CBSE Class 11 Physics Chapter 2 Notes: Units and Measurements

Chapter Introduction

Units and Measurements is the foundation of all quantitative physics. Every physical quantity you will ever calculate — from velocity to electric field — requires a unit, an understanding of errors, and dimensional consistency. This chapter is essential for the board exam (5–6 marks) and is also the basis of many JEE/NEET questions on error propagation and dimensional analysis.

Master three things from this chapter: (1) SI units and dimensional formulas, (2) error types and propagation, and (3) significant figure rules. These three topics appear in every physics paper without fail.

What this guide covers: SI base and derived units, measurement techniques, error types and propagation, significant figures, dimensional formulas, and dimensional analysis applications — all with worked numericals.

Key Definitions

• Unit: A standard quantity chosen to measure a physical quantity of the same kind.
• Fundamental (base) unit: A unit that cannot be expressed in terms of other units. SI defines seven.
• Derived unit: A unit expressed as a combination of base units (e.g., newton = kg m s^-2).
• Measurement: The comparison of a physical quantity with a standard unit.
• Error: The difference between the measured value and the true value of a quantity.
• Accuracy: Closeness of a measured value to the true value.
• Precision: Closeness of repeated measurements to each other (indicated by less scatter).
• Significant figures: The meaningful digits in a measured quantity, including the uncertain last digit.
• Dimension: The power to which a base quantity is raised to express a derived quantity.
• Least count: The smallest measurement that an instrument can reliably measure.
• Parallax error: Error due to observing a scale from an angle rather than directly from above.

Detailed Concepts

1. The International System of Units (SI)

Common supplementary units: radian (rad) for plane angle, steradian (sr) for solid angle.

Common derived units:

Prefixes for large and small quantities:

2. Measurement of Length, Mass and Time

Length measurement:

Vernier callipers: Least count = 1 MSD − 1 VSD. If N divisions of vernier scale = (N − 1) divisions of main scale, then LC = (1/N) MSD. Typical LC = 0.01 cm.

Screw gauge: Least count = Pitch / Number of divisions on circular scale. Typical pitch = 1 mm, 100 divisions → LC = 0.001 cm.

Mass measurement: Common balance (large masses), mass spectrograph (atomic masses), electron balance (lab precision).

Time measurement: Atomic clock (Cs-133, accuracy ~10^-12 s), quartz crystal clock (~10^-9 s), digital stopwatch (~0.01 s).

3. Errors in Measurement

Error representation:

• Absolute error: Δa_i = |a_mean − a_i| for each reading
• Mean absolute error: Δa_mean = (Σ|Δa_i|) / n
• Relative error (fractional error): Δa_mean / a_mean
• Percentage error: (Δa_mean / a_mean) × 100%

Error propagation rules — these are the most frequently tested formulas:

Key insight: In addition and subtraction, errors add (worst case). In multiplication and division, relative errors add. The power rule multiplies the relative error by the exponent.

4. Significant Figures

Rules for counting significant figures:

Rules for arithmetic operations:

• Addition/Subtraction: Result rounded to the least number of decimal places. Example: 4.36 + 2.1 = 6.46 → 6.5 (1 decimal place)
• Multiplication/Division: Result rounded to the least number of significant figures. Example: 2.5 × 3.42 = 8.55 → 8.6 (2 sig. figs.)

Rounding rules:

• If the digit to be dropped > 5, round up. (3.46 → 3.5)
• If the digit to be dropped < 5, round down. (3.41 → 3.4)
• If the digit to be dropped = 5 and the preceding digit is even, round down. (3.45 → 3.4)
• If the digit to be dropped = 5 and the preceding digit is odd, round up. (3.35 → 3.4)

5. Dimensions of Physical Quantities

6. Applications of Dimensional Analysis

1. Checking dimensional correctness: Principle of homogeneity — dimensions of every term on both sides must match. Example: Check s = ut + ½at²: [L] = [LT^-1][T] + [LT^-2][T²] → [L] = [L] + [L]. Correct.

2. Deriving relations: If a quantity depends on other quantities, write it as proportional to their powers, then solve for exponents using dimensional consistency.

Example: Derive the expression for the time period T of a simple pendulum. Assume T depends on length l and acceleration g: T ∝ l^a g^b Dimensions: [T] = [L]^a [LT^-2]^b = [L^a+b T^-2b] Equating: a + b = 0 and −2b = 1 → b = −½, a = ½ So T ∝ √(l/g) → T = k√(l/g) (k = 2π, found by experiment)

3. Unit conversion: If 1 newton = [MLT^-2], converting from SI to CGS: 1 N = 1 kg m s^-2 = 1000 g × 100 cm × s^-2 = 10⁵ dyne.

