Subject: Physics

Dimensions of a physical quantity are the powers to which fundamental quantities are to be raised to represent the quantity. The basic quantities with their symbols in square brackets are as follows:

$$[Length]=[L]$$

$$[Mass]=[M]$$

$$[Time]=[T]$$

$$[Temperature]=[K]or[\Theta]$$

$$[Current]=[A]0r[I]$$

$$[No.of Moles]=[N]$$

**Velocity**$$ V = \frac{displacement}{time} $$

$$ V = \frac{[L]}{[T]} $$

$$=[M^0L^1T^{-1}]$$

Dimensions of velocity are 0 in mass, 1 in length and -1 in time i.e. (0, 1, -1)

**Acceleration**$$ V = \frac{\text{change in velocity}}{\text{time taken}}$$

$$ V = \frac{displacement} {time \times time} $$

$$ V = \frac{[L^1]}{[T^2]} $$

$$=[M^0L^1T^{-2}]$$

Dimensions of acceleration are (0, 1, -2).

**Force**$$F = mass \times acceleration$$

$$ =mass \times \frac{\text{change in velocity}}{\text{time taken}}$$

$$ = mass\times \frac{displacement} {time \times time} $$

$$ =[M^1] \frac{[L^1]}{[T^2]} $$

$$=[M^1L^1T^{-2}]$$

Dimensions of force are (1, 1, -2).

It is the expression which shows how and which fundamental quantities are used in the representation of a physical quantity.

1) Velocity [M^{0} L^{1} T^{-1}]

2) Acceleration [M^{0} L^{1} T^{-2}]

3) Force [M^{1} L^{1} T^{-2}]

4) Energy [M^{1} L^{2} T^{-2}]

5) Power [M^{1} L^{2} T^{-3}]

6) Momentum [M^{1} L^{1} T^{-1}]

7) Pressure [M^{1} L^{-1} T^{-2}]

It is the equation obtained by equating a physical quantity with its dimensional formula.

1) Velocity [V] = [M^{0} L^{1} T^{-1}]

2) Acceleration[a] = [M^{0} L^{1} T^{-2}]

3) Force [F] = [M^{1} L^{1} T^{-2}]

4) Energy [E] = [M^{1} L^{2} T^{-2}]

5) Power [P] = [M^{1} L^{2} T^{-3}]

6) Momentum [P] = [M^{1} L^{1} T^{-1}]

7) Pressure [P] = [M^{1} L^{-1} T^{-2}]

**Dimensional Formulas of Some Physical Quantities**

S.N | Physical quantity | Relation with other physical quantities | Dimensional formula | SI-unit |

1. | Volume | length× breadth× height | [L] ×[L] ×[L]= [M | m |

2. | Velocity or speed | \(\frac{distance}{time}\) | = [M | ms |

3. | Momentum | mass × velocity | [M] × [LT | kgms |

4. | Force | mass × acceleration | [M] × [LT | N (newton) |

5. | Pressure | \(\frac{force}{area}\) | =[ML | Nm |

6. | Work | force × distance | [MLT | J (joule) |

7. | Energy | Work | [ML | J (joule) |

8. | Power | \(\frac{work}{time}\) | =[ML | W (watt) |

9. | Gravitational constant | \(\frac{force \times (distance)^2}{(mass)^2}\) | [M | Nm |

10. | Angle | \(\frac{arc}{radius}\) | Dimensionless | rad |

11. | Moment of inertia | mass × (distance) | [ML | Kgm |

12. | Angular momentum | moment of inertia × angular velocity | [ML | Kgm |

13. | Torque or couple | force × perpendicular distance | [MLT | Nm |

14. | Coefficient of viscosity | \(\frac{force}{\text {area} \times \text {velocity gradient}}\) | [ML | Dap (Dacapoise) |

15. | Frequency | \(\frac{1}{second}\) | [T | Hz |

It states that “The dimensions of fundamental quantities on a left-hand side of an equation must be equal to the dimensions of the fundamental quantities on the right-hand side of that equation.”

Physical quantities can be categorized into four types. They are:

**Dimensional variables**Those physical quantities which have dimensions but do not have fixed value are called dimensional variables. Examples: force, work, power, velocity etc.**Dimensionless variables**

Those physical quantities which have neither dimensions nor fixed value are called dimensionless variables.**Dimensional constant**Those physical quantities which possess dimensions and fixed value are called dimensional constant. Their examples are gravitational constant, velocity of light etc.**Dimensionless constant**Those physical quantities which do not possess dimensions but possess fixed value are called dimensionless constant. Examples are pi π, counting number etc.

- Principle of homogeneity states that “The dimensions of fundamental quantities on a left-hand side of an equation must be equal to the dimensions of the fundamental quantities on the right-hand side of that equation.”
- Dimensions of a physical quantity are the powers to which fundamental quantities are to be raised to represent the quantity.
- Dimensional formula is the expression which shows how and which fundamental quantities are used in the representation of a physical quantity.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

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