Units and Measurements revision notes

Key Concepts & Definitions

Measurement of any physical quantity involves comparing it with a basic, arbitrarily chosen, internationally accepted reference standard called a unit. The result of a measurement consists of a numerical measure accompanied by a unit.

Fundamental or Base Units:

The units for the fundamental or base quantities that cannot be derived from one another.

Derived Units:

Units of all other physical quantities that can be expressed as combinations of the base units.

System of Units:

A complete set of units, comprising both base and derived units. Historically, extensively used systems included the CGS (centimetre, gram, second), FPS/British (foot, pound, second), and MKS (metre, kilogram, second) systems.

Dimensions:

The physical nature of a quantity is described by its dimensions. The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.

Exact Numbers:

Multiplying or dividing factors that are not measured values but inherent to a formula (e.g., the 222 in s=2πrs = 2\pi rs=2πr, or the index nnn in T/nT/nT/n) are exact numbers. They have an infinite number of significant digits (e.g., 2.000...2.000...2.000...) and do not constrain the precision of calculated results.

Specific Distance Units:

Light year (ly): The distance light travels in one year in a vacuum. 1 ly=9.47×1015 m1 \text{ ly} = 9.47 \times 10^{15} \text{ m}1 ly=9.47×1015 m. Angstrom (A˚\text{\AA}A˚): A convenient unit for atomic scales. 1 A˚=10−10 m1 \text{ \AA} = 10^{-10} \text{ m}1 A˚=10−10 m. It is often used to denote the size of molecules or atoms, such as the hydrogen atom which is about 0.5 A˚0.5 \text{ \AA}0.5 A˚.

The International System of Units (SI)

The currently accepted system for international usage in scientific, technical, industrial, and commercial work is the Système International d’Unités (SI). It uses a decimal system, making conversions within the system highly convenient. The definitions of base units are continuously revised to keep up with measuring techniques leading to greater precision. The most recent revision was made by the General Conference on Weights and Measures in November 2018.

Seven SI Base Quantities and Units:

1. Length ($[L]$): metre (m). Defined by taking the fixed numerical value of the speed of light in vacuum ($c$) to be $299792458 \text{ m s}^{-1}$.
2. Mass ($[M]$): kilogram (kg). Defined by fixing the Planck constant ($h$) to $6.62607015 \times 10^{-34} \text{ J s}$ (or $\text{kg m}^2\text{s}^{-1}$).
3. Time ($[T]$): second (s). Defined by taking the fixed numerical value of the caesium frequency ($\Delta\nu_{Cs}$) to be $9192631770 \text{ Hz}$.
4. Electric Current ($[A]$): ampere (A). Defined by fixing the elementary charge ($e$) to $1.602176634 \times 10^{-19} \text{ C}$ (or $\text{A s}$).
5. Thermodynamic Temperature ($[K]$): kelvin (K). Defined by fixing the Boltzmann constant ($k$) to $1.380649 \times 10^{-23} \text{ J K}^{-1}$.
6. Amount of Substance ($[mol]$): mole (mol). One mole contains exactly $6.02214076 \times 10^{23}$ elementary entities (Avogadro number, $N_A$). The entities must be specified (atoms, molecules, ions, electrons, etc.).
7. Luminous Intensity ($[cd]$): candela (cd). Defined by fixing the luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12} \text{ Hz}$, $K_{cd}$, to $683 \text{ lm W}^{-1}$.

Two Supplementary Quantities (Dimensionless):

• Plane Angle ($d\theta$): radian (rad). The ratio of the length of arc ($ds$) to the radius ($r$). Both are dimensionless quantities.
• Solid Angle ($d\Omega$): steradian (sr). The ratio of the intercepted area ($dA$) of a spherical surface, described about the apex as the centre, to the square of its radius ($r$).

Significant Figures & Order of Magnitude

Every measurement involves errors. To indicate the precision of measurement, results are reported using significant figures, which consist of all reliably known digits plus the first uncertain digit. Precision depends on the least count of the measuring instrument. [JEE TIP] A choice of a change of units does NOT change the number of significant digits in a measurement.

Rules for Determining Significant Figures:

1. All non-zero digits are significant.
2. All zeros between two non-zero digits are significant, no matter where the decimal point is.
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., $0.\underline{00}2308$ has 4 sig figs).
4. The terminal or trailing zero(s) in a number without a decimal point are not significant (e.g., $12300 \text{ m}$ has 3 sig figs).
5. The trailing zero(s) in a number with a decimal point are significant (e.g., $3.500$ has 4 sig figs).
6. Multiplying or dividing factors that are exact numbers (e.g., the $2$ in $d = 2r$) have an infinite number of significant digits (can be written as $2.0, 2.000...$).

