Need for Measurement; Units of Measurement

Physics is fundamentally a quantitative science. The development of physics, and indeed all science, depends critically on accurate and precise measurements. To describe, understand, and predict natural phenomena, we must be able to quantify various physical attributes. For instance, to describe the motion of an object, we need to measure its position, velocity, and acceleration at different instants of time. To measure a physical quantity, a comparison is made with a reference standard of the same physical quantity. This reference standard is called a unit of the physical quantity.

A unit must possess two essential characteristics: it must be internationally accepted for uniformity, and it must be reproducible. For example, if we measure the length of a table, we state it as ‘2 meters’. Here, ‘2’ is the numerical value and ‘meter’ is the unit. Without a universally agreed-upon unit, such a measurement would be meaningless or lead to confusion across different contexts or countries. Therefore, the result of a measurement of a physical quantity is always expressed as the product of a numerical value and a unit.

Systems of Units; SI Units, Fundamental and Derived Units

Historically, scientists around the world used different systems of units for measurement, leading to inconsistencies and difficulties in exchanging scientific and technical data. Prominent examples of these older systems include the CGS system (Centimetre, Gram, Second), the FPS system (Foot, Pound, Second), and the MKS system (Metre, Kilogram, Second). To address these issues and promote international uniformity, the General Conference on Weights and Measures (CGPM) in 1960 adopted a universal system known as the International System of Units (Système Internationale d’Unités or SI). This system is now universally accepted and employed worldwide for scientific, technical, industrial, and commercial work. The SI system is based on a set of seven fundamental (base) units and two supplementary units, from which all other units are derived.

SI Base (Fundamental) Units

The SI system defines seven base quantities that are considered dimensionally independent. For each base quantity, there is a corresponding SI base unit, which is precisely defined to ensure high accuracy and reproducibility. All other physical quantities are expressed in terms of these base quantities.

It is worth noting that each of these units has a detailed, operational definition that establishes its value based on fundamental physical constants or highly stable phenomena.

SI Supplementary Units

In addition to the seven base units, the SI system includes two supplementary units:

• Radian (rad): The SI unit for plane angle. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
• Steradian (sr): The SI unit for solid angle. One steradian is the solid angle subtended at the centre of a sphere by a surface of the sphere having an area equal to the square of the radius of the sphere.

Derived Units

Derived units are those units that are obtained by multiplying or dividing the fundamental (base) units. Every physical quantity that is not a base quantity has a derived unit. For example, the unit of speed (distance/time) is meters per second (m/s), which combines the base units of length and time. Similarly, the unit of force (mass × acceleration) is kilogram meter per second squared (kg m/s²). To simplify expressions for commonly used derived quantities, the SI system assigns special names to many derived units. These special names are equivalent to the algebraic combination of base units but provide convenience in scientific communication.

• Newton (N): The SI unit of force, equivalent to kg m s^-2.
• Joule (J): The SI unit of work or energy, equivalent to N m or kg m² s^-2.
• Watt (W): The SI unit of power, equivalent to J s^-1 or kg m² s^-3.
• Pascal (Pa): The SI unit of pressure, equivalent to N m^-2 or kg m^-1 s^-2.
• Hertz (Hz): The SI unit of frequency, equivalent to s^-1.
• Coulomb (C): The SI unit of electric charge, equivalent to A s.
• Volt (V): The SI unit of electric potential difference, equivalent to J C^-1 or kg m² A^-1 s^-3.

The adoption of SI units has significantly reduced ambiguity and enhanced the efficiency and accuracy of scientific and technological endeavors globally.

Significant Figures

Every measurement is inherently approximate and carries some degree of uncertainty. The number of significant figures in a measured quantity or in a result of a calculation indicates the precision of the measurement and the reliability of the value. Significant figures are defined as all the digits in a number that are known with certainty, plus one final digit which is estimated or uncertain. By convention, a more precise measurement will have more significant figures.

Rules for Determining the Number of Significant Figures

To count the number of significant figures in a given numerical value, apply the following rules strictly as per NCERT:

1. All non-zero digits are significant.

   Example: 42.3 cm has 3 significant figures. 289.6 kg has 4 significant figures.

2. All zeros between two non-zero digits are significant. These are often called “sandwich zeros” or “enclosed zeros.”

   Example: 2007 m has 4 significant figures. 4.008 g has 4 significant figures.

3. Trailing zeros (zeros at the end of the number) are significant only if the number is written with a decimal point. If there is no decimal point, trailing zeros are usually considered ambiguous and are not significant.

   Example: 3.200 kg has 4 significant figures (the decimal point makes the zeros significant). 12.000 cm has 5 significant figures. However, 12000 kg (without a decimal point) has 2 significant figures (the trailing zeros are not significant). To avoid ambiguity for whole numbers, scientific notation is preferred. For example, 1.20 × 104 kg clearly shows 3 significant figures, whereas 1.2 × 104 kg shows 2 significant figures.

