Units, Dimensions and Measurements

A physical quantity is any property of matter or phenomenon that can be measured. Each measurement is expressed as the product of a numerical value and a unit: $Q = n \cdot u$. If $n_1 u_1 = n_2 u_2$ for the same physical quantity expressed in two unit systems, then $n_2 = n_1 \cdot (u_1 / u_2)$. This proportionality is the basis of all unit conversion.

The International System of Units (SI) adopted in 1960 is built on seven fundamental (base) quantities along with their base units:

Two supplementary units are also defined: the radian (rad) for plane angle and the steradian (sr) for solid angle. A radian is the angle subtended at the centre by an arc of length equal to the radius ($1 \text{ rad} \approx 57.2958^\circ$). A steradian is the solid angle subtended at the centre of a sphere by a surface area equal to the square of its radius; the total solid angle around a point is $4\pi$ sr.

All other quantities are derived quantities, obtained from the fundamental ones via mathematical operations. For example:

• Velocity: $v = \dfrac{\text{displacement}}{\text{time}}$, SI unit: $\text{m/s}$.
• Force: $F = ma$, SI unit: $\text{kg·m/s}^2 = $ newton (N).
• Energy: $E = Fd$, SI unit: $\text{kg·m}^2/\text{s}^2 = $ joule (J).
• Pressure: $P = F/A$, SI unit: $\text{N/m}^2 = $ pascal (Pa).
• Power: $P = E/t$, SI unit: $\text{J/s} = $ watt (W).

Coherent and non-coherent units: A unit is coherent within a system if it is derived using a product/quotient of base units with no extra numerical factor. The SI is a coherent system. Non-SI units (litre, electron-volt, atomic mass unit, light-year) are tolerated for convenience:

• $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$
• $1 \text{ u} = 1.661 \times 10^{-27} \text{ kg}$
• $1 \text{ ly} = 9.461 \times 10^{15} \text{ m}$
• $1 \text{ parsec} = 3.086 \times 10^{16} \text{ m}$
• $1 \text{ atm} = 1.013 \times 10^{5} \text{ Pa}$

SI Prefixes scale units by powers of 10:

The dimension of a physical quantity expresses how it depends on the seven base quantities. We write dimensions using bracketed base symbols: $$[M], [L], [T], [A], [K], [\text{mol}], [\text{cd}]$$

The dimensional formula of a derived quantity is the algebraic combination raised to appropriate exponents. For example:

• Area: $[L^2]$
• Volume: $[L^3]$
• Velocity: $[LT^{-1}]$
• Acceleration: $[LT^{-2}]$
• Force: $[MLT^{-2}]$
• Work/Energy: $[ML^2T^{-2}]$
• Power: $[ML^2T^{-3}]$
• Pressure: $[ML^{-1}T^{-2}]$
• Momentum: $[MLT^{-1}]$
• Angular momentum: $[ML^2T^{-1}]$
• Frequency: $[T^{-1}]$
• Charge: $[AT]$
• Electric field: $[MLT^{-3}A^{-1}]$
• Potential: $[ML^2T^{-3}A^{-1}]$
• Resistance: $[ML^2T^{-3}A^{-2}]$
• Magnetic field: $[MT^{-2}A^{-1}]$
• Planck's constant: $[ML^2T^{-1}]$
• Gravitational constant $G$: $[M^{-1}L^3T^{-2}]$

Principle of Dimensional Homogeneity: In any physically meaningful equation, every additive term must have the same dimensions. If $A = B + C$, then $[A] = [B] = [C]$. Arguments of transcendental functions ($\sin, \cos, e^x, \ln x$) must be dimensionless.

Three Main Uses of Dimensional Analysis:

1. Checking correctness of equations — every term must have identical dimensions.
2. Conversion of units between systems using $n_2 = n_1 \cdot (u_1/u_2)$.
3. Deriving relationships between quantities (up to a dimensionless constant).

Limitations:

• Cannot determine dimensionless multiplicative constants (e.g., the $\frac{1}{2}$ in $\frac{1}{2}mv^2$).
• Fails when a quantity depends on more than three other quantities (insufficient equations).
• Cannot decide between $\sin\theta$ and $\cos\theta$ since both are dimensionless.
• Cannot handle equations involving addition of quantities of the same dimension (e.g., $s = ut + \frac{1}{2}at^2$ has two terms of the same dimension — coefficients can't be found).

