class 11 notes

Class 11 Physics Notes Chapter 1 (Chapter 1) – Examplar Problems (English) Book

Philoid

Philoid

— 9 min read

Examplar Problems (English)
Alright class, let's get straight into Chapter 1 from your Physics Exemplar book. This chapter lays the foundation for everything else we'll study in physics, focusing on how we quantify the physical world. For your government exam preparation, mastering these fundamentals – units, dimensions, errors, and significant figures – is crucial as direct questions are often asked from this section.

Chapter 1: Units and Measurement - Detailed Notes

1. Physical Quantities:

  • Any quantity that can be measured is called a physical quantity.
  • Examples: Length, mass, time, force, velocity, temperature.
  • Types:
    • Fundamental (or Base) Quantities: These are independent quantities that are not defined in terms of other physical quantities. The internationally accepted system (SI) recognizes 7 fundamental quantities.
    • Derived Quantities: These quantities are derived from combinations (multiplication or division) of fundamental quantities. Examples: Speed (Distance/Time), Force (Mass × Acceleration), Density (Mass/Volume).

2. Systems of Units:

  • A unit is a standard reference used for measuring a physical quantity.
  • A complete set of units (both fundamental and derived) is called a system of units.
  • Common Systems:
    • FPS System: Foot (length), Pound (mass), Second (time). (British system)
    • CGS System: Centimetre (length), Gram (mass), Second (time). (Gaussian system)
    • MKS System: Metre (length), Kilogram (mass), Second (time).
    • SI System (Système Internationale d'Unités): The internationally accepted, rationalized, and coherent system based on the MKS system. It's the standard for scientific work.

3. The SI System:

  • Seven Fundamental Units:
    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:
    1. Plane Angle: radian (rad) - Defined as arc/radius. (Dimensionless)
    2. Solid Angle: steradian (sr) - Defined as area intercepted on sphere / (radius)². (Dimensionless)

4. Dimensions and Dimensional Analysis:

  • Dimensions: The powers to which the fundamental quantities (represented by symbols like M for Mass, L for Length, T for Time, K for Temperature, A for Current, etc.) must be raised to represent a physical quantity.
  • Dimensional Formula: An expression showing how and which fundamental quantities represent a physical quantity. Example: Dimensional formula of Velocity is [M⁰L¹T⁻¹] or simply [LT⁻¹].
  • Dimensional Equation: An equation obtained by equating a physical quantity with its dimensional formula. Example: [Force] = [MLT⁻²].
  • Important Dimensional Formulae:
    • Area: [L²]
    • Volume: [L³]
    • Density: [ML⁻³]
    • Velocity/Speed: [LT⁻¹]
    • Acceleration: [LT⁻²]
    • Force: [MLT⁻²]
    • Work/Energy/Torque: [ML²T⁻²]
    • Power: [ML²T⁻³]
    • Pressure/Stress: [ML⁻¹T⁻²]
    • Frequency: [T⁻¹]
    • Momentum/Impulse: [MLT⁻¹]
    • Angular Velocity: [T⁻¹]
    • Gravitational Constant (G): [M⁻¹L³T⁻²]
    • Planck's Constant (h): [ML²T⁻¹]
    • Coefficient of Viscosity (η): [ML⁻¹T⁻¹]
    • Surface Tension: [MT⁻²]
  • Principle of Homogeneity of Dimensions: A physical equation is dimensionally correct only if the dimensions of all the terms occurring on both sides of the equation are the same. This means you can only add or subtract quantities having the same dimensions.
  • Applications of Dimensional Analysis:
    1. Checking Dimensional Consistency: Verify if an equation is dimensionally correct (e.g., check v = u + at).
    2. Deriving Relationships: Deduce the relation between physical quantities if you know the factors on which a quantity depends (e.g., deriving the formula for the time period of a simple pendulum). Limitation: Cannot determine dimensionless constants.
    3. Converting Units: Convert the numerical value of a physical quantity from one system of units to another using the relation n₁u₁ = n₂u₂, where n is the numerical value and u is the unit (represented dimensionally).
  • Limitations: Cannot determine dimensionless constants (like 2π, 1/2), cannot derive formulae involving trigonometric, logarithmic, or exponential functions, fails if a quantity depends on more than the number of fundamental dimensions used (usually M, L, T).