4. Leading zeros (zeros appearing before the first non-zero digit) are never significant. These zeros simply indicate the position of the decimal point.

   Example: 0.00456 m has 3 significant figures. 0.0203 g has 3 significant figures (the leading zeros are not significant, but the zero between 2 and 3 is significant).

5. Exact numbers have an infinite number of significant figures. These include numbers obtained by counting (e.g., 2 chairs) or by definition (e.g., 1 inch = 2.54 cm exactly, 100 cm = 1 m exactly). Such numbers do not limit the number of significant figures in a calculation.

Rules for Arithmetic Operations with Significant Figures

When calculations are performed, the result must be reported with an appropriate number of significant figures, reflecting the precision of the input measurements. This prevents misrepresenting the accuracy of the final answer.

1. In multiplication or division, the final result should retain the same number of significant figures as the original number with the least number of significant figures.

   Example: Calculate the density of an object with mass = 4.237 g and volume = 2.51 cm³.

   • Mass: 4.237 g (4 significant figures)
   • Volume: 2.51 cm³ (3 significant figures)

   Density = Mass / Volume = 4.237 g / 2.51 cm³ = 1.6880478… g/cm³.

   Since the least precise input (volume) has 3 significant figures, the result must be rounded to 3 significant figures: 1.69 g/cm³.

2. In addition or subtraction, the final result should retain the same number of decimal places as the original number with the fewest decimal places. The number of significant figures in the total or difference is not the primary consideration; it’s the position of the last significant digit relative to the decimal point.

   Example: Add 3.45 g, 2.1 g, and 7.001 g.

   • 3.45 g (2 decimal places)
   • 2.1 g (1 decimal place)
   • 7.001 g (3 decimal places)

   Sum = 3.45 + 2.1 + 7.001 = 12.551 g.

   The number with the fewest decimal places is 2.1 g (1 decimal place). Therefore, the sum must be rounded to one decimal place: 12.6 g.

Rounding Off the Uncertain Digit: When rounding off, the digit immediately to be dropped determines the fate of the preceding digit:

• If the digit to be dropped is less than 5, the preceding digit is left unchanged. (e.g., 2.74 is rounded to 2.7)
• If the digit to be dropped is greater than 5, the preceding digit is increased by one. (e.g., 2.76 is rounded to 2.8)
• If the digit to be dropped is 5 followed by non-zero digits, the preceding digit is increased by one. (e.g., 2.751 is rounded to 2.8)
• If the digit to be dropped is 5 or 5 followed by zeros, and the preceding digit is even, it is left unchanged. (e.g., 2.450 is rounded to 2.4)
• If the digit to be dropped is 5 or 5 followed by zeros, and the preceding digit is odd, it is increased by one. (e.g., 2.350 is rounded to 2.4)

Dimensions of Physical Quantities

The dimensions of a physical quantity are the powers to which the base units are raised to represent that quantity. Essentially, dimensions express the fundamental nature of a physical quantity in terms of its dependence on the base quantities (Mass [M], Length [L], Time [T], Electric Current [A], Thermodynamic Temperature [K], Amount of Substance [mol], and Luminous Intensity [cd]), irrespective of the specific system of units employed. For example, velocity is always length divided by time, so its dimension is [L T^-1], whether measured in m/s or km/h.

A dimensional formula is an expression that shows how and which of the base quantities are involved in a derived quantity. For instance, the dimensional formula for force is [M L T^-2]. A dimensional equation is an equation obtained by equating a physical quantity with its dimensional formula. For example, the dimensional equation for energy is [E] = [M L² T^-2].

Based on their dimensions, physical quantities can be categorized as:

• Dimensional Constants: These are physical quantities that have specific dimensions but possess fixed, constant numerical values. Examples include the Universal Gravitational Constant (G), which has dimensions [M^-1 L³ T^-2], Planck’s Constant (h) with dimensions [M L² T^-1], and the speed of light in vacuum (c) with dimensions [L T^-1].
• Dimensionless Quantities: These are quantities that have no dimensions, meaning their dimensional formula is effectively [M⁰ L⁰ T⁰]. They are pure numbers and are often ratios of two quantities with the same dimensions. Examples include angle (ratio of arc length to radius), strain (ratio of change in length to original length), relative density (ratio of densities), and all trigonometric ratios (sin θ, cos θ, etc.). Dimensionless quantities remain unchanged regardless of the system of units chosen.

The fundamental guiding principle in dimensional analysis is the Principle of Homogeneity of Dimensions, which states that the dimensions of all the terms on both sides of a physical equation must be identical. This principle implies that physical quantities can only be added or subtracted if they have the same dimensions.

