9.1 Physical Quantities, SI Units, Errors & Dimensional Analysis

Physical Quantities, SI Units, Errors & Dimensional Analysis

In the Intermediate Physics Part I syllabus and the University of Engineering and Technology (UET) ECAT entry test, physical measurements and dimensional analysis represent essential foundational concepts. In the ECAT examination, questions from this domain test quantitative error propagation under speed conditions, unit conversions, and the dimensional verification of physical laws without handheld calculators. Mastering significant figure conventions, uncertainty propagation formulas, and dimensional homogeneity ensures immediate scoring advantages.

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1. The International System of Units (SI)

In 1960, the Eleventh General Conference on Weights and Measures established the International System of Units (SI) (Le Système International d'Unités). The SI framework organizes all measurable physical quantities into three distinct categories: base units, supplementary units, and derived units.

The Seven Base Physical Quantities

Base quantities are functionally independent physical quantities that cannot be defined in terms of other physical quantities. The SI system is founded on exactly seven base units:

1. Length: Meter ($\text{m}$) — defined by the distance light travels in vacuum during a time interval of $\frac{1}{299,792,458}$ of a second.
2. Mass: Kilogram ($\text{kg}$) — historically defined by the international prototype cylinder, now defined by fixing the numerical value of the Planck constant $h = 6.62607015 \times 10^{-34}\text{ J}\cdot\text{s}$.
3. Time: Second ($\text{s}$) — defined by the duration of $9,192,631,770$ periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.
4. Electric Current: Ampere ($\text{A}$) — defined by taking the fixed numerical value of the elementary charge $e = 1.602176634 \times 10^{-19}\text{ C}$.
5. Thermodynamic Temperature: Kelvin ($\text{K}$) — defined by fixing the Boltzmann constant $k = 1.380649 \times 10^{-23}\text{ J}\cdot\text{K}^{-1}$.
6. Amount of Substance: Mole ($\text{mol}$) — containing exactly $6.02214076 \times 10^{23}$ elementary entities (Avogadro's number $N_A$).
7. Luminous Intensity: Candela ($\text{cd}$) — the luminous intensity of a source emitting monochromatic radiation of frequency $540 \times 10^{12}\text{ Hz}$ with a radiant intensity of $\frac{1}{683}\text{ W}\cdot\text{sr}^{-1}$.

Supplementary Units: Radian and Steradian

Many Intermediate textbooks still list two purely geometric units as supplementary units, although the General Conference on Weights and Measures abolished the supplementary class in 1995 and now treats them as dimensionless derived units. These quantities are dimensionless ratios of geometric metrics:

1. Radian ($\text{rad}$): The measure of a plane angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle:

$\theta = \frac{s}{r}$

Because the circumference of a full circle is $2\pi r$, a full revolution subtends:

$\theta_{\text{circle}} = \frac{2\pi r}{r} = 2\pi \text{ rad} = 360^\circ \implies 1 \text{ rad} = \left(\frac{180}{\pi}\right)^\circ \approx 57.3^\circ$

1. Steradian ($\text{sr}$): The measure of a solid angle subtended at the center of a sphere by a spherical surface area equal to the square of the radius of the sphere:

$\Omega = \frac{A}{r^2}$

Because the total surface area of a complete sphere is $4\pi r^2$, the total solid angle subtended by a closed sphere around its center is:

$\Omega_{\text{sphere}} = \frac{4\pi r^2}{r^2} = 4\pi \text{ sr} \approx 12.57 \text{ sr}$

Derived Units in Physics

Derived units are formed through algebraic multiplication or division of one or more base units. Standard derived units frequently tested on the ECAT include:

• Force (Newton): $1\text{ N} = 1\text{ kg}\cdot\text{m}\cdot\text{s}^{-2}$
• Work / Energy (Joule): $1\text{ J} = 1\text{ N}\cdot\text{m} = 1\text{ kg}\cdot\text{m}^2\cdot\text{s}^{-2}$
• Power (Watt): $1\text{ W} = 1\text{ J}\cdot\text{s}^{-1} = 1\text{ kg}\cdot\text{m}^2\cdot\text{s}^{-3}$
• Pressure / Stress (Pascal): $1\text{ Pa} = 1\text{ N}\cdot\text{m}^{-2} = 1\text{ kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$
• Electric Charge (Coulomb): $1\text{ C} = 1\text{ A}\cdot\text{s}$
• Electric Potential Difference (Volt): $1\text{ V} = 1\text{ J}\cdot\text{C}^{-1} = 1\text{ kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}$
• Electric Resistance (Ohm): $1\ \Omega = 1\text{ V}\cdot\text{A}^{-1} = 1\text{ kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}$

