class 11 notes

Class 11 Physics Notes Chapter 2 (Units and measurements) – Physics Part-I Book

Philoid

Philoid

— 9 min read

Physics Part-I
Alright class, let's delve into Chapter 2: Units and Measurements. This chapter forms the bedrock of physics. Understanding how we measure quantities and express them accurately is crucial, not just for your Class 11 studies, but also for various government exams where fundamental physics concepts are tested.

Chapter 2: Units and Measurements - Detailed Notes

1. Physical Quantities:

  • Any quantity that can be measured is called a physical quantity.
  • Examples: Length, mass, time, force, velocity, temperature.
  • Measurement involves comparing a physical quantity with a standard unit.
  • Measurement = Numerical Value (n) × Unit (u)

2. Need for Units:

  • To express any measurement of a physical quantity meaningfully and consistently.
  • To compare different measurements of the same physical quantity.
  • To establish relationships between different physical quantities (formulas).

3. Systems of Units:

  • CGS System: Centimetre (length), Gram (mass), Second (time).
  • FPS System: Foot (length), Pound (mass), Second (time). (British System)
  • MKS System: Metre (length), Kilogram (mass), Second (time).
  • SI System (Système Internationale d'Unités): The internationally accepted system, a rationalized and extended version of the MKS system.

4. Fundamental (Base) Quantities and SI Units:

  • These are the elementary quantities that are independent of each other.

  • The SI system defines seven fundamental quantities:

    S.No. Fundamental Quantity SI Unit Symbol
    1 Length metre m
    2 Mass kilogram kg
    3 Time second s
    4 Electric Current ampere A
    5 Thermodynamic Temperature kelvin K
    6 Amount of Substance mole mol
    7 Luminous Intensity candela cd

  • Two Supplementary Units:

    • Plane Angle: radian (rad) - Defined as arc/radius (dθ = ds/r)
    • Solid Angle: steradian (sr) - Defined as area intercepted / (radius)² (dΩ = dA/r²)

5. Derived Quantities and Units:

  • Quantities that can be expressed in terms of fundamental quantities.
  • Their units are derived from the fundamental units.
  • Examples:
    • Area: Length × Breadth (m × m = m²)
    • Volume: Length × Breadth × Height (m × m × m = m³)
    • Density: Mass / Volume (kg / m³ = kg m⁻³)
    • Velocity: Displacement / Time (m / s = m s⁻¹)
    • Acceleration: Velocity / Time (m s⁻¹ / s = m s⁻²)
    • Force: Mass × Acceleration (kg × m s⁻² = kg m s⁻² or Newton (N))
    • Work/Energy: Force × Displacement (N × m = kg m² s⁻² or Joule (J))
    • Power: Work / Time (J / s = kg m² s⁻³ or Watt (W))
    • Pressure: Force / Area (N / m² = kg m⁻¹ s⁻² or Pascal (Pa))

6. Measurement of Length:

  • Direct Methods: Using instruments like:
    • Metre scale (Least count 1 mm or 0.1 cm)
    • Vernier Callipers (Least count typically 0.1 mm or 0.01 cm)
    • Screw Gauge (Least count typically 0.01 mm or 0.001 cm)
  • Indirect Methods (for large distances):
    • Parallax Method: Used for measuring distances to planets or stars.
      • Parallax is the apparent shift in the position of an object with respect to the background when viewed from two different positions.
      • The distance between the two observation points is called the basis (b).
      • The angle subtended by the basis at the object is the parallactic angle (θ) (in radians).
      • Distance (D) = Basis (b) / Parallactic Angle (θ) => D = b/θ (where θ is small and in radians)
  • Units for Very Small and Very Large Distances:
    • 1 fermi (f) = 10⁻¹⁵ m (nuclear size)
    • 1 angstrom (Å) = 10⁻¹⁰ m (atomic size)
    • 1 nanometer (nm) = 10⁻⁹ m
    • 1 Astronomical Unit (AU) = Average distance between Earth and Sun ≈ 1.496 × 10¹¹ m
    • 1 Light Year (ly) = Distance travelled by light in vacuum in one year = 9.46 × 10¹⁵ m
    • 1 Parsec (pc) = Distance at which an arc of length 1 AU subtends an angle of 1 second of arc = 3.08 × 10¹⁶ m ≈ 3.26 ly
    • Order: 1 pc > 1 ly > 1 AU

