Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit . The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit.
Physical quantities are broadly classified into two categories:
A complete set of these units, both the base units and derived units, is known as the system of units .
Historically, three main systems of units were extensively used in different countries:
The system of units which is at present internationally accepted for measurement is the Système International d’Unités (French for International System of Units), abbreviated as SI . It was developed by the Bureau International des Poids et Mesures (BIPM) in 1971 and recently revised by the General Conference on Weights and Measures in November 2018.
The SI is founded on seven base quantities, defined as follows:
| Base Quantity | Unit Name | Symbol | Definition and Constants |
|---|---|---|---|
| Length | metre | m | Defined by taking the fixed numerical value of the speed of light in vacuum $c$ to be 299792458 when expressed in the unit m s⁻¹, where the second is defined in terms of the caesium frequency $\Delta \nu_{Cs}$. |
| Mass | kilogram | kg | Defined by taking the fixed numerical value of the Planck constant $h$ to be 6.62607015 $\times$ 10⁻³⁴ when expressed in the unit J s, which is equal to kg m² s⁻¹, where the metre and second are defined in terms of $c$ and $\Delta \nu_{Cs}$. |
| Time | second | s | Defined by taking the fixed numerical value of the caesium frequency $\Delta \nu_{Cs}$, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s⁻¹. |
| Electric Current | ampere | A | Defined by taking the fixed numerical value of the elementary charge $e$ to be 1.602176634 $\times$ 10⁻¹⁹ when expressed in the unit C, which is equal to A s, where the second is defined in terms of $\Delta \nu_{Cs}$. |
| Thermodynamic Temperature | kelvin | K | Defined by taking the fixed numerical value of the Boltzmann constant $k$ to be 1.380649 $\times$ 10⁻²³ when expressed in the unit J K⁻¹, which is equal to kg m² s⁻² K⁻¹, where the kilogram, metre and second are defined in terms of $h$, $c$ and $\Delta \nu_{Cs}$. |
| Amount of Substance | mole | mol | One mole contains exactly 6.02214076 $\times$ 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, $N_A$, when expressed in the unit mol⁻¹ and is called the Avogadro number. |
| Luminous Intensity | candela | cd | Defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 $\times$ 10¹² Hz, $K_{cd}$, to be 683 when expressed in the unit lm W⁻¹, which is equal to cd sr W⁻¹, or cd sr kg⁻¹ m⁻² s³, where the kilogram, metre and second are defined in terms of $h$, $c$ and $\Delta \nu_{Cs}$. |
Besides the seven base units, there are two supplementary units defined for dimensionless geometric quantities:
Several non-SI units are commonly used and retained for general use:
| Name | Symbol | Value in SI Unit |
|---|---|---|
| minute | min | 60 s |
| hour | h | 3600 s |
| day | d | 86400 s |
| year | y | 3.156 $\times$ 10⁷ s |
| degree | ° | 1° = $\pi / 180$ rad |
| litre | L | 1 dm³ = 10⁻³ m³ |
| tonne | t | 10³ kg |
| carat | c | 200 mg |
| bar | bar | 0.1 MPa = 10⁵ Pa |
| curie | Ci | 3.7 $\times$ 10¹⁰ s⁻¹ |
| roentgen | R | 2.58 $\times$ 10⁻⁴ C/kg |
| quintal | q | 100 kg |
| barn | b | 100 fm² = 10⁻²⁸ m² |
| are | a | 1 dam² = 10² m² |
| hectare | ha | 1 hm² = 10⁴ m² |
| standard atmospheric pressure | atm | 101325 Pa = 1.013 $\times$ 10⁵ Pa |
Every measurement involves errors. The reported result of a measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. These reliable digits plus the first uncertain digit are known as significant figures or significant digits.
Mathematical Problem Statement: Find the density of an object of mass 4.237 g and volume 2.51 cm³ to appropriate significant figures.
Step-by-Step Algorithmic Solution Trace:
- Identify the significant figures in the given values: Mass = 4.237 g (4 significant figures), Volume = 2.51 cm³ (3 significant figures).
- The least number of significant figures is 3 (from the volume).
- Calculate the raw density: $$\text{Density} = \frac{4.237\text{ g}}{2.51\text{ cm}^3} \approx 1.6880478\text{ g cm}^{-3}$$
- Round off the result to 3 significant figures.
Final Evaluation Output: $1.69\text{ g cm}^{-3}$
Mathematical Problem Statement: Sum the numbers 436.32 g, 227.2 g, and 0.301 g.
Step-by-Step Algorithmic Solution Trace:
- Identify the decimal places: 436.32 g (2 decimal places), 227.2 g (1 decimal place), 0.301 g (3 decimal places).
- The least precise measurement has 1 decimal place (227.2 g).