Order of Magnitude & Scientific Notation: To remove ambiguities regarding trailing zeros, express measurements in scientific notation: $a \times 10^b$.

• $a$ is a number between 1 and 10, and every digit in $a$ is significant.
• $b$ is the order of magnitude.
• JEE TIPTrap 1 - Rounding for Order of Magnitude: By convention, round $a$ to $1$ for $a \le 5$, and round $a$ to $10$ for $5 < a \le 10$. Example: $1.28 \times 10^7 \text{ m}$ is order of magnitude $10^7 \text{ m}$ ($b=7$).

Rules for Arithmetic Operations & Rounding Off

The result of a calculation must reflect the uncertainties of the original measured values and cannot be more precise than the least precise original value.

Arithmetic Rules:

• Multiplication/Division: The final result must retain as many significant figures as the original number with the least significant figures. (e.g., $4.237 \text{ g} \div 2.51 \text{ cm}^3 = 1.69 \text{ g cm}^{-3}$ since $2.51$ has 3 sig figs).
• Addition/Subtraction: The final result must retain as many decimal places as the original number with the least decimal places. (e.g., $436.32 \text{ g} + 227.2 \text{ g} + 0.301 \text{ g} = 663.8 \text{ g}$ since $227.2$ has only 1 decimal place).

Rounding Off Uncertain Digits:

1. If the insignificant digit to be dropped is $> 5$, raise the preceding digit by $1$.
2. If the insignificant digit to be dropped is $< 5$, leave the preceding digit unchanged.
3. JEE TIPTrap 2 - Dropping exactly 5: If the digit to be dropped is exactly $5$, look at the preceding digit. If it is even, drop the $5$ and leave the digit unchanged (e.g., $2.745 \rightarrow 2.74$). If it is odd, raise the preceding digit by $1$ (e.g., $2.735 \rightarrow 2.74$).

Dimensions and Dimensional Analysis

The dimensions of a physical quantity describe its nature in terms of the seven base dimensions: $[L], [M], [T], [A], [K], [cd], [mol]$. Enclosing a quantity in brackets $[X]$ means "the dimensions of $X$".

• Dimensional Formula: The expression showing which base quantities represent the dimensions of a quantity (e.g., Volume is $[M^0 L^3 T^0]$).
• Dimensional Equation: Equating a physical quantity with its dimensional formula (e.g., $[F] = [M L T^{-2}]$).

Applications of Dimensional Analysis:

1. Checking Dimensional Consistency (Homogeneity): We can only add or subtract physical quantities having the same dimensions (Principle of Homogeneity). The dimensions on both sides of a mathematical equation must be identical.
   • Example: For $x = x_0 + v_0 t + \frac{1}{2} a t^2$, every term evaluates to the dimension $[L]$, making it dimensionally correct.
2. Deducing Relations Among Physical Quantities: By assuming a product-type dependence on linearly independent variables (e.g., $T = k l^x g^y m^z$), we can equate dimensions on both sides to solve for the exponents $x, y, z$.

Conditions & Limitations

• Limitations of Dimensional Consistency:JEE TIPA dimensionally correct equation is not necessarily physically exact. It cannot guarantee correctness because dimensional analysis ignores dimensionless quantities (like pure numbers $\frac{1}{2}, \pi$) and dimensionless functions. However, a dimensionally wrong equation must definitely be wrong.
• Limitations of Deducing Relations: The method of dimensions cannot obtain the value of the dimensionless constant ($k$). It only works if the dependence is a product-type dependence and usually fails if the physical quantity depends on more than three linearly independent variables. It does not distinguish between the physical quantities having the exact same dimensions.
• Dimensionless Arguments:JEE TIPThe arguments (inputs) of special functions such as trigonometric functions ($\sin \theta$, $\cos \theta$), logarithmic functions ($\ln x$), and exponential functions ($e^x$) must be completely dimensionless pure numbers. Ratios of similar physical quantities like angle ($L/L$) and refractive index ($v_{vac}/v_{med}$) are also dimensionless.

