Class 12 Physics Notes Chapter 1 (Chapter 1) – Lab Manual (English) Book

Detailed Notes with MCQs of the foundational aspects covered in Chapter 1 of your Physics Lab Manual. This chapter is crucial not just for your board practicals but also forms the basis for understanding accuracy and precision in measurements, which is often tested in various government exams.

Chapter 1: Introduction to Experimental Physics & Error Analysis

1. The Essence of Experimentation in Physics:

• Physics is an experimental science. Experiments help verify theoretical concepts, determine physical constants, and understand relationships between different physical quantities.
• A typical experiment involves:
  • Aim: Clearly stating the objective.
  • Apparatus: Listing the required equipment.
  • Theory/Formula Used: The underlying principle and mathematical formula.
  • Procedure: Step-by-step method to perform the experiment.
  • Observations: Recording measurements systematically, usually in a tabular format with proper units.
  • Calculations: Processing the observed data using the formula.
  • Result: The final outcome of the experiment, stated clearly with appropriate units.
  • Precautions: Steps taken to minimize errors.
  • Sources of Error: Identifying potential reasons for inaccuracies.

2. Errors in Measurement:
No measurement is perfectly accurate. The difference between the true value and the measured value of a quantity is called an error.

• Systematic Errors: These errors tend to occur in one direction (either positive or negative) and affect the accuracy of the measurement. They have assignable causes and can, in principle, be eliminated or corrected for.

  • Instrumental Errors: Due to imperfect design, calibration, or zero error in the instrument (e.g., a metre scale worn off at the zero mark, zero error in Vernier callipers or screw gauge).
  • Imperfection in Experimental Technique or Procedure: Following a wrong procedure, not setting up the apparatus correctly (e.g., parallax error in reading a scale, not accounting for heat loss in calorimetry).
  • Personal Errors: Due to individual bias, lack of proper setting of the apparatus, or carelessness in taking observations (e.g., consistently reading a value slightly higher or lower).
  • Errors due to External Conditions: Changes in temperature, pressure, humidity, voltage fluctuations, etc., affecting the measurement.
  • Least Count Error: Associated with the resolution (the smallest value that can be measured) of the instrument. The least count error belongs to the category of random errors but can be limited by using higher precision instruments.
    • Least Count (LC): The smallest measurement that can be made accurately using an instrument.
    • Example: LC of a standard metre scale = 1 mm or 0.1 cm. LC of typical Vernier Callipers = 0.01 cm. LC of typical Screw Gauge = 0.001 cm.

• Random Errors: These errors occur irregularly and are random with respect to sign and size. They arise due to unpredictable fluctuations in experimental conditions, personal judgment variations, etc. They affect the precision of measurements.

  • Minimization: Random errors can be minimized by repeating the observation several times and taking the arithmetic mean of all observations. The mean value is likely to be very close to the true value.

3. Representing Errors:
Let a physical quantity be measured 'n' times. Let the measured values be a₁, a₂, a₃, ..., a<0xE2><0x82><0x99>.

• Arithmetic Mean (a<0xE2><0x82><0x98><0xE2><0x82><0x97>): The best estimate of the true value.
  a<0xE2><0x82><0x98><0xE2><0x82><0x97> = (a₁ + a₂ + ... + a<0xE2><0x82><0x99>) / n = (1/n) Σ a<0xE1><0xB5><0xA2> (where i goes from 1 to n)
• Absolute Error (Δa<0xE1><0xB5><0xA2>): The magnitude of the difference between the true value (or mean value) and the individual measured value.
  Δa₁ = |a<0xE2><0x82><0x98><0xE2><0x82><0x97> - a₁|, Δa₂ = |a<0xE2><0x82><0x98><0xE2><0x82><0x97> - a₂|, ..., Δa<0xE2><0x82><0x99> = |a<0xE2><0x82><0x98><0xE2><0x82><0x97> - a<0xE2><0x82><0x99>|
• Mean Absolute Error (Δa<0xE2><0x82><0x98><0xE2><0x82><0x97>): The arithmetic mean of the magnitudes of absolute errors in all measurements.
  Δa<0xE2><0x82><0x98><0xE2><0x82><0x97> = (Δa₁ + Δa₂ + ... + Δa<0xE2><0x82><0x99>) / n = (1/n) Σ |Δa<0xE1><0xB5><0xA2>|
  The final result of the measurement is often expressed as: a = a<0xE2><0x82><0x98><0xE2><0x82><0x97> ± Δa<0xE2><0x82><0x98><0xE2><0x82><0x97>
• Relative Error (or Fractional Error): The ratio of the mean absolute error to the mean value.
  Relative Error = Δa<0xE2><0x82><0x98><0xE2><0x82><0x97> / a<0xE2><0x82><0x98><0xE2><0x82><0x97>
• Percentage Error: The relative error expressed in percent.
  Percentage Error = (Δa<0xE2><0x82><0x98><0xE2><0x82><0x97> / a<0xE2><0x82><0x98><0xE2><0x82><0x97>) × 100%