Metric Prefixes and Scientific Notation

In standard scientific notation, numbers are expressed as $M \times 10^n$, where $1 \le M < 10$ and $n \in \mathbb{Z}$. Standard decimal SI prefixes must be applied without compounding (for instance, $10^{-12}\text{ F}$ is written as $1\text{ pF}$, never as $1\ \mu\mu\text{F}$):

• Pico ($p = 10^{-12}$), Nano ($n = 10^{-9}$), Micro ($\mu = 10^{-6}$), Milli ($m = 10^{-3}$), Centi ($c = 10^{-2}$), Deci ($d = 10^{-1}$)
• Kilo ($k = 10^3$), Mega ($M = 10^6$), Giga ($G = 10^9$), Tera ($T = 10^{12}$)

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2. Experimental Errors & Uncertainties

Every physical measurement contains experimental doubt, categorized formally into errors and uncertainties. An error represents the numerical deviation between a measured value and the true physical value. An uncertainty quantifies the estimated range of values within which the true value is asserted to lie.

Systematic Errors vs. Random Errors

The Arithmetic Mean for Random Error Reduction

For $N$ independent measurements $x_1, x_2, \dots, x_N$, the best estimate of the true value is the arithmetic mean:

$\overline{x} = \frac{1}{N}\sum_{i=1}^N x_i$

The absolute deviation of each reading is $d_i = |x_i - \overline{x}|$, and the mean uncertainty is given by the average deviation:

$\Delta x = \frac{1}{N}\sum_{i=1}^N |x_i - \overline{x}|$

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3. Precision vs. Accuracy

A critical distinction in physics is that between precision and accuracy:

1. Precision: Relates to the sensitivity and least count of the measuring instrument. Absolute uncertainty is numerically equal to the least count of the instrument. An instrument with a smaller least count produces higher precision.
   • Meter rod least count: $0.1\text{ cm} = 1\text{ mm}$
   • Vernier caliper least count: $0.01\text{ cm} = 0.1\text{ mm}$
   • Micrometer screw gauge least count: $0.001\text{ cm} = 0.01\text{ mm}$
   • Spherometer least count: $0.001\text{ cm} = 0.01\text{ mm}$
2. Accuracy: Relates to how close the measured value is to the accepted true value. In quantitative error analysis, accuracy is inversely proportional to the fractional uncertainty or percentage uncertainty:

$\text{Fractional Uncertainty} = \frac{\text{Absolute Uncertainty (Least Count)}}{\text{Measured Quantity}} = \frac{\Delta x}{x}$

$\text{Percentage Uncertainty} = \frac{\Delta x}{x} \times 100\%$

ECAT Conceptual Trap: A measurement with a micrometer screw gauge ($0.25\text{ cm} \pm 0.001\text{ cm}$) has high precision (least count $= 0.001\text{ cm}$) and a percentage uncertainty of $\frac{0.001}{0.25} \times 100\% = 0.4\%$. If the instrument carries an uncorrected zero error of $+0.05\text{ cm}$, the result is highly precise yet completely inaccurate. Precision does not guarantee accuracy.

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4. Mathematical Propagation of Uncertainties

When combining measured values in physical equations, their uncertainties propagate according to specific algebraic rules:

1. Addition and Subtraction

When quantities are added or subtracted, their absolute uncertainties are added:

$\text{If } z = x + y \quad \text{or} \quad z = x - y, \quad \text{then } \Delta z = \Delta x + \Delta y$

Worked Example: Let initial position be $x_1 = (10.5 \pm 0.1)\text{ cm}$ and final position be $x_2 = (26.8 \pm 0.1)\text{ cm}$. The displacement is:

$\Delta x_{\text{disp}} = x_2 - x_1 = (26.8 - 10.5) \pm (0.1 + 0.1) = (16.3 \pm 0.2)\text{ cm}$

Even though the quantities were subtracted, their absolute uncertainties added.