5. Errors in Measurement:

  • Accuracy: How close a measured value is to the true value.
  • Precision: How close repeated measurements are to each other (resolution or limit of the instrument).
  • Error: The difference between the measured value and the true value.
  • Types of Errors:
    • Systematic Errors: Errors that tend to be in one direction (either positive or negative). Causes: Instrumental errors (e.g., zero error), imperfections in technique, personal bias, external factors (temperature, pressure changes). Can often be minimized or corrected.
    • Random Errors: Errors that occur irregularly due to unpredictable fluctuations. Can be minimized by repeating measurements and taking the mean.
  • Calculating Errors:
    • Let measurements be a₁, a₂, ..., aₙ.
    • True Value (approx): Arithmetic Mean, a_mean = (a₁ + a₂ + ... + aₙ) / n
    • Absolute Error: Δaᵢ = |a_mean - aᵢ| (Magnitude of difference)
    • Mean Absolute Error: Δa_mean = (|Δa₁| + |Δa₂| + ... + |Δaₙ|) / n
    • Relative Error: δa = Δa_mean / a_mean
    • Percentage Error: Relative Error × 100% = (Δa_mean / a_mean) × 100%
  • Combination of Errors:
    • Sum (Z = A + B): ΔZ = ΔA + ΔB (Absolute errors add up)
    • Difference (Z = A - B): ΔZ = ΔA + ΔB (Absolute errors still add up)
    • Product (Z = A × B): ΔZ/Z = ΔA/A + ΔB/B (Relative errors add up)
    • Division (Z = A / B): ΔZ/Z = ΔA/A + ΔB/B (Relative errors add up)
    • Power (Z = Aⁿ): ΔZ/Z = n (ΔA/A) (Relative error is multiplied by the power)
    • General Case (Z = AᵖB<0xC> / Dʳ): ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C) + s(ΔD/D)

6. Significant Figures:

  • Digits in a measured value that are known reliably plus one uncertain digit. They indicate the precision of a measurement.
  • Rules for Counting Significant Figures:
    1. All non-zero digits are significant. (e.g., 123.45 has 5 SF)
    2. Zeros between non-zero digits are significant. (e.g., 1002.5 has 5 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., 0.0035 has 2 SF)
    4. Trailing zeros (zeros to the right of the last non-zero digit) in a number with a decimal point are significant. (e.g., 1.230 has 4 SF; 0.0500 has 3 SF)
    5. Trailing zeros in a number without a decimal point are ambiguous. Use scientific notation to avoid ambiguity. (e.g., 2300 m could have 2, 3, or 4 SF. Better written as 2.3 × 10³ m (2 SF), 2.30 × 10³ m (3 SF), or 2.300 × 10³ m (4 SF)).
    6. Exact numbers (like counting numbers or defined constants like π or factors in formulae) have infinite significant figures.
  • Rules for Arithmetic Operations:
    1. Addition/Subtraction: The result should be rounded off to the same number of decimal places as the number with the least number of decimal places. (e.g., 12.1 + 1.345 + 0.02 = 13.465 ≈ 13.5)
    2. Multiplication/Division: The result should be rounded off to the same number of significant figures as the number with the least number of significant figures. (e.g., 2.1 × 1.23 = 2.583 ≈ 2.6)
  • Rounding Off Rules:
    1. If the digit to be dropped is < 5, leave the preceding digit unchanged. (e.g., 3.42 rounded to 2 SF is 3.4)
    2. If the digit to be dropped is > 5, increase the preceding digit by 1. (e.g., 3.47 rounded to 2 SF is 3.5)
    3. If the digit to be dropped is 5:
      • Increase the preceding digit by 1 if it is odd. (e.g., 3.35 rounded to 2 SF is 3.4)
      • Leave the preceding digit unchanged if it is even. (e.g., 3.45 rounded to 2 SF is 3.4) (This convention avoids bias in rounding).