Standard Derivations & Step-by-Step Problem Solving

1. Derivation of the Time Period of a Simple Pendulum using Dimensional Analysis:

• Assume the time period $T$ depends on length $l$, mass of bob $m$, and acceleration due to gravity $g$ as a product relationship: $T = k \cdot l^x \cdot g^y \cdot m^z$ (where $k$ is a dimensionless constant).
• Write the dimensions for both sides of the equation: $[M^0 L^0 T^1] = [L]^x \cdot [L T^{-2}]^y \cdot [M]^z$.
• Combine the base dimensions on the RHS: $[M^0 L^0 T^1] = [M^z] \cdot [L^{x+y}] \cdot [T^{-2y}]$.
• Equate the exponents from both sides: For $[M]$: $0 = z$; For $[L]$: $0 = x + y$; For $[T]$: $1 = -2y$.
• Solve the linear equations: $y = -1/2$, hence $x = 1/2$. $z = 0$.
• Substitute the exponents back into the relation: $T = k \cdot l^{1/2} \cdot g^{-1/2} \implies T = k \sqrt{\frac{l}{g}}$. (Advanced analysis proves $k = 2\pi$, which dimensional analysis cannot deduce).

2. Calculating Uncertainty in Arithmetic Results (Propagation of Error): When combining measured values via multiplication, relative errors are added to find the final uncertainty.

• Example: If a sheet has length $l = 16.2 \pm 0.1 \text{ cm}$ ($\pm 0.6\%$) and breadth $b = 10.1 \pm 0.1 \text{ cm}$ ($\pm 1\%$):
  1. Multiply the base values: $A = 16.2 \times 10.1 = 163.62 \text{ cm}^2$.
  2. Add the relative percentage errors: $0.6\% + 1\% = 1.6\%$.
  3. Convert total relative error back to absolute error: $1.6\%$ of $163.62 = 2.6 \text{ cm}^2$.
  4. Quote the final result: $A = 164 \pm 3 \text{ cm}^2$.

3. Checking the Relativistic Mass Equation: Albert Einstein's special relativity relates moving mass $m$ to rest mass $m_0$ and speed $v$. If a student writes $m = m_0 / (1 - v^2)^{1/2}$. By the Principle of Homogeneity, the term $(1 - v^2)$ is dimensionally inconsistent because $1$ is a dimensionless constant but $v^2$ has dimensions $[L^2 T^{-2}]$. To make it dimensionless, divide $v^2$ by $c^2$. The correct relation is $m = m_0 / (1 - v^2/c^2)^{1/2}$.

Previous Year JEE Topics

1. Dimensional Analysis of Unknown Functions:JEE TIPQuestions frequently provide a complex formula like $y = A \sin(Bt - Cx)$ or $P = P_0 e^{-\alpha t^2 / x}$ and ask for the dimensions of $A, B, C$ or $\alpha$. Remember the rule: arguments of trig and exponential functions must be entirely dimensionless ($[M^0 L^0 T^0]$).
2. Order of Magnitude Estimation: Calculating ratios of macroscopic dimensions (like Earth's diameter, $1.28 \times 10^7 \text{ m}$) to microscopic ones (like a hydrogen atom diameter, $1.06 \times 10^{-10} \text{ m}$), demonstrating that macroscopic objects are often ~17 orders of magnitude larger.
3. Arithmetic with Mixed Precision: Evaluating density from mass and volume, requiring the strict application of multiplication/division significant figure rules (retaining the lowest count of sig figs).
4. Checking Equations with Added/Subtracted Terms: Using homogeneity to verify expressions like $K = \frac{1}{2}mv^2 + ma$. Since $[mv^2]$ represents energy $[M L^2 T^{-2}]$ and $[ma]$ represents force $[M L T^{-2}]$, they cannot be added together, making the equation definitively invalid.

JEE Traps

• [JEE TIP] Trap 1 - Significant Unit Conversions:

  • Misconception: Shifting a measurement between different physical units (such as converting from meters to kilometers) alters the number of significant figures in the data.
  • Correct Understanding: Changing units never alters the number of significant figures of a measurement. A reading of $2.308 \text{ cm}$ contains exactly 4 significant figures; rewriting it as $0.00002308 \text{ km}$ via decimal shifting does not add or remove accuracy, leaving it with the exact same 4 significant figures.

• [JEE TIP] Trap 2 - The Arithmetic Sum Precision Rule:

  • Misconception: When adding or subtracting two values ($A + B$), the final result's significant figures are limited by the term with the fewest overall significant figures, matching the multiplication rule.
  • Correct Understanding: For addition and subtraction, the final precision is strictly limited by the term with the least number of decimal places, completely ignoring the total count of significant figures. For example, adding $123.4$ (one decimal place) to $0.0078$ must be rounded to exactly one decimal place, even though the terms have vastly different significant counts.

