## Introduction

A scientific study involves the systematic and organized study of a relationship between variables, generally two.

For example, a physicist wants to learn the relationship between the depth of water from the surface and liquid pressure; a biologist wants to find out the relationship between calorie intake and sleepiness in human beings; and a chemist wants to investigate the relationship between the surface area of reactants and the rate of a chemical reaction.

Evidently, we study the cause-and-effect relationship between two variables in real life. In the above examples, the depth of water, calorie intake, and the surface area of reactants are the causes; whereas liquid pressure, sleepiness in human beings, and the rate of a chemical reaction are their respective effects.

However, each scientific study comes with several constants too.

## Variables of Scientific Research

## a) Independent variable

An independent variable is a physical quantity that changes during the study and brings some changes in another physical quantity.

## b) Dependent variable

A dependent variable is a physical quantity that changes during the study and its changes depend upon the independent variable.

## c) Controlled variable

A controlled variable is a physical quantity that does not change during the study and has little to no effect on the study's observation.

## Types of units

### Fundamental units

Units of the fundamental physical quantities are called fundamental units. These are independent of other physical units and have their own identity. There are only seven fundamental units in the SI system of measurement.

Fundamental Quantity | Fundamental Unit | Symbol |
---|---|---|

Mass | Kilogram | kg |

Length | Metre | m |

Time | Second | s |

Temperature | Kelvin | K |

Current | Ampere | A |

Luminous Intensity | Candela | cd |

Amount of substance | Mole | mol |

Table: Physical quantity, Fundamental unit, and Symbol

### Derived units

Units of the derived physical quantities are called derived units. These are dependent on other physical units and do not have their own identity. There are hundreds and thousands of derived units in the SI system of measurement.

Derived Quantity | Derived Unit | Symbol | Fundamental Units |
---|---|---|---|

Area | squared meters | \( \rm m^{2} \) | m |

Volume | cubic meters | \( \rm m^{3} \) | m |

Density | kilogram-per-cubic-meters | \( \rm kgm^{-3} \) | kg, m |

Velocity | meter-per-second | \( \rm ms^{-1} \) | m, s |

Momentum | kilogram-metre-per-second | \( \rm kgms^{-1} \) | kg, m, s |

Force | Newton | \( \rm kgms^{-2} \) | kg, m, s |

Pressure | Pascal | \( \rm kgm^{-1}s^{-2} \) | kg, m, s |

Table: Physical quantity, Derived unit, Symbol, Fundamental unit symbol

## Analysis of unit-wise equation

In physics or dimensional analysis, physicists validate the truthfulness of a physical equation by comparing the units of each term involved in it. This process is known as the analysis of unit-wise equations.

For example, one can tell that the following physical equation is false by just looking at the units of the individual terms involved:

\[ \rm v^{2} = u^{2} + 2a \]

In this equation, we are trying to add the square of velocities, represented by v and u, whose units are \( \rm m^{2}s^{-2} \) with a multiple of acceleration whose unit is \( \rm ms^{-2} \). Clearly, their units are not the same. Hence, the physical equation is not true.

Analysis of unit-wise equation serves as a useful tool to check if a physical equation is true just by looking at the units of the terms involved in the equation, as explained above. However, this process cannot verify if the physical equation is true in the real world.

For example, one can argue that the following physical equation is true by stating the correctness of the analysis of unit-wise equations:

\[ \rm v^{2} = u^{2} + as \]

However, the physical equation produces inaccurate results when applied to the real world. It is missing an important coefficient in the final term. To improve the accuracy of the above equation, one needs to add 2 as the coefficient of the final term. When we rewrite the equation, we get

\[ \rm v^{2} = u^{2} + 2as \]

The analysis of the unit-wise equation validates the above physical relationship. Additionally, experiments conducted in the laboratory also validate the accuracy of the above physical equation.

**When does a physical relationship become a physical equation?**

When a physical relationship maintains accuracy in the analysis of unit-wise equations and laboratory experiments, it becomes eligible for being a physical equation.

**When can we add or subtract two or more physical quantities?**

When each physical quantities have the same units, we can add or subtract them. For example, we can add or subtract velocity with the product of acceleration and time because they have the same units, i.e., m/s. However, we cannot add or subtract velocity with distance as the two do not have the same units, i.e., the unit of velocity is m/s and the unit of distance is m.

**When can we multiply or divide two or more physical quantities?**

There is no limitation when we multiply or divide two or more physical quantities. We can multiply or divide two or more physical quantities that have or do not have the same units.