7. Measurement Instruments (Brief Overview):

  • Least Count (LC): The smallest value that can be measured accurately by an instrument.
  • Vernier Callipers: Used to measure length accurately. LC = Value of 1 Main Scale Division (MSD) - Value of 1 Vernier Scale Division (VSD). Often, LC = (Value of smallest division on main scale) / (Total number of divisions on Vernier scale).
  • Screw Gauge (Micrometer): Used to measure small lengths like diameter of a wire. LC = Pitch / (Number of divisions on circular scale). Pitch = distance moved by screw for one complete rotation of the circular scale.
  • Zero Error: Error present when the instrument should read zero. It can be positive or negative. Corrected Reading = Observed Reading - Zero Error (with sign).

Practice MCQs for Exam Preparation

  1. The dimensional formula for Gravitational Constant (G) is:
    a) [MLT⁻²]
    b) [M⁻¹L³T⁻²]
    c) [ML²T⁻²]
    d) [M⁻¹L²T⁻³]

  2. If Force (F), Velocity (V) and Time (T) are taken as fundamental quantities, the dimensions of Mass would be:
    a) [FVT⁻¹]
    b) [FV⁻¹T]
    c) [FVT]
    d) [FV⁻¹T⁻¹]

  3. Which of the following pairs does NOT have the same dimensions?
    a) Impulse and Momentum
    b) Work and Torque
    c) Surface Tension and Force
    d) Planck's Constant and Angular Momentum

  4. The density of a material in the CGS system is 4 g/cm³. In a system of units where the unit of length is 10 cm and the unit of mass is 100 g, the value of density of the material will be:
    a) 0.04
    b) 0.4
    c) 40
    d) 400

  5. A physical quantity P is related to four observables a, b, c, and d as follows: P = a³b² / (√c d). The percentage errors of measurement in a, b, c, and d are 1%, 3%, 4%, and 2%, respectively. What is the percentage error in the quantity P?
    a) 13%
    b) 12%
    c) 10%
    d) 14%

  6. The number of significant figures in the measurement 0.06900 is:
    a) 5
    b) 4
    c) 3
    d) 2

  7. The length, breadth, and thickness of a rectangular sheet are 4.234 m, 1.005 m, and 2.01 cm, respectively. The volume of the sheet, considering significant figures, should be:
    a) 0.0855 m³
    b) 0.086 m³
    c) 0.08554 m³
    d) 8.55 × 10⁻² m³

  8. In an experiment, the refractive index of glass was observed to be 1.45, 1.56, 1.54, 1.44, 1.54, and 1.53. The mean absolute error in the observations is:
    a) 0.04
    b) 0.03
    c) 0.50
    d) 1.51

  9. The equation of state for a real gas is given by (P + a/V²)(V - b) = RT. The dimensions of the constant 'a' are:
    a) [ML⁵T⁻²]
    b) [ML⁻¹T⁻²]
    c) [ML³T⁻²]
    d) [M L⁻⁵ T²]

  10. The least count of a Vernier callipers is 0.01 cm. If the zero error is +0.03 cm, and the main scale reading is 4.2 cm with the 6th Vernier division coinciding, the correct length is:
    a) 4.26 cm
    b) 4.29 cm
    c) 4.23 cm
    d) 4.17 cm

Answers to MCQs:

