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Units and Measurements

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Chapter Notes

Overview
This chapter introduces the fundamental concepts of measurement in physics, including the International System of Units (SI), significant figures, and dimensional analysis. It exists to establish a common language for quantifying physical phenomena and to equip students with the tools to express measurements precisely, check the consistency of equations, and derive relationships between physical quantities. The one big idea a student must walk away with is that every physical measurement has an associated unit and a limited precision, and that dimensional analysis is a powerful method to verify and derive physical equations without detailed knowledge of the underlying constants.
Key Concepts (13)
Unit
A certain basic, arbitrarily chosen, internationally accepted reference standard used for comparison in measurement of a physical quantity.
e.g. The metre is the unit of length; the kilogram is the unit of mass.
Fundamental or base units
Units for the fundamental or base quantities, such as metre, kilogram, second, ampere, kelvin, mole, and candela.
e.g. The metre (m) for length, kilogram (kg) for mass, and second (s) for time are base units.
Derived units
Units of all other physical quantities expressed as combinations of the base units.
e.g. The unit of speed is metre per second (m/s), derived from length and time.
System of units
A complete set of units, both base units and derived units, used for measurement.
e.g. The International System of Units (SI) is the globally accepted system of units.
SI (Système International d'Unités)
The internationally accepted system of units based on seven base units, revised by the General Conference on Weights and Measures in November 2018.
e.g. In SI, length is measured in metres, mass in kilograms, and time in seconds.
Radian
The unit for plane angle, defined as the ratio of the length of arc ds to the radius r.
e.g. A full circle has an angle of 2π radians.
Steradian
The unit for solid angle, defined as the ratio of the intercepted area dA of a spherical surface to the square of its radius r.
e.g. The solid angle subtended by a sphere at its centre is 4π steradians.
Significant figures
The reliable digits plus the first uncertain digit in a measured or reported number.
e.g. The period 1.62 s has three significant figures: 1 and 6 are reliable, 2 is uncertain.
Order of magnitude
The exponent (or power) of 10 when a number is expressed in scientific notation as a × 10^b, where a is between 1 and 10, and b is an integer.
e.g. The diameter of the Earth (1.28×10^7 m) has an order of magnitude of 7.
Dimensions of a physical quantity
The powers (or exponents) to which the base quantities are raised to represent that quantity.
e.g. The dimensions of velocity are [L T⁻¹], meaning one dimension in length and -1 dimension in time.
Dimensional formula
An expression which shows how and which of the base quantities represent the dimensions of a physical quantity.
e.g. The dimensional formula of force is [M L T⁻²].
Dimensional equation
An equation obtained by equating a physical quantity with its dimensional formula.
e.g. [F] = [M L T⁻²] is the dimensional equation for force.
Principle of homogeneity of dimensions
The principle that physical quantities can be added or subtracted only if they have the same dimensions, and that the dimensions of each term on both sides of a mathematical equation must be the same.
e.g. In the equation x = x₀ + v₀t + ½at², each term has the dimension of length [L].
Formulas & Equations (7)
Density = Mass / Volume
Reads as: Density equals mass divided by volume.
Variables: Mass (m) in kg, Volume (V) in m³, Density (ρ) in kg/m³.
Use when: Calculating density from measured mass and volume, especially when determining significant figures in the result.
⚠ Reporting density with more significant figures than the least precise measurement; e.g., for 5.74 g and 1.2 cm³, students often write 4.7833 g/cm³ instead of 4.8 g/cm³.
Surface area of a cube = 6 × (side)²
Reads as: Surface area of a cube equals six times the square of the side length.
Variables: Side length (l) in m, Surface area (A) in m².
Use when: Calculating the total surface area of a cube from a measured side length, applying significant figure rules.
⚠ Not rounding the final area to the same number of significant figures as the side length; e.g., for side 7.203 m, writing 311.299254 m² instead of 311.3 m².
Volume of a cube = (side)³
Reads as: Volume of a cube equals side length cubed.
Variables: Side length (l) in m, Volume (V) in m³.
Use when: Calculating the volume of a cube, applying significant figure rules.
⚠ Not rounding the final volume to the correct number of significant figures; e.g., for side 7.203 m, writing 373.714754 m³ instead of 373.7 m³.
T = k l^x g^y m^z