Buckingham π-theorem (advanced): If a physical quantity depends on $n$ variables with $k$ fundamental dimensions, then there are $n - k$ independent dimensionless groups that govern the relationship.

Significant Figures (SF) in a measurement are the digits known with certainty plus the first uncertain digit. They reflect the precision of the measuring instrument.

Rules for counting SF:

1. All non-zero digits are significant. (`23.45` has 4 SF)
2. Zeros between non-zero digits are significant. (`2003` has 4 SF)
3. Leading zeros are not significant. (`0.0023` has 2 SF)
4. Trailing zeros in a number with a decimal point are significant. (`2.300` has 4 SF)
5. Trailing zeros in a whole number without a decimal point are ambiguous; scientific notation removes the ambiguity. (`300` → `3 × 10²` is 1 SF; `3.00 × 10²` is 3 SF)

Rules for arithmetic operations:

• Addition/subtraction: result has the same number of decimal places as the term with the fewest decimal places. ($2.34 + 1.2 = 3.5$)
• Multiplication/division: result has the same number of significant figures as the term with the fewest SF. ($2.0 \times 3.14 = 6.3$)

Rounding off: If the digit to be dropped is < 5, round down; if > 5, round up; if exactly 5, round to make the last retained digit even (banker's rounding).

Types of errors:

• Systematic errors — consistent biases (zero error, calibration error, environmental factors). Can be eliminated by correction.
• Random errors — unpredictable fluctuations. Reduced by repeated measurements and averaging.
• Gross errors — careless mistakes (misreading, calculation error). Avoidable.

Quantitative error definitions: For $n$ measurements $a_1, a_2, \ldots, a_n$ of a quantity:

• Mean (best estimate): $\bar{a} = \dfrac{1}{n}\sum_{i=1}^{n} a_i$
• Absolute error in the $i$-th measurement: $\Delta a_i = |a_i - \bar{a}|$
• Mean absolute error: $\overline{\Delta a} = \dfrac{1}{n}\sum |\Delta a_i|$
• Relative error: $\dfrac{\overline{\Delta a}}{\bar{a}}$
• Percentage error: $\dfrac{\overline{\Delta a}}{\bar{a}} \times 100\%$

Propagation of errors (for small errors $\Delta x \ll x$):

The absolute errors add in subtraction, which is why subtracting nearly equal numbers amplifies relative error — a key practical concern.

Vernier Calipers measure lengths up to a precision of typically $0.1$ mm or $0.05$ mm using a secondary (vernier) scale that slides along a primary (main) scale.

Vernier Constant (Least Count): $$\text{LC} = \text{(value of 1 MSD)} - \text{(value of 1 VSD)}$$ where MSD = main scale division and VSD = vernier scale division. If $n$ VSD coincide with $(n-1)$ MSD, then: $$\text{LC} = \frac{1 \text{ MSD}}{n}$$

For a typical vernier with $10$ VSD = $9$ MSD and $1$ MSD = $1$ mm: $\text{LC} = 0.1$ mm.

Reading: Total reading = (Main scale reading) + (vernier division coinciding) × (LC) ± (zero error).

Zero error occurs when the vernier shows a non-zero reading with the jaws fully closed:

• Positive zero error: zero of vernier is to the right of main scale zero → subtract from observed reading.
• Negative zero error: zero of vernier is to the left of main scale zero → add to observed reading.

Screw Gauge (Micrometer) measures very small lengths (typically up to $0.001$ cm = $0.01$ mm precision) using the principle that the rotation of a screw produces a linear translation equal to the pitch per revolution.

Pitch: linear distance moved by the spindle in one complete rotation = (distance moved) / (number of rotations).

Least Count: $$\text{LC} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}$$

For a typical screw gauge with pitch = $1$ mm and $100$ circular divisions: $\text{LC} = 0.01$ mm.

Reading: Total = (Linear/pitch scale reading) + (circular scale reading) × (LC) ± (zero error).

Backlash error: Caused by play between screw and nut; minimized by always rotating the screw in the same direction during measurement.

Travelling Microscope is used for measuring small distances with vernier-style precision in two dimensions; useful for measuring focal lengths, refractive indices, and small displacements.