4. Combination of Errors:
When a result depends on calculations involving multiple measurements, the errors in individual measurements combine.

• Error in Sum or Difference: If Z = A + B or Z = A - B, then the maximum absolute error in Z is ΔZ = ΔA + ΔB.
• Error in Product or Quotient: If Z = A × B or Z = A / B, then the maximum relative error in Z is (ΔZ / Z) = (ΔA / A) + (ΔB / B).
• Error in Case of a Measured Quantity Raised to a Power: If Z = Aⁿ, then the relative error in Z is (ΔZ / Z) = n (ΔA / A).
• General Case: If Z = (Aᵖ B<0xE1><0xB5><0xA1>) / (Cʳ), then the maximum relative error is (ΔZ / Z) = p(ΔA / A) + q(ΔB / B) + r(ΔC / C).

5. Significant Figures:
Significant figures indicate the precision of a measurement. They are the reliable digits plus the first uncertain digit.

• Rules for determining significant figures:
  1. All non-zero digits are significant. (e.g., 123.45 has 5 s.f.)
  2. All zeros occurring between two non-zero digits are significant. (e.g., 102.005 has 6 s.f.)
  3. If the number is less than 1, the zero(s) on the right of the decimal point but to the left of the first non-zero digit are not significant. (e.g., 0.00305 has 3 s.f. - 3, 0, 5)
  4. For numbers without a decimal point, the terminal or trailing zero(s) are not significant unless specified by the measurement precision (using scientific notation is better). (e.g., 12300 m has 3 s.f. To indicate 5 s.f., write 1.2300 × 10⁴ m).
  5. For numbers with a decimal point, the trailing zero(s) are significant. (e.g., 1.2300 has 4 s.f.; 0.030400 has 5 s.f.)
• Rules for Arithmetic Operations with Significant Figures:
  1. Addition/Subtraction: The final result should retain as many decimal places as are there in the number with the least decimal places. (e.g., 12.1 + 1.03 + 0.013 = 13.143 ≈ 13.1)
  2. Multiplication/Division: The final result should retain as many significant figures as are there in the original number with the least significant figures. (e.g., 4.12 × 0.11 = 0.4532 ≈ 0.45)
• Rounding Off:
  1. If the digit to be dropped is > 5, the preceding digit is increased by 1.
  2. If the digit to be dropped is < 5, the preceding digit is left unchanged.
  3. If the digit to be dropped is 5, the preceding digit is increased by 1 if it is odd, and left unchanged if it is even. (e.g., 2.745 ≈ 2.74; 2.735 ≈ 2.74)

6. Graph Plotting:
Graphs are powerful tools for visualizing relationships between variables.

• Choosing Axes: Independent variable usually on the x-axis, dependent variable on the y-axis. Label axes clearly with quantity and unit.
• Choosing Scale: Select a scale such that the plotted points occupy a significant portion (at least 2/3rd) of the graph paper. The scale should be convenient (e.g., 1 unit = 1, 2, 5, 10... cm). Mention the scale on the graph.
• Plotting Points: Plot points accurately using a sharp pencil (use symbols like ⊕ or ⊗).
• Drawing the Best-Fit Line/Curve: Draw a smooth line (straight or curved) that passes through, or as close as possible to, the maximum number of plotted points, with roughly equal numbers of points distributed on either side of the line. Do not simply connect the dots.
• Calculating Slope (for straight-line graphs): Choose two points on the best-fit line (preferably far apart) and use the formula: Slope (m) = (y₂ - y₁) / (x₂ - x₁). Include units for the slope if applicable.

