Short Notes: Units and Measurements
1.1 SI Units and Dimensions
Physical quantity is a property of a system that can be measured (for example, length, mass, time). Unit is a standard amount of a physical quantity used to express measurements of that quantity. The SI (Système International) is the internationally accepted system of units based on seven base quantities and their corresponding base units.
| Physical Quantity | SI Unit (Symbol) |
|---|---|
| Length | metre (m) |
| Mass | kilogram (kg) |
| Time | second (s) |
| Electric current | ampere (A) |
| Thermodynamic temperature | kelvin (K) |
| Amount of substance | mole (mol) |
| Luminous intensity | candela (cd) |
Derived quantities are expressed in terms of base quantities. The dimension of a physical quantity expresses its dependence on the base quantities and is written using symbols M (mass), L (length) and T (time) for mechanical quantities. A dimensional formula is written as [M^a L^b T^c], where a, b, c are integers (or zero).
1.2 Dimensional Formulas
The dimensional formula of a quantity shows how that quantity depends on the base quantities. Dimensional analysis is useful to check the correctness of equations, to derive relations up to a dimensionless constant, and to convert units.
| Physical Quantity | Dimensional Formula |
|---|---|
| Force | [M L T⁻²] |
| Work, Energy | [M L² T⁻²] |
| Power | [M L² T⁻³] |
| Pressure | [M L⁻¹ T⁻²] |
| Momentum | [M L T⁻¹] |
| Angular Momentum | [M L² T⁻¹] |
| Torque | [M L² T⁻²] |
| Frequency | [T⁻¹] |
| Velocity | [L T⁻¹] |
| Acceleration | [L T⁻²] |
Example - derive the dimensional formula of pressure using the relation pressure = force / area.
\[ \text{Force} = [M L T^{-2}] \]
\[ \text{Area} = [L^2] \]
\[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} = [M L T^{-2}] \div [L^2] \]
\[ \text{Pressure} = [M L^{-1} T^{-2}] \]
1.3 Significant Figures Rules
Significant figures express the precision of a measured quantity. They include all digits that are known reliably plus one final digit that is uncertain.
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros (zeros to the left of the first non-zero digit) are not significant; they only locate the decimal point.
- Trailing zeros after a decimal point are significant.
- For addition and subtraction: the result should be reported with the same number of decimal places as the least precise measurement (the measurement with the fewest decimal places).
- For multiplication and division: the result should have the same number of significant figures as the measurement with the fewest significant figures.
Examples:
- 0.004560 has four significant figures: 4, 5, 6 and the trailing zero after the decimal point.
- 1200 (without a decimal point) has two significant figures if written as 1.2 × 10³ or may be ambiguous; use scientific notation to indicate significant figures clearly.
1.4 Error Analysis
Error is the difference between a measured value and the true (or accepted) value. Errors quantify the uncertainty in measurements.
| Error Type | Formula |
|---|---|
| Absolute error | Δa = a_measured - a_true |
| Mean absolute error | Δa_mean = (|Δa₁| + |Δa₂| + ... + |Δa_n|) / n |
| Relative error | Δa / a_mean |
| Percentage error | (Δa / a_mean) × 100% |
Example - a length is measured as 12.03 m while the true length is 12.00 m. Find the absolute and percentage error.
Sol.
\[ \text{Absolute error } \Delta a = 12.03 - 12.00 = 0.03 \text{ m} \]
\[ \text{Relative error} = \frac{0.03}{12.00} = 0.0025 \]
\[ \text{Percentage error} = 0.0025 \times 100\% = 0.25\% \]
1.5 Error Propagation
When a quantity depends on measured quantities, its uncertainty depends on the uncertainties of those measurements. If measured quantities are independent and errors are small, relative (fractional) errors add according to the following rules (keeping only first-order terms):
- Sum/Difference: For \(A = B \pm C\), \(\Delta A = \Delta B + \Delta C\).
- Product: For \(A = B \times C\), \(\dfrac{\Delta A}{A} = \dfrac{\Delta B}{B} + \dfrac{\Delta C}{C}\).
- Quotient: For \(A = \dfrac{B}{C}\), \(\dfrac{\Delta A}{A} = \dfrac{\Delta B}{B} + \dfrac{\Delta C}{C}\).
- Power: For \(A = B^{n}\), \(\dfrac{\Delta A}{A} = |n|\,\dfrac{\Delta B}{B}\).
Example - find the fractional and percentage error in area \(A = l \times b\) when \(l = 2.00 \pm 0.01\ \text{m}\) and \(b = 3.00 \pm 0.02\ \text{m}.\)
Sol.
\[ A = l \times b \]
\[ \text{Fractional error in } A = \frac{\Delta A}{A} = \frac{\Delta l}{l} + \frac{\Delta b}{b} \]
\[ \frac{\Delta l}{l} = \frac{0.01}{2.00} = 0.005 \]
\[ \frac{\Delta b}{b} = \frac{0.02}{3.00} \approx 0.0066667 \]
\[ \frac{\Delta A}{A} = 0.005 + 0.0066667 = 0.0116667 \]
\[ \text{Percentage error} = 0.0116667 \times 100\% \approx 1.167\% \]
\[ \text{So } \Delta A \approx 0.011667 \times A \text{ (fractional uncertainty) } \]
Practical notes: use scientific notation to indicate significant figures clearly; always state both the measured value and its uncertainty (for example, \( (2.00 \pm 0.01)\ \text{m} \)). Dimensional analysis cannot determine dimensionless constants (like 2π) and cannot distinguish additive constants; it is a consistency check and a tool for deriving scaling relations.
Summary: Know the seven SI base units and their symbols; be able to write dimensional formulas and use dimensional analysis to check equations; apply significant-figure rules for reporting results; compute absolute, relative and percentage errors; and use error-propagation rules to estimate uncertainties in derived quantities.
FAQs on Short Notes: Units and Measurements
| 1. What are the fundamental units of measurement in the International System of Units (SI)? |  |
| 2. Why is it essential to use standard units in scientific measurements? | |
| 3. How do you convert units from one system to another, such as from centimetres to metres? | |
| 4. What is the difference between accuracy and precision in measurements? | |
| 5. What role do significant figures play in measurements? | |