Reads as: Time period T equals a dimensionless constant k times length to the power x times acceleration due to gravity to the power y times mass to the power z.
Variables: T = time period (s), l = length (m), g = acceleration due to gravity (m/s²), m = mass (kg), k = dimensionless constant.
Use when: Deriving the expression for the time period of a simple pendulum using dimensional analysis.
⚠ Forgetting that the dimensionless constant (k) cannot be determined by dimensional analysis; students often try to find its value from dimensions.
T = 2π √(l/g)
Reads as: Time period T equals 2π times the square root of length divided by acceleration due to gravity.
Variables: T = time period (s), l = length of pendulum (m), g = acceleration due to gravity (m/s²).
Use when: Calculating the time period of a simple pendulum for small oscillations.
⚠ Forgetting the factor 2π or incorrectly writing the square root; also, confusing l (length) with 1 (one).
1 J = 1 kg m² s⁻²
Reads as: One joule equals one kilogram metre squared per second squared.
Variables: J = joule (unit of energy), kg = kilogram, m = metre, s = second.
Use when: Converting units of energy or checking dimensional consistency of energy equations.
⚠ Writing the exponent incorrectly, e.g., kg m s⁻² instead of kg m² s⁻².
1 N = 1 kg m s⁻²
Reads as: One newton equals one kilogram metre per second squared.
Variables: N = newton (unit of force), kg = kilogram, m = metre, s = second.
Use when: Converting units of force or checking dimensional consistency of force equations.
⚠ Confusing the unit of force with the unit of energy; e.g., writing N = kg m² s⁻².
How It Works
Determining the number of significant figures in a measurement
  1. Identify all non-zero digits; they are always significant.
  2. Identify zeros between two non-zero digits; they are always significant.
  3. For numbers less than 1, zeros to the left of the first non-zero digit are not significant.
  4. Trailing zeros in a number without a decimal point are not significant.
  5. Trailing zeros in a number with a decimal point are significant.
  6. To avoid ambiguity, report the measurement in scientific notation (a × 10^b), where all digits in 'a' are significant.
💡 Changing units does not change the number of significant figures. For example, 4.700 m = 470.0 cm = 0.004700 km all have four significant figures.
Rounding off uncertain digits
  1. If the digit to be dropped is more than 5, increase the preceding digit by 1.
  2. If the digit to be dropped is less than 5, leave the preceding digit unchanged.
  3. If the digit to be dropped is exactly 5, check the preceding digit: if it is even, drop the 5 and leave the preceding digit unchanged; if it is odd, increase the preceding digit by 1.
💡 For multi-step calculations, retain one extra digit in intermediate steps to avoid rounding errors, then round off at the end.
Arithmetic operations with significant figures
  1. For multiplication or division: the final result should have as many significant figures as the original number with the least significant figures.
  2. For addition or subtraction: the final result should have as many decimal places as the number with the least decimal places.
💡 Do not apply the multiplication/division rule to addition/subtraction. For example, 12.9 g – 7.06 g = 5.8 g (one decimal place), not 5.84 g.
Checking dimensional consistency of an equation
  1. Write the dimensions of each term on both sides of the equation using base quantities [L], [M], [T], etc.
  2. Simplify the dimensions of each term.
  3. If the dimensions of all terms are the same, the equation is dimensionally correct.
  4. If the dimensions are not the same, the equation is definitely wrong.
💡 A dimensionally correct equation is not necessarily correct (e.g., it may have a wrong dimensionless constant), but a dimensionally incorrect equation is always wrong.
Deducing relation among physical quantities using dimensional analysis
  1. Identify the physical quantity (say Q) whose relation is to be found.
  2. List the quantities on which Q depends (say A, B, C).
  3. Assume Q = k A^x B^y C^z, where k is a dimensionless constant.
  4. Write the dimensional formulae for Q, A, B, and C.
  5. Equate the dimensions on both sides to form equations for the exponents x, y, z.
  6. Solve for x, y, z.
  7. Write the final relation, noting that the constant k cannot be determined by this method.
💡 This method works only if the dependence is of a product type and involves up to three linearly independent variables. It cannot give the value of dimensionless constants.
Exam Traps — Don't Make These Mistakes
⚠ Significant figures in subtraction
✗ 12.9 g – 7.06 g = 5.84 g
→ The rule for subtraction is based on decimal places, not significant figures. 12.9 has one decimal place, so the result must be rounded to one decimal place.
✓ 12.9 g – 7.06 g = 5.8 g
⚠ Trailing zeros without decimal point
✗ 12300 cm has five significant figures.