• [JEE TIP] Trap 3 - The Terminal Five Even-Odd Rounding Boundary:

  • Misconception: When rounding off a number where the drop digit is exactly 5 followed by no other digits, you always automatically round up to the next highest integer.
  • Correct Understanding: If the insignificant digit to be dropped is exactly 5, you must apply the even-odd rule: round to the nearest even number. Under this standard convention, $2.745$ rounds down to $2.74$ (since 4 is even), whereas $2.735$ rounds up to $2.74$ (since 3 is odd).

• [JEE TIP] Trap 4 - The Whole Number Trailing Zero Illusion:

  • Misconception: Trailing zeros in a whole number measurement lacking a visible decimal point (such as $12300 \text{ m}$) are always significant because they communicate the structural magnitude of the measurement.
  • Correct Understanding: Trailing zeros in a number without a decimal point are strictly non-significant. A reading of $12300 \text{ m}$ contains only 3 significant figures. To make those trailing zeros significant, the measurement must be explicitly written with a decimal point ($12300.\text{ m}$) or expressed unambiguously using scientific notation ($1.2300 \times 10^4 \text{ m}$).

• [JEE TIP] Trap 5 - The Dimensional Validity Sufficiency Fallacy:

  • Misconception: If a derivation or equation is verified to be dimensionally correct, it is guaranteed to be a physically valid representation of the exact natural relationship.
  • Correct Understanding: Dimensional consistency is merely a preliminary elimination test. A dimensionally wrong equation is always physically impossible, but a dimensionally correct equation is not necessarily physically true. Dimensional analysis cannot detect missing dimensionless constant multipliers, log terms, or verify the actual physical interaction of the variables.

• [JEE TIP] Trap 6 - The Order of Magnitude Numerical Threshold:

  • Misconception: The order of magnitude for both $4.7 \times 10^5$ and $6.2 \times 10^5$ is identical and simply extracted from the exponent as $10^5$.
  • Correct Understanding: To determine the true order of magnitude ($10^b$) from a number expressed in scientific notation ($a \times 10^b$), the coefficient $a$ is compared to a rounding threshold of $5$ (or $\sqrt{10} \approx 3.16$ depending on the specific textbook, though NCERT explicitly uses $5$). Because $6.2 > 5$, the coefficient is rounded up to $10$, transforming the order of magnitude to $10 \times 10^5 = 10^6$. Conversely, $4.7 \le 5$ rounds to $1$, leaving its order at $10^5$.

• [JEE TIP] Trap 7 - The Mechanics Dimensional Variable Ceiling:

  • Misconception: You can apply standard dimensional analysis to deduce the complete algebraic formula for a variable that simultaneously depends on four or more linearly independent parameters.
  • Correct Understanding: Deducing formulas via dimensions requires setting up and solving systems of linear exponent equations. In mechanics, you only have three fundamental dimensions available ($M, L,$ and $T$). Consequently, you can solve for a maximum of three independent exponents. If a quantity depends on four variables, you will get an underdetermined system that cannot be resolved without extra physical arguments.

• [JEE TIP] Trap 8 - Decimal Leading Zero Significance:

  • Misconception: Leading zeros that appear after a decimal point but before the first non-zero digit (such as the zeros in $0.002308$) are significant because they occupy a decimal position.
  • Correct Understanding: For any number less than 1, zeros positioned to the right of the decimal point but to the left of the first non-zero digit are completely non-significant. They serve merely as structural placeholders to orient the decimal point. Therefore, $0.002308$ contains exactly 4 significant figures ($2, 3, 0,$ and $8$).

• [JEE TIP] Trap 9 - The Special Function Argument Dimension Rule:

  • Misconception: The input arguments inside special functions like $\sin(\theta)$, $\ln(x)$, or $e^y$ can possess physical dimensions depending on the units chosen for the system.
  • Correct Understanding: The inputs (arguments) and outputs of all trigonometric, logarithmic, and exponential functions must strictly be pure, dimensionless numbers. If an argument contains physical variables (e.g., $e^{-kt}$), the product of those variables must combine to perfectly cancel out all dimensions ($[M^0 L^0 T^0]$), requiring the dimension of $k$ to be $[T^{-1}]$.

• [JEE TIP] Trap 10 - Pure Fractional Number Blindness:

  • Misconception: If a formula is presented with an incorrect numerical fraction—such as kinetic energy given as $K = \frac{3}{16}mv^2$—it can be immediately ruled out as dimensionally invalid.
  • Correct Understanding: Dimensional analysis is completely blind to pure numbers and dimensionless scalar coefficients. Both $\frac{1}{2}mv^2$ and $\frac{3}{16}mv^2$ evaluate to the exact same dimensional formula: $[M L^2 T^{-2}]$. You can never use dimensional equations alone to isolate or verify the accuracy of numerical coefficients.