- Calculate the raw sum: $$436.32\text{ g} + 227.2\text{ g} + 0.301\text{ g} = 663.821\text{ g}$$
- Round off the sum to 1 decimal place.
Final Evaluation Output: $663.8\text{ g}$
The rules for rounding off numbers to appropriate significant figures are:
When combining measurements with uncertainties, the following principles apply:
Mathematical Problem Statement: Each side of a cube is measured to be 7.203 m. Find the total surface area and the volume of the cube to appropriate significant figures.
Step-by-Step Algorithmic Solution Trace:
- The measured length has 4 significant figures (7.203 m).
- Calculate the surface area: $$\text{Surface Area} = 6 \times (7.203\text{ m})^2 = 6 \times 51.883209\text{ m}^2 = 311.299254\text{ m}^2$$
- Round off the surface area to 4 significant figures: 311.3 m².
- Calculate the volume: $$\text{Volume} = (7.203\text{ m})^3 = 373.714754\text{ m}^3$$
- Round off the volume to 4 significant figures: 373.7 m³.
Final Evaluation Output: Surface Area = $311.3\text{ m}^2$, Volume = $373.7\text{ m}^3$
The nature of a physical quantity is described by its dimensions . All physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental or base quantities, denoted with square brackets $[ ]$:
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
For example, the dimensions of volume are $[L^3]$, which can be written as $[M^0 L^3 T^0]$. The dimensions of force are $[M L T^{-2}]$, meaning force has 1 dimension in mass, 1 dimension in length, and -2 dimensions in time.
The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity.
An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.
| Physical Quantity | Dimensional Formula | Dimensional Equation |
|---|---|---|
| Volume ($V$) | $[M^0 L^3 T^0]$ | $[V] = [M^0 L^3 T^0]$ |
| Speed or Velocity ($v$) | $[M^0 L T^{-1}]$ | $[v] = [M^0 L T^{-1}]$ |
| Acceleration ($a$) | $[M^0 L T^{-2}]$ | $[a] = [M^0 L T^{-2}]$ |
| Force ($F$) | $[M L T^{-2}]$ | $[F] = [M L T^{-2}]$ |
| Mass Density (${\rho}$) | $[M L^{-3} T^0]$ | $[\rho] = [M L^{-3} T^0]$ |
Dimensional analysis is a powerful tool used to check the correctness of mathematical expressions and to deduce relations among physical quantities. It is based on the principle of homogeneity of dimensions , which states that physical quantities can be added or subtracted only if they have the same dimensions.
If the dimensions of all terms on both sides of an equation are not the same, the equation is wrong. However, dimensional consistency does not guarantee that an equation is physically correct, as it cannot account for dimensionless constants or functions (like trigonometric, logarithmic, or exponential functions, which must be dimensionless).
Mathematical Problem Statement: Check the dimensional consistency of the equation: $$x = x_0 + v_0 t + \frac{1}{2} a t^2$$ where $x$ and $x_0$ are distances, $v_0$ is velocity, $a$ is acceleration, and $t$ is time.
Step-by-Step Algorithmic Solution Trace:
- Write the dimensions of each term:
- $[x] = [L]$
- $[x_0] = [L]$
- $[v_0 t] = [L T^{-1}] [T] = [L]$
- $[\frac{1}{2} a t^2] = [L T^{-2}] [T^2] = [L]$
- Since each term on both sides of the equation has the same dimension of length $[L]$, the equation is dimensionally correct.
Final Evaluation Output: The equation is dimensionally consistent.
By knowing the dependence of a physical quantity on other quantities, we can derive its mathematical relation using the method of dimensions (up to three independent variables).
Mathematical Problem Statement: Derive the expression for the time period $T$ of a simple pendulum, assuming it depends on the length of the string $l$, mass of the bob $m$, and acceleration due to gravity $g$.
Step-by-Step Algorithmic Solution Trace:
- Express the dependence as a product: $$T = k l^x g^y m^z$$ where $k$ is a dimensionless constant.
- Substitute the dimensions of each quantity: $$[M^0 L^0 T^1] = [L]^x [L T^{-2}]^y [M]^z = [M^z L^{x+y} T^{-2y}]$$
- Equate the exponents of $[M]$, $[L]$, and $[T]$ on both sides:
- For $[M]$: $z = 0$
- For $[T]$: $-2y = 1 \implies y = -\frac{1}{2}$
- For $[L]$: $x + y = 0 \implies x = -y = \frac{1}{2}$
- Substitute the values of $x$, $y$, and $z$ back into the equation: $$T = k l^{1/2} g^{-1/2} m^0 = k \sqrt{\frac{l}{g}}$$
- Using experimental analysis, the constant $k$ is found to be $2\pi$.
Final Evaluation Output: $T = 2\pi \sqrt{\frac{l}{g}}$