→ Trailing zeros in a number without a decimal point are not significant. 12300 cm has only three significant figures (1, 2, 3).
✓ 12300 cm has three significant figures. To indicate more, use scientific notation: 1.2300 × 10^4 cm has five significant figures.
⚠ Confusing dimensional correctness with actual correctness
✗ Since the equation ½ m v² = m g h is dimensionally correct, it must be the correct equation for kinetic energy.
→ Dimensional consistency is a necessary but not sufficient condition for correctness. The equation could have a wrong dimensionless constant or be dimensionally correct but physically meaningless.
✓ The equation is dimensionally correct, but this does not guarantee it is the correct physical equation. The actual definition of kinetic energy is K = ½ m v².
⚠ Adding quantities with different dimensions
✗ K = (1/2)mv² + ma is a valid expression for kinetic energy.
→ The term (1/2)mv² has dimensions [M L² T⁻²] while ma has dimensions [M L T⁻²]. Quantities with different dimensions cannot be added.
✓ K = (1/2)mv² is the correct expression. The term ma cannot be added to it because of dimensional inconsistency.
⚠ Rounding off when the digit is exactly 5
✗ 2.745 rounded to three significant figures is 2.75.
→ When the digit to be dropped is exactly 5, the rule is to look at the preceding digit. If it is even, leave it; if odd, increase it by 1. Here, the preceding digit is 4 (even), so the 5 is dropped and the digit remains 4.
✓ 2.745 rounded to three significant figures is 2.74.
Compare & Contrast
Definition and origin
Fundamental (base) units
Derived units
Fundamental units are arbitrarily chosen and defined for base quantities (e.g., metre, kilogram), while derived units are expressed as combinations of base units (e.g., m/s for speed).
Type of angle measured
Radian
Steradian
Radian measures plane angle (ratio of arc length to radius), while steradian measures solid angle (ratio of intercepted area to square of radius). Both are dimensionless.
Exam Focus
Marks pattern: 1-2 MCQs (1 mark each) on SI units or significant figures, 1 short answer (2-3 marks) on dimensional analysis or significant figure rules, 1 long answer (5 marks) on deriving a relation using dimensional analysis or a numerical problem involving significant figures.
Most asked: Significant figures and arithmetic operations with them · Dimensional analysis: checking consistency of equations and deriving relations · SI base units and their definitions · Order of magnitude and scientific notation · Rounding off uncertain digits
Don't skip: Deriving the time period of a simple pendulum using dimensional analysis (Example 1.5 style); Checking dimensional consistency of a given equation (e.g., s = ut + ½at²); Numerical problems involving significant figures in multiplication/division and addition/subtraction
One-Line Revision (read aloud before the exam)
  1. Measurement of any physical quantity involves comparison with an internationally accepted reference standard called a unit.
  2. The SI system has seven base units: metre, kilogram, second, ampere, kelvin, mole, and candela.
  3. Derived units are combinations of base units, e.g., speed is m/s.
  4. Radian and steradian are dimensionless units for plane and solid angles respectively.
  5. Significant figures include all digits known reliably plus the first uncertain digit.
  6. All non-zero digits are significant; zeros between non-zero digits are significant.
  7. Trailing zeros in a number without a decimal point are not significant.
  8. In multiplication or division, the result should have as many significant figures as the least precise measurement.
  9. In addition or subtraction, the result should have as many decimal places as the number with the least decimal places.
  10. When rounding off, if the digit to be dropped is more than 5, increase the preceding digit by 1.
  11. If the digit to be dropped is exactly 5, and the preceding digit is even, leave it unchanged; if odd, increase it by 1.
  12. Dimensions of a physical quantity are the powers to which base quantities are raised.
  13. The principle of homogeneity states that only quantities with the same dimensions can be added or subtracted.
  14. Dimensional analysis can check the consistency of equations but cannot determine dimensionless constants.
  15. A dimensionally correct equation is not necessarily correct, but a dimensionally incorrect equation is always wrong.
  16. The time period of a simple pendulum is T = 2π √(l/g), derived using dimensional analysis.
  17. Order of magnitude is the exponent of 10 when a number is expressed in scientific notation.
  18. Scientific notation removes ambiguity about significant figures in trailing zeros.
  19. Exact numbers (like 2 in 2π) have infinite significant figures.
  20. The density of a substance is mass per unit volume, and its value must be reported with appropriate significant figures.