Multiple Choice Questions (MCQs):

1. 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) A metre scale (LC = 0.1 cm)
   b) Vernier callipers (LC = 0.01 cm)
   c) A screw gauge (LC = 0.001 cm)
   d) A measuring tape (LC = 0.1 cm)
   Answer: b) Vernier callipers (LC = 0.01 cm) (The measurement is precise to the second decimal place)

2. In an experiment, the readings of the period of oscillation of a simple pendulum were found to be 2.63 s, 2.56 s, 2.42 s, 2.71 s, and 2.80 s. The mean absolute error is:
   a) 0.11 s
   b) 0.22 s
   c) 0.01 s
   d) 1.1 s
   Answer: a) 0.11 s (Mean = (2.63+2.56+2.42+2.71+2.80)/5 = 13.12/5 = 2.624 ≈ 2.62 s. Absolute errors: |2.62-2.63|=0.01, |2.62-2.56|=0.06, |2.62-2.42|=0.20, |2.62-2.71|=0.09, |2.62-2.80|=0.18. Mean Absolute Error = (0.01+0.06+0.20+0.09+0.18)/5 = 0.54/5 = 0.108 ≈ 0.11 s)

3. The number of significant figures in the measurement 0.007800 kg is:
   a) 2
   b) 3
   c) 4
   d) 6
   Answer: c) 4 (7, 8, and the two trailing zeros after the decimal are significant).

4. The length and breadth of a rectangular sheet are 16.2 cm and 10.1 cm, respectively. The area of the sheet in appropriate significant figures is:
   a) 163.62 cm²
   b) 163.6 cm²
   c) 163.7 cm²
   d) 164 cm²
   Answer: d) 164 cm² (Area = 16.2 cm × 10.1 cm = 163.62 cm². Both measurements have 3 significant figures. The result should be rounded to 3 significant figures, which is 164 cm²).

5. A physical quantity P is related to four observables a, b, c, and d as follows: P = a³b² / (√c d). The percentage errors in the measurement of a, b, c, and d are 1%, 3%, 4%, and 2%, respectively. What is the percentage error in the quantity P?
   a) 12%
   b) 13%
   c) 14%
   d) 15%
   Answer: b) 13% (%Error in P = 3(%Error in a) + 2(%Error in b) + (1/2)(%Error in c) + 1(%Error in d) = 3(1%) + 2(3%) + (1/2)(4%) + 1(2%) = 3% + 6% + 2% + 2% = 13%).

6. An error that occurs due to a faulty calibration of an instrument is called:
   a) Random error
   b) Personal error
   c) Systematic error
   d) Least count error
   Answer: c) Systematic error (Specifically, an instrumental error, which is a type of systematic error).

7. While plotting a graph, the best-fit line:
   a) Must pass through the origin.
   b) Must pass through all the plotted points.
   c) Should connect the first and the last point.
   d) Should represent the general trend of the data, passing as close as possible to the points.
   Answer: d) Should represent the general trend of the data, passing as close as possible to the points.

8. The least count of a standard laboratory screw gauge is typically:
   a) 0.1 cm
   b) 0.01 cm
   c) 0.001 cm
   d) 0.0001 cm
   Answer: c) 0.001 cm (Which is equal to 0.01 mm).

9. When adding 9.8 m, 15.26 m, and 0.321 m, the result expressed in correct significant figures is:
   a) 25.381 m
   b) 25.38 m
   c) 25.4 m
   d) 25 m
   Answer: c) 25.4 m (Sum = 25.381 m. The number with the least decimal places is 9.8 m (one decimal place). The result must be rounded to one decimal place).

10. Random errors can be minimized by:
    a) Using a more precise instrument.
    b) Correcting for zero error.
    c) Taking multiple readings and calculating their mean.
    d) Avoiding parallax error.
    Answer: c) Taking multiple readings and calculating their mean. (Using a precise instrument reduces least count error; correcting zero error and avoiding parallax address systematic errors).

Remember to thoroughly understand these concepts, as they apply to every experiment you will perform. Good luck with your preparation!