  1. b) [M⁻¹L³T⁻²]
  2. b) [FV⁻¹T] (Hint: F = Ma = M(V/T) => M = FT/V = F V⁻¹ T)
  3. c) Surface Tension ([MT⁻²]) and Force ([MLT⁻²])
  4. c) 40 (Hint: Use n₁u₁ = n₂u₂. Density = Mass/Length³. n₂ = n₁ (M₁/M₂) (L₁/L₂)⁻³. Here n₁=4, M₁=1g, L₁=1cm, M₂=100g, L₂=10cm. n₂ = 4 (1/100) (1/10)⁻³ = 4 (1/100) (1000) = 40)
  5. a) 13% (Hint: %Error in P = 3(%err a) + 2(%err b) + (1/2)(%err c) + 1(%err d) = 3(1) + 2(3) + (1/2)(4) + 1(2) = 3 + 6 + 2 + 2 = 13%)
  6. b) 4 (Trailing zeros after decimal are significant)
  7. b) 0.086 m³ (Hint: Convert thickness to meters: 2.01 cm = 0.0201 m. Volume = 4.234 × 1.005 × 0.0201 = 0.085528... m³. Least SF is 3 (from 0.0201 m). Rounding to 3 SF gives 0.0855 m³. Correction: 2.01 cm has 3 SF. 4.234 has 4 SF. 1.005 has 4 SF. Least SF = 3. Result should have 3 SF. 0.0855 m³. Let me recheck the calculation and rounding. 4.234 * 1.005 * 0.0201 = 0.0855289... Rounding to 3 SF: 0.0855. Wait, let's re-read the options. Option b) 0.086 m³. Why? Let's check the rounding rule if the digit to be dropped is 5 followed by non-zero digits. 0.08552... The digit to drop is 2 (after the third SF '5'). Since 2 < 5, it should remain 0.0855. Let's assume the question intended rounding to 2 SF, then it would be 0.086. Or maybe there's a typo in the question or options. Given the standard options, 0.086 m³ might be expected if rounding rules are applied slightly differently or if one input value was intended differently (e.g., 2.0 cm -> 2 SF). However, strictly by rules with the given numbers, 0.0855 m³ (3 SF) is correct. Let's choose the closest option often intended in exams, which might round 0.0855 to 0.086 if intermediate rounding occurred or if only 2 SF were considered for 2.01 cm. Let's stick to the strict rule: 3 SF -> 0.0855. Option (a) is 0.0855 m³. Let's select (a). Rechecking common exam practices, sometimes the rounding of 5 is simplified. If we round 0.08552 to 2 SF, it's 0.086. If we round to 3 SF, it's 0.0855. Let's assume option (b) implies rounding to 2 SF, which is inconsistent with the input data having minimum 3 SF. Option (a) 0.0855 m³ seems the most correct based on 3 SF. Let's re-evaluate option (d) 8.55 x 10^-2 m^3, which is the same as (a). Let's assume option (b) is the intended answer due to potential rounding variations or simpler exam logic. Let's stick with the most precise application of rules: 3 SF -> 0.0855 m³. So (a) or (d). Let's pick (a). Self-correction: Let's re-examine the question. 2.01 cm has 3 SF. 4.234 m has 4 SF. 1.005 m has 4 SF. The minimum number of SF is 3. Volume = 4.234 * 1.005 * (0.0201) = 0.0855289... Rounding to 3 SF gives 0.0855 m³. Options are a) 0.0855, b) 0.086, c) 0.08554, d) 8.55 x 10^-2. Both (a) and (d) represent 0.0855 m³. Let's select (a) as the primary representation.
  8. a) 0.04 (Hint: Mean = (1.45+1.56+1.54+1.44+1.54+1.53)/6 = 9.06/6 = 1.51. Absolute errors: |1.51-1.45|=0.06, |1.51-1.56|=0.05, |1.51-1.54|=0.03, |1.51-1.44|=0.07, |1.51-1.54|=0.03, |1.51-1.53|=0.02. Mean Absolute Error = (0.06+0.05+0.03+0.07+0.03+0.02)/6 = 0.26/6 ≈ 0.0433... ≈ 0.04)
  9. a) [ML⁵T⁻²] (Hint: By homogeneity, P and a/V² must have the same dimensions. [a] = [P][V²] = [ML⁻¹T⁻²][(L³)²] = [ML⁻¹T⁻²][L⁶] = [ML⁵T⁻²])
  10. c) 4.23 cm (Hint: Observed Reading = MSR + VSD × LC = 4.2 cm + 6 × 0.01 cm = 4.2 + 0.06 = 4.26 cm. Correct Reading = Observed Reading - Zero Error = 4.26 cm - (+0.03 cm) = 4.23 cm)

Make sure you understand the concepts behind each question, not just the answers. Revise these notes thoroughly, especially the dimensional formulae, error combination rules, and significant figure rules. Good luck with your preparation!

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