MCQs — 10 Questions with Answers

Q1. Which of the following is NOT a base quantity in the SI system?

  • A. Thermodynamic temperature
  • B. Electric current
  • C. Velocity ✓
  • D. Amount of substance

Answer: C — Velocity is a derived quantity (m/s) obtained from base quantities length and time, whereas thermodynamic temperature, electric current, and amount of substance are the seven SI base quantities.

Q2. The plane angle dθ is defined as the ratio of arc length ds to radius r. What is its unit and dimension?

  • A. metre, dimension [L]
  • B. radian, dimensionless ✓
  • C. degree, dimension [L]
  • D. steradian, dimension [L²]

Answer: B — Plane angle θ = ds/r is a ratio of two lengths, making it dimensionless; its unit is the radian (rad).

Q3. A length is measured as 25.7 cm using a ruler. How many significant figures does this measurement have?

  • A. 1
  • B. 2
  • C. 3 ✓
  • D. 4

Answer: C — The measurement 25.7 cm has three significant figures: digits 2 and 5 are certain, and digit 7 is the first uncertain digit.

Q4. Which of the following statements about SI system is correct? (I) It uses decimal system. (II) It was standardised in 1971.

  • A. Both (I) and (II) are correct ✓
  • B. Only (I) is correct
  • C. Only (II) is correct
  • D. Neither (I) nor (II) is correct

Answer: A — SI system uses a decimal (base-10) system for easy conversions, and was developed by BIPM in 1971 with revisions in 2018.

Q5. In a measurement reported as 287.5 cm, which digit represents the first uncertain digit?

  • A. 2
  • B. 8
  • C. 7
  • D. 5 ✓

Answer: D — The digits 2, 8, and 7 are certain from the measuring instrument, while 5 is the first uncertain digit; the measurement has four significant figures.

Q6. Convert 5 kg from SI to CGS system. What is the mass in grams?

  • A. 50 g
  • B. 500 g
  • C. 5000 g ✓
  • D. 50,000 g

Answer: C — Since 1 kg = 1000 g, 5 kg = 5 × 1000 = 5000 g in the CGS system.

Q7. The solid angle dΩ is defined as dA/r². If dA = 2 m² and r = 2 m, calculate the solid angle in steradians.

  • A. 0.25 sr
  • B. 0.5 sr ✓
  • C. 1 sr
  • D. 2 sr

Answer: B — Solid angle Ω = dA/r² = 2/(2)² = 2/4 = 0.5 steradian; solid angle is also dimensionless like plane angle.

Q8. Which statement correctly distinguishes between the CGS and FPS systems? (I) CGS uses metre while FPS uses foot. (II) Both systems have second as the unit of time.

