications, (3) 3-mark numerical problems with units, (4) 5-mark long-answer derivations followed by application. The MyAiSchool lesson tags each Competency-Based Question (CBQ) with Bloom level (L1-L6) so students know how to study for each weight.
How can students use the interactive simulation effectively?
The interactive simulation in the Units and Si System lesson allows students to adjust input parameters (sliders or selectors) and see physical quantities update in real time. To use it effectively: (1) try extreme values to understand limiting cases, (2) compare with the analytical formula, (3) check unit consistency, (4) test special configurations from worked examples. The simulation reinforces conceptual intuition that pure formula manipulation cannot.
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