7. Measurement of Mass:

  • Mass is a fundamental property of matter, measuring its inertia. SI unit: kg.
  • Measured using a common balance (comparing with known masses).
  • For atomic/molecular masses, unified atomic mass unit (u) is used.
    • 1 u = (1/12)th of the mass of a Carbon-12 atom ≈ 1.66 × 10⁻²⁷ kg

8. Measurement of Time:

  • SI unit: second (s).
  • Based on the atomic standard of time using Cesium-133 atomic clocks. One second is defined as the duration of 9,192,631,770 periods of radiation corresponding to the transition between two hyperfine levels of the ground state of the Cs-133 atom. These clocks are extremely accurate.

9. Accuracy, Precision, and Errors in Measurement:

  • Accuracy: How close a measured value is to the true or accepted value.
  • Precision: How close multiple measurements of the same quantity are to each other (resolution or limit of the instrument).
    • High precision does not guarantee high accuracy.
  • Error: The difference between the measured value and the true value. Error = Measured Value - True Value.
  • Types of Errors:
    • Systematic Errors: Errors that tend to be in one direction (either positive or negative). They have assignable causes.
      • Instrumental Errors: Due to faulty instrument calibration or design (e.g., zero error).
      • Imperfection in Experimental Technique: Following wrong procedures.
      • Personal Errors: Bias or carelessness of the observer (e.g., parallax error in reading a scale).
      • Errors due to External Causes: Changes in temperature, pressure, humidity, etc.
      • Minimization: Improving instruments, refining techniques, removing personal bias, applying corrections.
    • Random Errors: Errors that occur irregularly and are random in sign and size. Caused by unpredictable fluctuations.
      • Minimization: Taking multiple observations and calculating the arithmetic mean. The mean value is likely closer to the true value.

10. Estimation of Errors:

  • Let measurements be a₁, a₂, a₃, ..., aₙ.
  • Arithmetic Mean (a_mean): Best estimate of the true value.
    • a_mean = (a₁ + a₂ + ... + aₙ) / n = (1/n) Σ aᵢ
  • Absolute Error (Δaᵢ): Magnitude of the difference between the mean value and each individual measurement.
    • Δa₁ = |a_mean - a₁|, Δa₂ = |a_mean - a₂|, ..., Δaₙ = |a_mean - aₙ|
  • Mean Absolute Error (Δa_mean): Arithmetic mean of the absolute errors.
    • Δa_mean = (|Δa₁| + |Δa₂| + ... + |Δaₙ|) / n = (1/n) Σ |Δaᵢ|
    • The final result is often expressed as: a = a_mean ± Δa_mean
  • Relative Error (or Fractional Error): Ratio of the mean absolute error to the mean value.
    • Relative Error = Δa_mean / a_mean
  • Percentage Error: Relative error expressed as a percentage.
    • Percentage Error = (Δa_mean / a_mean) × 100%

11. Combination of Errors:

  • If Z = A + B or Z = A - B: ΔZ = ΔA + ΔB (Absolute errors add up)
  • If Z = A × B or Z = A / B: ΔZ/Z = ΔA/A + ΔB/B (Relative errors add up)
  • If Z = Aⁿ: ΔZ/Z = n (ΔA/A) (Relative error is multiplied by the power)
  • General Case: If Z = (Aᵖ B<0xC2><0xAA>) / (Cʳ), then ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C)