  • A. Both (I) and (II) are correct
  • B. Only (I) is correct
  • C. Only (II) is correct ✓
  • D. Neither (I) nor (II) is correct

Answer: C — Statement (I) is incorrect: CGS uses centimetre, not metre; statement (II) is correct: both CGS and FPS use second as the unit of time.

Q9. A student reports a measurement as 45.0 cm but the measuring instrument has a precision of ±0.1 cm. Which reporting is most appropriate?

  • A. 45 cm (2 significant figures)
  • B. 45.0 cm (3 significant figures) ✓
  • C. 45.00 cm (4 significant figures)
  • D. 45.000 cm (5 significant figures)

Answer: B — The instrument's precision of ±0.1 cm means the measurement can be reported to the nearest 0.1 cm, giving 3 significant figures as 45.0 cm.

Q10. The Planck constant h is defined as 6.62607015 × 10⁻³⁴ J·s. In how many significant figures is this value expressed, and why is such high precision necessary?

  • A. 5 figures; precision not critical for Class 11 physics
  • B. 9 figures; precision needed for fundamental constant definition in modern SI ✓
  • C. 3 figures; older convention still valid
  • D. 7 figures; arbitrary choice by BIPM

Answer: B — The Planck constant is expressed with 9 significant figures because the 2018 SI revision fixed this value as a fundamental constant defining the kilogram, requiring extremely high precision for consistency.

Flashcards

What is a fundamental unit?

A fundamental (base) unit is the standard reference for a base quantity like length (metre), mass (kilogram), or time (second) from which all other units are derived.

Define a derived unit with one example.

A derived unit is obtained by combining two or more base units; for example, velocity has units of m/s (combining metres and seconds).

How many base quantities are there in SI?

There are seven base quantities in SI: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

What are significant figures?

Significant figures are all the reliable (certain) digits in a measured value plus the first uncertain digit, indicating the precision of the measurement.

In the measurement 1.62 s, which digits are certain and which is uncertain?

The digits 1 and 6 are certain (reliable), while the digit 2 is uncertain; the measurement has three significant figures.

Why was SI system adopted internationally?

SI system uses a decimal (base-10) scheme making conversions simple, and provides a standardised set of units accepted worldwide for scientific and commercial work.

What is the symbol and unit for plane angle?

The plane angle is measured in radians (rad), defined as the ratio of arc length to radius, and is a dimensionless quantity.

Distinguish between CGS and SI for length.

CGS system uses centimetre as the unit of length while SI system uses metre; 1 metre = 100 centimetres.

What does the first uncertain digit represent in a measurement?

The first uncertain digit is the rightmost digit in a measured value where doubt exists due to the limitations of the measuring instrument's precision.

Name two systems of units used before SI was adopted.

The CGS (centimetre–gram–second) and FPS (foot–pound–second) systems were widely used before SI became the international standard.

Important Board Questions

Define significant figures and explain why the measurement 1.62 s has three significant figures, not two or four. [2 marks]

Significant figures = all certain digits + first uncertain digit. In 1.62 s, digits 1 and 6 are certain (read directly from scale), digit 2 is the first uncertain digit (estimated between scale marks).

Explain why the SI system was adopted internationally instead of CGS or FPS. Give two advantages of using SI for scientific work. Also convert 500 centimetres (CGS) to metres (SI). [5 marks]

SI advantages: (1) decimal system simplifies conversions, (2) standardised globally for science and commerce. Conversion: 1 m = 100 cm, so 500 cm = 500/100 = 5 m. Show working: identify conversion factor first, then multiply by it.

The seven SI base quantities and their units are fundamental to all physical measurements. Describe the definition of the metre in modern SI and explain how it differs from the historical definition. Why is it important that base units are defined in terms of universal physical constants? Derive the relationship between plane angle (radian) and solid angle (steradian) using their mathematical definitions. [6 marks]

Modern metre definition: fixed numerical value of speed of light c = 299,792,458 m/s (involves caesium frequency). Historical: arbitrary reference standard (platinum bar). Importance: ensures reproducibility and consistency as technology improves. Plane angle: θ = s/r (dimensionless); solid angle: Ω = A/r² (dimensionless). Show that both are ratios of geometric quantities with matching dimensions.