Dimensional Analysis and its Applications

Dimensional analysis is a powerful technique based on the Principle of Homogeneity of Dimensions. It serves as an invaluable tool for physicists and engineers in several ways, primarily for checking consistency and establishing relationships between physical quantities.

Applications of Dimensional Analysis

1. To Check the Dimensional Consistency (Correctness) of a Given Equation:

   The principle of homogeneity requires that the dimensions of each term on both sides of a physical equation must be the same. If the dimensions on both sides of an equation are not the same, the equation is incorrect. This provides a quick and robust method to verify the fundamental validity of a derived or proposed formula. It is important to note, however, that while dimensional consistency is a necessary condition for a correct equation, it is not a sufficient one, as numerical factors (like 1/2 or π) cannot be determined by this method.

   Example: Check the dimensional consistency of the equation for kinetic energy, K = 1/2 mv2.

   • Dimension of Kinetic Energy (K): [M L² T^-2] (same as work/energy).
   • Dimension of mass (m): [M]
   • Dimension of velocity (v): [L T^-1]
   • Dimension of v²: ([L T^-1])² = [L² T^-2]
   • Dimension of the term mv2: [M] × [L² T^-2] = [M L² T^-2]

   The numerical factor 1/2 is dimensionless. Since the dimensions of both sides are [M L² T^-2], the equation is dimensionally consistent.

2. To Derive Relationship Between Physical Quantities:

   Dimensional analysis can be used to derive a relationship between physical quantities if it is known that a particular quantity depends on certain other quantities. This is achieved by assuming a power law dependence and equating the powers of the fundamental dimensions on both sides of the proportional relationship.

   Example: Derive the expression for the time period (T) of a simple pendulum, assuming it depends on the mass of the bob (m), the length of the pendulum (l), and the acceleration due to gravity (g).

   We assume the relation is of the form: T ∝ m^a l^b g^c

   Writing this in terms of their dimensions:

   [M0 L0 T1] = [M]a [L]b [L T-2]c

   Combining the powers of M, L, T on the right side:

   [M0 L0 T1] = [Ma L(b+c) T-2c]

   Equating the powers of M, L, and T on both sides:

   1. For Mass (M): a = 0
   2. For Length (L): b + c = 0 → b = -c
   3. For Time (T): 1 = -2c → c = -1/2

   Substituting c = -1/2 into b = -c, we get b = 1/2.

   Thus, T ∝ m0 l1/2 g-1/2, which simplifies to T ∝ √(l/g).

   Dimensional analysis successfully determines the dependence on length and gravity but cannot determine the numerical constant (which is 2π for a simple pendulum).

3. To Convert a Physical Quantity from One System of Units to Another:

   This application is based on the principle that the magnitude of a physical quantity remains the same regardless of the system of units used. If n₁ is the numerical value and U₁ is the unit in one system, and n₂ and U₂ are the corresponding values in another system, then n1U1 = n2U2. By expressing the units in terms of their fundamental dimensions, the conversion factor can be found.

   Example: Convert 1 Newton (SI unit of force) into Dynes (CGS unit of force).

   Dimension of Force: [M L T-2]

   In SI system (System 1): M₁ = 1 kg, L₁ = 1 m, T₁ = 1 s

   In CGS system (System 2): M₂ = 1 g, L₂ = 1 cm, T₂ = 1 s

   We use the relation: n2 = n1 (M1/M2)a (L1/L2)b (T1/T2)c. Here a=1, b=1, c=-2 (from the dimensions of force).

   n2 = 1 × (1 kg / 1 g)1 × (1 m / 1 cm)1 × (1 s / 1 s)-2

   n2 = 1 × (1000 g / 1 g) × (100 cm / 1 cm) × (1)

   n2 = 1 × 1000 × 100 = 100,000 = 105

   Therefore, 1 Newton = 10⁵ Dynes.

Limitations of Dimensional Analysis

Despite its utility, dimensional analysis has certain inherent limitations:

• It cannot determine dimensionless constants of proportionality. The value of such constants (e.g., 1/2, 2π, etc.) must be found experimentally or through theoretical derivations.
• It cannot derive relations involving trigonometric, exponential, or logarithmic functions, as the arguments of these functions are dimensionless, and their series expansions involve infinite terms.
• It cannot derive relations if a physical quantity depends on more than three independent physical quantities (when using only M, L, T as fundamental dimensions). For example, if a formula involves five variables, dimensional analysis alone cannot determine the relationship uniquely.
• It cannot distinguish between physical quantities that have the same dimensions. For instance, both work and torque have the same dimensions ([M L2 T-2]), but they are distinct physical quantities.
• It cannot derive equations that involve the sum or difference of quantities. For instance, while it can check the dimensional consistency of an equation like v = u + at, it cannot derive its exact additive form, as dimensional analysis primarily works for products and quotients of powers of quantities.

Notwithstanding these limitations, dimensional analysis remains a powerful and practical tool for initial checks and understanding relationships in physics.