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Unit 1: Physical Quantities

Physics - Class 11

This chapter explores the concepts of precision and significant figures in measurements, detailing rules for identifying and rounding significant digits. It also covers dimensions and dimensional analysis, showing how to check equations, derive relationships, convert units, and understand dimensionless quantities.

No MCQ questions available for this chapter.

Unit 1: Physical Quantities

Precision and Significant Figures

Definition of Precision

Precision refers to the degree of exactness with which a measurement is expressed. It is determined by the smallest division or least count of the measuring instrument used. For example, a ruler marked in millimetres allows measurements to be precise to the nearest 0.1 cm, while a vernier caliper can achieve precision up to 0.01 mm. Higher precision implies that repeated measurements of the same quantity will yield values that are closely clustered together.

What Are Significant Figures?

Significant figures (often abbreviated as sig figs) are the digits in a measured quantity that carry reliable information about its magnitude. They include all non‑zero digits, any zeros that lie between non‑zero digits, and trailing zeros that appear after a decimal point. Leading zeros, which merely indicate the position of the decimal point, are not considered significant.

Rules for Identifying Significant Figures

  1. All non‑zero digits are significant. Example: 123.45 has five significant figures.
  2. Zeros between non‑zero digits are significant. Example: 1002 has four significant figures.
  3. Leading zeros are not significant. Example: 0.00456 has three significant figures (4, 5, 6).
  4. Trailing zeros in a number containing a decimal point are significant. Example: 78.00 has four significant figures.
  5. Trailing zeros in a whole number without a decimal point are ambiguous. To avoid ambiguity, scientific notation is used. Example: 1500 could be 2, 3, or 4 sig figs; writing 1.5×10³ (2 sig figs) or 1.500×10³ (4 sig figs) clarifies the intention.

Examples of Significant Figures

Number Significant Figures Explanation
2.34 3 All digits are non‑zero.
0.0056 2 Leading zeros are not significant; only 5 and 6 count.
100.0 4 Zeros between 1 and the decimal point are significant; the trailing zero after the decimal is also significant.
4500 Ambiguous (2‑4) Without a decimal point, the trailing zeros’ significance is unclear.
4.500×10³ 4 Scientific notation makes all digits explicit.

Rounding to a Specified Number of Significant Figures

When a calculation yields a result with more significant figures than warranted by the input data, the answer must be rounded. The rounding procedure follows the standard rule: if the digit immediately after the last retained significant figure is 5 or greater, increase the last retained figure by one; otherwise, leave it unchanged.

Example: Multiply 4.56 (3 sig figs) by 1.4 (2 sig figs). The raw product is 6.384. Since the factor with the fewest sig figs (1.4) has 2 sig figs, the result must be rounded to 2 sig figs: 6.4.

Additional rounding examples:

  • 0.004567 rounded to 2 sig figs → 0.0046
  • 12345 rounded to 3 sig figs → 1.23×10⁴
  • 9.999 rounded to 2 sig figs → 1.0×10¹

Applications of Precision and Significant Figures

In experimental physics, the number of significant figures reported in a measurement communicates the confidence in that value. When combining measurements through addition, subtraction, multiplication, or division, the result must not imply greater precision than the least precise input. This practice prevents the propagation of unwarranted certainty and ensures that conclusions drawn from data are scientifically sound.

Dimensions and Dimensional Analysis

Concept of Dimensions

The dimension of a physical quantity expresses how it is built from the seven base quantities of the International System of Units (SI). Each base quantity is assigned a symbol: Length [L], Mass [M], Time [T], Temperature [K], Electric Current [A], Amount of Substance [mol], and Luminous Intensity [cd]. In most introductory treatments, the first five are sufficient.

A derived quantity’s dimension is written as a product of powers of these base symbols. For instance, speed is distance divided by time, giving it the dimension [L T⁻¹].

Base Quantities and Their Symbols

Base Quantity Symbol SI Unit
Length [L] metre (m)
Mass [M] kilogram (kg)
Time [T] second (s)
Temperature [K] kelvin (K)
Electric Current [A] ampere (A)

Dimensional Formula

The dimensional formula (or dimensional equation) of a physical quantity shows the powers to which each base quantity must be raised to represent it. It is conventionally expressed in the MLT notation (mass, length, time) when temperature and current are not involved.