12. Significant Figures:

  • Digits in a measured value that are known reliably plus one uncertain digit. They indicate the precision of a measurement.
  • Rules for determining significant figures:
    1. All non-zero digits are significant. (e.g., 285 has 3 sf)
    2. Zeros between two non-zero digits are significant. (e.g., 2005 has 4 sf)
    3. Leading zeros (zeros to the left of the first non-zero digit) are not significant. They only indicate the position of the decimal point. (e.g., 0.0032 has 2 sf)
    4. Trailing zeros (zeros to the right of the decimal point) are significant. (e.g., 4.700 has 4 sf)
    5. Trailing zeros in a whole number without a decimal point are ambiguous. Use scientific notation to avoid ambiguity. (e.g., 3500 m could have 2, 3, or 4 sf. Writing 3.5 × 10³ m implies 2 sf; 3.50 × 10³ m implies 3 sf; 3.500 × 10³ m implies 4 sf).
    6. Exact numbers (like counting numbers or numbers in formulas like 2 in 2πr) have infinite significant figures.
  • Rules for Arithmetic Operations:
    1. Addition/Subtraction: The final result should retain the same number of decimal places as the number with the fewest decimal places.
      • Example: 12.11 + 18.0 + 1.012 = 31.122. Result should be rounded to 31.1 (one decimal place as in 18.0).
    2. Multiplication/Division: The final result should retain the same number of significant figures as the number with the fewest significant figures.
      • Example: 4.237 × 2.51 = 10.63487. Result should be rounded to 10.6 (three significant figures as in 2.51).
  • Rounding Off:
    1. If the digit to be dropped is > 5, increase the preceding digit by 1. (e.g., 2.746 -> 2.75)
    2. If the digit to be dropped is < 5, leave the preceding digit unchanged. (e.g., 2.743 -> 2.74)
    3. If the digit to be dropped is 5:
      • If the preceding digit is even, leave it unchanged. (e.g., 2.745 -> 2.74)
      • If the preceding digit is odd, increase it by 1. (e.g., 2.735 -> 2.74) (This rule ensures unbiased rounding over many operations).

13. Dimensions of Physical Quantities:

  • Dimensions are the powers to which the fundamental units (Mass [M], Length [L], Time [T], etc.) must be raised to represent a derived physical quantity.
  • Dimensional Formula: An expression showing how and which fundamental quantities represent a physical quantity. Example: Dimensional formula of Velocity is [M⁰ L¹ T⁻¹].
  • Dimensional Equation: An equation obtained by equating a physical quantity with its dimensional formula. Example: [v] = [M⁰ L¹ T⁻¹].

14. Common Dimensional Formulas:

  • Velocity (v) = [L T⁻¹]
  • Acceleration (a) = [L T⁻²]
  • Force (F) = [M L T⁻²]
  • Work/Energy (W/E) = [M L² T⁻²]
  • Power (P) = [M L² T⁻³]
  • Pressure (P) = [M L⁻¹ T⁻²]
  • Density (ρ) = [M L⁻³]
  • Frequency (ν) = [T⁻¹]
  • Momentum (p) = [M L T⁻¹]
  • Angular Velocity (ω) = [T⁻¹]
  • Torque (τ) = [M L² T⁻²] (Same as Work/Energy)
  • Stress = [M L⁻¹ T⁻²] (Same as Pressure)
  • Strain = [M⁰ L⁰ T⁰] (Dimensionless)
  • Modulus of Elasticity = [M L⁻¹ T⁻²] (Same as Stress/Pressure)
  • Surface Tension = [M T⁻²]
  • Coefficient of Viscosity (η) = [M L⁻¹ T⁻¹]
  • Gravitational Constant (G) = [M⁻¹ L³ T⁻²]
  • Planck's Constant (h) = [M L² T⁻¹]
  • Specific Heat Capacity = [L² T⁻² K⁻¹]
  • Electric Charge (q) = [A T]
  • Electric Potential (V) = [M L² T⁻³ A⁻¹]
  • Resistance (R) = [M L² T⁻³ A⁻²]
  • Capacitance (C) = [M⁻¹ L⁻² T⁴ A²]

15. Dimensional Analysis and its Applications:

  • Based on the Principle of Homogeneity: Dimensions of all terms on both sides of a valid physical equation must be the same.
  • Applications:
    1. Checking Dimensional Consistency: Verify if an equation is dimensionally correct. (e.g., check v = u + at). Note: A dimensionally correct equation may not be physically correct (missing dimensionless constants).
    2. Deducing Relations among Physical Quantities: If a quantity depends on other quantities, we can find a possible relation between them (up to a dimensionless constant). (e.g., deriving T ∝ √(l/g) for a simple pendulum).
    3. Converting Units: Convert a physical quantity from one system of units to another. Formula: n₂ = n₁ [M₁/M₂]ᵃ [L₁/L₂]ᵇ [T₁/T₂]ᶜ
  • Limitations of Dimensional Analysis:
    1. Cannot determine dimensionless constants (like k, 2π).
    2. Cannot be used if the relation involves trigonometric, logarithmic, or exponential functions.
    3. Cannot be used if a quantity depends on more than three fundamental quantities (M, L, T - unless other dimensions like A, K are involved).
    4. Cannot distinguish between quantities having the same dimensions (e.g., Work and Torque).
    5. Doesn't tell if a quantity is a scalar or a vector.