Limitations of dimensional analysis:

• Cannot determine dimensionless constants (numerical factors like ½, π, 2π).
• Cannot distinguish between quantities with the same dimensions (e.g., work and torque both have [ML²T^-2]).
• Cannot handle trigonometric, exponential, or logarithmic functions.
• Cannot be used for equations with more than one term on the same side with unknown constants.
• Cannot determine whether a quantity is a scalar or vector.

Important Numericals

Numerical 1: Error Propagation in Multiplication

The resistance R = V/I, where V = (100 ± 5) V and I = (10 ± 0.2) A. Find the percentage error in R.

Numerical 2: Error Propagation in Combined Formula

The physical quantity Q = A²B³/C, where A = (10.0 ± 0.1), B = (5.0 ± 0.2), C = (2.0 ± 0.1). Find the percentage error in Q.

Numerical 3: Significant Figures

Compute (2.5 × 1.24) + 3.5 with proper significant figures.

Numerical 4: Dimensional Analysis

Check the dimensional correctness of T = 2π√(l/g).

Numerical 5: Unit Conversion

Convert 1 J (SI) to erg (CGS).

Numerical 6: Dimensional Formula Derivation

Using dimensional analysis, find the expression for the escape velocity v in terms of gravitational constant G, mass M, and radius R.

NCERT Exercise Reference

Common Mistakes

1. Counting leading zeros as significant: 0.0045 has 2 significant figures, not 4. Only the 4 and 5 count. 2. Adding errors for multiplication: In multiplication, you add relative errors, not absolute errors. ΔZ/Z = ΔA/A + ΔB/B, not ΔZ = ΔA + ΔB. 3. Forgetting power rule: If Z = A², then ΔZ/Z = 2(ΔA/A), not ΔA/A. The exponent multiplies the relative error. 4. Confusing accuracy and precision: A set of measurements can be very close together (precise) but far from the true value (inaccurate). Think of a dartboard — all darts clustered together but far from the bullseye is precise but not accurate. 5. Dimensional analysis gives only proportionality: You cannot find the constant 2π in T = 2π√(l/g) using dimensions alone. The constant comes from calculus/experiment. 6. Rounding at intermediate steps: Always round only the final answer, not intermediate calculations. Rounding at intermediate steps accumulates error. 7. Subtraction error trap: When two nearly equal numbers are subtracted, the relative error can be very large. Example: (100.5 − 99.5) = 1.0, but each has error ±0.1, so result is 1.0 ± 0.2 → 20% error!

Exam Tips

• Memorise the 7 SI base units and their symbols — this is a guaranteed 1-mark question.
• Know the dimensional formulas for at least 15 common quantities (force, energy, power, pressure, G, h, surface tension, viscosity, specific heat, gas constant).
• Error propagation rules (addition vs. multiplication) are frequently tested — practise at least 5 numericals.
• Significant figure rules appear as short questions — be precise about leading vs. trailing zeros.
• For dimensional analysis questions, always write the dimensional formula of every term explicitly before equating powers.
• In the board exam, show all steps in numericals: write the formula, substitute values with units, and state the answer with correct significant figures and units.
• Deriving relations using dimensional analysis is a 3-mark question — remember you can only get the proportionality, not the dimensionless constant.
• The general error formula for Z = A^pB^q/C^r is the most powerful single formula — memorise it.

Quick Revision Notes

• 7 SI base units: m, kg, s, A, K, mol, cd.
• Derived units: N = kg m s^-2, J = kg m² s^-2, Pa = kg m^-1 s^-2, W = kg m² s^-3.
• Error types: systematic (consistent, correctable) vs. random (unpredictable, reduced by averaging).
• Error propagation: addition → add absolute errors; multiplication → add relative errors; power → multiply relative error by exponent.
• General rule: Z = A^pB^q/C^r → ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C).
• Significant figures: count non-zero digits + zeros between them + trailing zeros after decimal. Leading zeros do not count.
• Rounding: >5 up, <5 down, =5 use odd-even rule.
• Dimensional formulas: force [MLT^-2], energy [ML²T^-2], power [ML²T^-3], pressure [ML^-1T^-2], G [M^-1L³T^-2], h [ML²T^-1].
• Dimensional analysis: checks correctness, derives relations, converts units — but cannot find dimensionless constants or distinguish same-dimension quantities.
• Limitations: same dimensions for work and torque; cannot handle trig/exp/log; gives only proportionality.
• Vernier LC = 1 MSD − 1 VSD; Screw gauge LC = pitch / divisions on circular scale.