Examples:

  • Force: F = ma → dimension [M L T⁻²] (mass × acceleration).
  • Work/Energy: W = F·d → dimension [M L² T⁻²].
  • Power: P = W/t → dimension [M L² T⁻³].
  • Pressure: P = F/A → dimension [M L⁻¹ T⁻²].
  • Charge: Q = I·t → dimension [A T].

Uses of Dimensional Analysis

  1. Checking Dimensional Correctness: An equation must be dimensionally homogeneous; every term must have the same dimensions. If not, the equation is physically incorrect. Example: The equation s = ut + ½at² is dimensionally correct because each term has dimension [L].
  2. Deriving Relationships: By assuming a product of powers of relevant variables, dimensional analysis can yield the form of a physical law up to a dimensionless constant. Example: For the period T of a simple pendulum depending on length l and gravitational acceleration g, we set T ∝ l^a g^b. Solving [T] = [l]^a [g]^b gives a = ½, b = –½, leading to T = k√(l/g).
  3. Converting Units: Dimensional analysis provides a systematic way to convert a quantity from one unit system to another by multiplying with appropriate conversion factors that are dimensionally equal to one. Example: Converting 100 km/h to m/s: 100 km/h × (1000 m/1 km) × (1 h/3600 s) = 27.8 m/s.

Dimensionless Quantities

Some physical quantities have no dimensions; their dimensional formula is [1] (i.e., all exponents are zero). These are called dimensionless quantities. They often arise as ratios of similar quantities or as arguments of transcendental functions. Common examples include:

  • Angle measured in radians: θ = arc length / radius → [L/L] = [1].
  • Strain: ε = ΔL / L → [L/L] = [1].
  • Refractive index: n = speed of light in vacuum / speed in medium → [(L/T)/(L/T)] = [1].
  • Poisson’s ratio, coefficient of friction, Mach number.

Because they carry no units, dimensionless numbers are pure numbers and can be compared across different systems of units.

Limitations of Dimensional Analysis

While powerful, dimensional analysis has certain constraints:

  • Cannot Determine Dimensionless Constants: The method yields the form of a relationship but not the numerical value of any dimensionless multiplier (e.g., the 2π in the period of a mass‑spring system). Such constants must be found experimentally or from deeper theory.
  • Fails for Exponential, Logarithmic, and Trigonometric Functions: These functions require dimensionless arguments; dimensional analysis cannot deduce the functional form when the unknown quantity appears inside such functions.
  • Limited to Products of Powers: It assumes that the relationship between variables is a product of powers. If the true law involves sums or differences of terms with different dimensions, the method may give misleading results.

Practical Example: Deriving the Formula for the Speed of a Wave on a String

Consider a wave on a string under tension F with linear mass density μ (mass per unit length). We hypothesize that the wave speed v depends on F and μ only: v ∝ F^a μ^b. Writing dimensions:

[v] = [L T⁻¹], [F] = [M L T⁻²], [μ] = [M L⁻¹].

Thus:

[L T⁻¹] = [M L T⁻²]^a [M L⁻¹]^b = M^{a+b} L^{a‑b} T^{-2a}.

Equating exponents:

  • For M: 0 = a + b → b = –a
  • For L: 1 = a – b → substituting b = –a gives 1 = a – (‑a) = 2a → a = ½, b = –½
  • For T: ‑1 = –2a → a = ½ (consistent)

Hence v ∝ F^{1/2} μ^{-1/2} or v = k √(F/μ). Experiments show that the dimensionless constant k = 1, giving the well‑known formula v = √(F/μ).

Summary Table of Common Dimensions

Quantity Symbol Dimensional Formula
Displacement s [L]
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]
Charge Q [A T]
Voltage V [M L² T⁻³ A⁻¹]
Resistance R [M L² T⁻³ A⁻²]
Angle (rad) θ [1]
Refractive Index n [1]

Conclusion

Understanding precision and significant figures ensures that measurements are reported honestly, reflecting the true limits of experimental apparatus. Mastery of dimensional analysis provides a powerful tool for verifying equations, deriving plausible relationships, converting units, and recognizing the role of dimensionless quantities. Together, these topics form the foundation for quantitative reasoning in physics and are indispensable for solving problems at the Class 11 level and beyond.

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