Assertion-Reason Questions for Units and Measurements

Q1. Assertion (A):

The metre is defined by taking the fixed numerical value of the speed of light in vacuum to be 299792458 when expressed in the unit m s⁻¹.

Reason (R):

The speed of light is a fundamental constant in physics and provides a precise, reproducible standard for defining length.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The assertion accurately states the 2018 SI definition of the metre based on the speed of light. The reason correctly explains why this definition is used—the speed of light is a fundamental constant that is invariant and can be measured with extremely high precision, making it an ideal reference standard for length measurement. The reason is the correct explanation for the assertion.

Q2. Assertion (A):

Significant figures include all reliable digits plus the first uncertain digit in a measurement.

Reason (R):

The number of significant figures determines the precision and accuracy of a measurement.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true—significant figures are defined as reliable digits plus the first uncertain digit. The reason is also true—significant figures do relate to measurement quality. However, the reason does not explain WHY we include all reliable digits plus the first uncertain digit. The reason is true but not the correct explanation for the assertion. Significant figures are included this way because we want to report the measurement honestly with one uncertain digit to show the measurement's limit of precision.

Q3. Assertion (A):

The radian and steradian are SI base units used for measuring angles.

Reason (R):

Both radian and steradian are dimensionless quantities defined as ratios of physical quantities with the same dimension.

Answer: D — A is false but R is true — The assertion is false—radian and steradian are supplementary SI units, not base units. There are only seven SI base units. The reason is true—both radian (ratio of arc length to radius) and steradian (ratio of area to radius squared) are indeed dimensionless quantities. The assertion is incorrect because radian and steradian are not classified as base units, even though the reason correctly describes their nature as dimensionless quantities.

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Case-Based Questions (CBQ) for Units and Measurements

Case Study:

A physics teacher in Delhi conducted an experiment to verify the accuracy of different measuring instruments. Three students measured the diameter of a steel ball using: (1) a ruler with graduation of 1 mm, (2) a vernier caliper with least count 0.1 mm, and (3) a screw gauge with least count 0.01 mm. Student 1 recorded 24 mm, Student 2 recorded 24.2 mm, and Student 3 recorded 24.15 mm. All three measurements were taken multiple times with consistent results.

Q1. How many significant figures does each measurement contain?

  • A. 24 mm has 2 sig figs; 24.2 mm has 3 sig figs; 24.15 mm has 4 sig figs ✓
  • B. 24 mm has 1 sig fig; 24.2 mm has 2 sig figs; 24.15 mm has 3 sig figs
  • C. All three have 2 significant figures
  • D. 24 mm has 3 sig figs; 24.2 mm has 4 sig figs; 24.15 mm has 5 sig figs

Answer: A

Q2. Which instrument provides the best measurement and why?

  • A. The ruler, because it is the simplest instrument
  • B. The screw gauge, because it has the smallest least count and provides the highest precision ✓
  • C. The vernier caliper, because it is most commonly used
  • D. All three are equally good since they measure the same object

Answer: B

Q3. If these measurements are used to calculate the volume of the sphere (V = 4/3 πr³), which measurement should be used for the most accurate result?

  • A. 24 mm, for simplicity
  • B. 24.2 mm, as it represents a compromise
  • C. 24.15 mm, because it has the most significant figures from the most precise instrument ✓
  • D. All would give the same volume

Answer: C

Q4. What is the fundamental principle guiding the choice of measurement precision in experimental physics?

  • A. Use the simplest measurement to avoid errors
  • B. Use the measurement with the most digits
  • C. Use the measurement from the instrument with the smallest least count and best precision ✓
  • D. All measurements are equally valid regardless of instrument

Answer: C

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