# Units and Dimensions

Every measurement has two parts. The first is a **number (n)** and the next is a **unit (u)**. Q = nu. For example, Length of an object = 35 cm. The number expressing the magnitude of a physical quantity is inversely proportional to the unit selected.

If n_{1} and n_{2} are the numerical values of a physical quantity corresponding to the units u_{1} and u_{2}, then n_{1}u_{1} = n_{2}u_{2}. For example, 2.8 m = 280 cm; 6.2 kg = 6200 g

The quantities that are independent of other quantities are called **fundamental quantities**. The units that are used to measure these fundamental quantities are called **fundamental units**.

There are four systems of units:

- C.G.S - Centimetre, Gram, Second
- M.K.S - Metre, Kilogram, Second
- F.P.S - Foot, Pound, Second
- SI

The quantities that are derived using the fundamental quantities are called **derived quantities**. The units that are used to measure these derived quantities are called **derived units**.

**Fundamental Physical Quantities in SI System**

Physical Quantity |
Unit |
Symbol |

Length | metre | m |

Mass | kilogram | kg |

Time | second | s |

Electric current | ampere | A |

Thermodynamic temperature | kelvin | K |

Intensity of light | candela | cd |

Quantity of substance | mole | mol |

**Supplementary Quantities in SI System**

Quantity |
Unit |
Symbol |

Plane angle | radian | rad |

Solid angle | steradian | sr |

**Metre:** A metre is equal to 1650763.73 times the wavelength of the light emitted in vacuum due to electronic transition from 2p^{10} state to 5d^{5} state in Krypton-86. But in 1983, 17th General Assembly of weights and measures, adopted a new definition for the metre in terms of velocity of light. According to this definition, metre is defined as the distance travelled by light in vacuum during a time interval of 1/299, 792, 458 of a second.

**Kilogram:** The mass of a cylinder of platinum-iridium alloy kept in the International Bureau of weights and measures preserved at Serves near Paris is called one kilogram.

**Second:** The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of caesium-133 atom is called one second.

**Ampere:** The current which when flowing in each of two parallel conductors of infinite length and negligible cross-section and placed one metre apart in vacuum, causes each conductor to experience a force of 2x10^{-7} newton per metre of length is known as one ampere.

**Kelvin:** The fraction of 1/273.16 of the thermodynamic temperature of the triple point of water is called kelvin.

**Candela:** The luminous intensity in the perpendicular direction of a surface of a black body of area 1/600000 m^{2} at the temperature of solidifying platinum under a pressure of 101325 Nm^{-2} is known as one candela.

**Mole:** The amount of a substance of a system which contains as many elementary entities as there are atoms in 12x10^{3} kg of carbon-12 is known as one mole.

**Radian:** The angle made by an arc of the circle equivalent to its radius at the centre is known as radian. 1 radian = 57°17'45''.

**Steradian:** The angle subtended at the centre by one square metre area of the surface of a sphere of radius one metre is known as steradian.

**Dimensions** of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity. The expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is called the **dimensional formula** of that quantity.

If Q is the unit of a derived quantity represented by Q = M^{a}L^{b}T^{c}, then it is called dimensional formula and the exponents a, b and c are called the dimensions.

**Dimensional Constants:** The physical quantities which have dimensions and have a fixed value are called dimensional constants. For example, Gravitational constant (G), Planck’s constant (h), Universal gas constant (R), Velocity of light in vacuum (C).

**Dimensionless quantities** are those which do not have dimensions but have a fixed value.