2. Multiplication and Division

When quantities are multiplied or divided, their fractional or percentage uncertainties are added:

$\text{If } z = x \times y \quad \text{or} \quad z = \frac{x}{y}, \quad \text{then } \frac{\Delta z}{z} = \frac{\Delta x}{x} + \frac{\Delta y}{y}$

$\text{Percentage Uncertainty in } z = (\% \text{ uncertainty in } x) + (\% \text{ uncertainty in } y)$

To find the absolute uncertainty $\Delta z$:

$\Delta z = z \times \left(\frac{\Delta x}{x} + \frac{\Delta y}{y}\right)$

3. Exponents and Power Factors

When a measured quantity is raised to a power $n$, the percentage uncertainty is multiplied by that power exponent:

$\text{If } z = x^n, \quad \text{then } \frac{\Delta z}{z} = n \times \frac{\Delta x}{x}$

For a generalized composite formula $z = \frac{x^a y^b}{w^c}$:

$\frac{\Delta z}{z} = a\left(\frac{\Delta x}{x}\right) + b\left(\frac{\Delta y}{y}\right) + c\left(\frac{\Delta w}{w}\right)$

ECAT Speed Rule: Uncertainties always accumulate. Even if a quantity appears in the denominator raised to a power (such as $w^c$), its percentage uncertainty is added, never subtracted.

4. Uncertainty in Timing Experiments (Oscillations)

In timing experiments (such as measuring the time period $T$ of a simple pendulum), timing $N$ complete vibrations reduces the absolute uncertainty in the time period:

$\Delta T = \frac{\text{Least Count of Stopwatch}}{N}$

For example, if 30 vibrations take $54.6\text{ s}$ measured with a stopwatch of least count $0.1\text{ s}$, then:

$T = \frac{54.6\text{ s}}{30} = 1.82\text{ s}, \quad \Delta T = \frac{0.1\text{ s}}{30} = 0.0033\text{ s} \approx 0.003\text{ s}$

$T = (1.820 \pm 0.003)\text{ s}$

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5. Dimensional Analysis and Homogeneity

The dimension of any physical quantity represents its qualitative dependence on the fundamental dimensions: Mass $[M]$, Length $[L]$, Time $[T]$, Electric Current $[A]$, and Thermodynamic Temperature $[K]$.

Dimensional Formulas Table

The Principle of Homogeneity of Dimensions

The Principle of Homogeneity states that an equation is dimensionally correct if and only if every additive term on both sides possesses identical fundamental dimensions:

$[\text{LHS}] = [\text{RHS}]$

Verification Case 1: Simple Pendulum Formula

Verify $T = 2\pi\sqrt{\frac{l}{g}}$:

• Dimensions of LHS: $[T]$
• Dimensions of RHS: $[2\pi]$ is dimensionless ($[M^0 L^0 T^0]$). Thus:

$[\text{RHS}] = \left[\frac{l}{g}\right]^{1/2} = \left[\frac{L}{L T^{-2}}\right]^{1/2} = [T^2]^{1/2} = [T]$

Since $[\text{LHS}] = [\text{RHS}] = [T]$, the formula is dimensionally consistent.

Verification Case 2: Speed of Sound in a Medium

Verify $v = \sqrt{\frac{E}{\rho}}$, where $E$ is modulus of elasticity ($[M L^{-1} T^{-2}]$) and $\rho$ is density ($[M L^{-3}]$):

$[\text{RHS}] = \left[\frac{M L^{-1} T^{-2}}{M L^{-3}}\right]^{1/2} = [L^{-1 - (-3)} T^{-2}]^{1/2} = [L^2 T^{-2}]^{1/2} = [L T^{-1}]$

Since $[v] = [L T^{-1}]$, $[\text{LHS}] = [\text{RHS}]$.

Deriving Empirical Formulas via Dimensional Analysis

To derive an empirical formula for a physical quantity $Q$ depending on variables $x, y, z$:

1. Express relation as $Q = k x^a y^b z^c$, where $k$ is a dimensionless constant.
2. Write dimensional equations for all quantities.
3. Equate exponents of $M, L, T$ on both sides to solve the system of linear equations for $a, b, c$.

Limitations of Dimensional Analysis

1. Dimensionless Constants: It provides no information about dimensionless proportionality factors (such as $2\pi, \frac{1}{2}, 6\pi$).
2. Transcendental Functions: It cannot analyze equations containing trigonometric functions ($\sin\theta, \cos\theta$), exponential terms ($e^{kx}$), or logarithmic expressions ($\ln x$), because their arguments must be strictly dimensionless.
3. Physical Distinction: It cannot distinguish between distinct physical quantities sharing the same dimensions (such as scalar Work and vector Torque, both possessing $[M L^2 T^{-2}]$).
4. Variable Constraint: If a mechanical quantity depends on more than three mutually independent variables, dimensional analysis cannot determine unique exponents using only $M, L, T$.