Multiple Choice Questions (MCQs)

  1. Which of the following is NOT a fundamental quantity in the SI system?
    (a) Mass
    (b) Length
    (c) Force
    (d) Time
    Answer: (c) Force (Force is a derived quantity, F = ma)

  2. The dimensional formula for Gravitational Constant (G) is:
    (a) [M L T⁻²]
    (b) [M⁻¹ L³ T⁻²]
    (c) [M L² T⁻²]
    (d) [M L⁻¹ T⁻¹]
    Answer: (b) [M⁻¹ L³ T⁻²] (From F = Gm₁m₂/r²)

  3. A student measures the length of a rod and writes it as 3.50 cm. Which instrument might have been used to measure it?
    (a) Metre scale (least count 0.1 cm)
    (b) Vernier callipers (least count 0.01 cm)
    (c) Screw gauge (least count 0.001 cm)
    (d) Any of these
    Answer: (b) Vernier callipers (The measurement has precision up to the second decimal place, matching the typical least count of vernier callipers).

  4. The number of significant figures in 0.06900 is:
    (a) 5
    (b) 4
    (c) 2
    (d) 3
    Answer: (b) 4 (Leading zeros are not significant, trailing zeros after the decimal are significant).

  5. If the percentage error in the measurement of radius (r) of a sphere is 2%, then the percentage error in the calculation of its volume will be:
    (a) 2%
    (b) 4%
    (c) 6%
    (d) 8%
    Answer: (c) 6% (Volume V = (4/3)πr³. ΔV/V = 3 (Δr/r). % error in V = 3 × (% error in r) = 3 × 2% = 6%).

  6. Which of the following pairs has the same dimensions?
    (a) Force and Work
    (b) Work and Torque
    (c) Pressure and Force
    (d) Power and Energy
    Answer: (b) Work and Torque (Both have dimensions [M L² T⁻²]).

  7. The unit 'parsec' is used to measure:
    (a) Time
    (b) Angle
    (c) Very small distances
    (d) Astronomical distances
    Answer: (d) Astronomical distances

  8. According to the principle of homogeneity of dimensions, which of the following is correct?
    (a) An equation must be dimensionally correct to be physically correct.
    (b) A dimensionally correct equation is always physically correct.
    (c) Dimensions of all terms in a physical equation must be the same.
    (d) Dimensional analysis can determine the value of dimensionless constants.
    Answer: (c) Dimensions of all terms in a physical equation must be the same.

  9. The sum of numbers 436.32, 227.2, and 0.301 in appropriate significant figures is:
    (a) 663.821
    (b) 663.8
    (c) 663.82
    (d) 664
    Answer: (b) 663.8 (In addition, the result must be rounded to the least number of decimal places, which is one in 227.2).

  10. If momentum (P), area (A) and time (T) are taken to be fundamental quantities, then the dimensional formula for energy is:
    (a) [P A T⁻¹]
    (b) [P A¹ᐟ² T⁻¹]
    (c) [P² A T]
    (d) [P A⁻¹ T]
    Answer: (b) [P A¹ᐟ² T⁻¹]

    • Energy [E] = [M L² T⁻²]
    • Momentum [P] = [M L T⁻¹]
    • Area [A] = [L²]
    • Time [T] = [T]
    • From [P], [M] = [P L⁻¹ T]. Substitute in [E]:
    • [E] = [P L⁻¹ T] [L²] [T⁻²] = [P L T⁻¹]
    • From [A], [L] = [A¹ᐟ²]. Substitute in [E]:
    • [E] = [P] [A¹ᐟ²] [T⁻¹] = [P A¹ᐟ² T⁻¹]

Remember to thoroughly revise these concepts, especially dimensional analysis, error calculation, and significant figures, as questions from these topics are frequently asked in competitive exams. Good luck!

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