# Dimensions of a Physical Quantity

1. In all the systems of units, the derived units of all the physical quantities can be expressed in terms of the fundamental units of mass, length and time, raised to some power. The powers to witch fundamentals units must be raised in order to express a physical quantity, are called its dimensions. To make it clear, consider the physical quantity density which is defined as mass per unit volume.

Hence, *density=mass/volume=M/L ^{3}=[M1L^{3}T^{0}]*

2. a) Above expression shows that dimensions of density are 1 in mass, -3 in lengths and 0 in time.

c) Dimensional formula for density is [ML^{-3}] or [ Ml^{-3}T^{0}].

d) Dimensional equation for density is [d] =[ML^{-3}T^{0}].

### Dimensional formula of some physical quantities in mechanics and their units in different systems

Physical Quantity | Formula | Dimensional Formulae | S.I Unit |

Area | Length × breadth | [L^{2}] |
m^{2} |

Volume | Length × breadth × height | [L^{3}] |
m^{3} |

Density | Mass/vol. | [ML^{-3}] |
Kg/m^{3} |

Speed or velocity | Distance/time | [LT^{-1}] |
m/s |

Acceleration | Velocity/time | [LT^{-2}] |
m/s^{2} |

Force | Mass × acc | [MLT^{-2}] |
Newton |

Work | Force × distance(FsCosθ) | [ML2T^{-2}] |
Kg.m^{2}/s^{2} or jule |

Energy(Mechanical heat, light etc) | 1/2mv^{2} or mc2 |
[ML^{2}T^{-2}] |
N ×m |

Power | dW/dt | [ML^{2}T^{-3}] |
J/s or watt |

Momentum or Impulse | mV | [MLT^{-1}] |
Kg-m/sec |

Momentum of force or Torque | Force × distance | [ML^{2}T^{-2}] |
N ×m |

Angle or solid angle | Arc/radius | [M^{0}L^{0}T^{0}] |
Radians |

Stress | Force/Area | [ML^{-1}T^{-2}] |
N/m^{2} |

Strain | Change in dimensions/original dimensions | [M^{0}L^{0}T^{0}] |
No unit |

Coefficient of Elasticity | Stress/Strain | [ML^{-1}T^{-2}] |
N/m^{2} |

Surface Tension | Energy/length | [ML^{0}T^{-2}] |
N/m |

Surface Energy | Energy/Area | [ML^{0}T^{-2}] |
J/m^{2} |

Velocity gradient | Velocity/distance | [ML^{0}T^{-1}] |
s^{-1} |

Coefficient of viscosity | Force/Area × velocity gradient | [ML^{-1}T^{-1}] |
Kgm/m/sec |

Radius of gyration | Distance | [M^{0}LT^{0}] |
m |

Moment of Inertia | Mass ×(Radius of gyration)^{2} |
[ML^{2}] |
Kg × m^{2} |

Angular velocity | Angle/Time | [T^{-1}] |
Radian/sec |

Frequency | 1/T | [T^{-1}]] |
Cps or Hertz |

Angular acceleration | Angular velocity/time | [M^{0}L^{0}T^{-2}] |
Radian/sec^{2} |

Angular momentum | Moment of inertia × velocity (Iω) | [ML^{2}T^{-1}] |
Kgm^{2}s^{-1} |

Pressure | Force/Area | [ML^{-1}T^{-2}] |
N/m^{2} |

** If anyone ask you:**

**What do you mean** by** the dimension of a physical quantity?**

You can simply tell him:

The *dimensions of a physical quantity* are the powers to which the fundamental units are raised to represent a physical quantity.