**Deducing Relation Among The Physical Quantities: **Dimensional analysis can be used to infer the relationship between physical quantities. To do this, we must understand dependency and other related ideas using an illustration. Assume, for instance, that we wish to transfer a box from point A to point B. To accomplish this, we must exert some force on the box to get it to the desired location. The weight of the box (say M), the distance (say L) from point A to point B, and the amount of time (say T) needed to move the box will all contribute to calculating the amount of force (say F). Therefore, we can state that force is a function of mass, length, distance travelled, and time. This has the following numerical expression:

F=fM,L,T

We will now discover how to calculate the exact formula for doing that.

F = kM^{a}L^{b}T^{c}……………………Equation (1)

[MLT^{-2}] = [M]^{a}[L]^{b}[T]^{c} = [M^{a}L^{b}T^{c}]

Let’s take a = 1,

b = 1,

c = -2

And put it in equation (1)

F = kML/T^{2}

Table of Contents

## Concept Used

The concepts listed below are used to deduce the relationship between physical quantities:

- Measurement Units: Units are the numerical measures used to express the outcome of a physical quantity measurement.

- Physical Quantities: These are the quantities that characterise the laws governing the physical universe and can be measured with an instrument. Fundamental physical quantities are those that are independent of all other physical quantities. (As an example: mass, length, time, etc.)

- Physical quantity dimensions: The powers (or exponents, or indices) to which the basic units of physical quantities are increased are known as dimensions.

- Dimensional Analysis: Homogeneity Principle: Every term in an equation ought to have the same dimensions. As stated differently, we can add or subtract comparable physical amounts. Dimensional analysis refers to the above-mentioned assertion that addition and subtraction can be performed as needed to determine the dimensions of each term in an equation.

## Steps for Creating an Equation

The procedures to be taken to create an equation for determining the relationship between the physical quantities are as follows:

- Think about the powers of each physical quantity (such as a, b, c, and so forth).
- On both sides, write the dimensional formula.
- To determine the powers’ values, compare them.
- Enter the new power values in the equation above.
- To get the needed equation, simplify it.

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## Deducing Relation Among The Physical Quantities: Examples

**Example 1: **Finding the expression for a basic pendulum’s oscillation period.

Time T can be written quantitatively as the function of the bob’s mass (M), length (L), and gravity (g) as follows:

T = f(M,L,g)

T = kM^{x}L^{y}g^{z}……………….Equation (1)

[M^{0}L^{0}T^{1}] = [M^{1}L^{0}T^{0}]^{x}[M^{0}L^{1}T^{0}]^{y}[M^{0}L^{1}T^{-2}]^{z}

[M^{0}L^{0}T^{1}] = [M]^{x}[L]^{y=z}[T]^{-2z}

When simplifying, the values are:

0 = x,

0 = y + z,

1 = -2z

Therefore,

x = 0,

y = ½,

z = -½

Now, putting values in equation 1, we get,

T = kM^{0}L^{1/2}g^{-½}

This is the relationship between the length, mass, gravity, and period stated quantitatively.

**Example 2: Mass-Energy Equivalence**

It is possible to express energy (E) numerically as the function of mass (m) and speed of light (c), by:

E = f(m,c)

E = kM^{a}c^{b}………………Equation (1)

[ML^{2}T^{-2}] = [M^{1}L^{0}T^{0}]^{a}[M^{0}L^{1}T^{-1}]^{b}

[ML^{2}T^{-2}] = [M^{a}L^{b}T^{-b}]

When simplified, we get,

1 = a,

2 = b,

-2 = -b

Therefore,

a = 0,

b = 2

When put the values in equation (1),

E = kmc^{2}

E = mc^{2}

This is the energy, mass, and speed of light relationship expressed quantitatively.

## Conclusion

We can sum up the primary points by saying that dimensional analysis or the concept of homogeneity, is applied to infer the relationship between physical quantities. This article teaches students how to recognise and establish relationships between a variety of physical parameters, including mass, length, volume, temperature, gravity, speed, and distance.

## FAQs

**What is a fundamental physical quantity?**

Physical quantities classified as fundamental are those that are independent of all other physical quantities. (For instance: mass, length, temperature, duration, etc.)

**What is meant by units of measurement?**

The numerical measures used to express the outcome of measuring a physical quantity are called units of measurement.

**What is the definition of dimensional analysis?**

Every term in an equation ought to have the same dimensions. As stated differently, humans can add and subtract comparable physical amounts. Dimensional analysis refers to the above-mentioned assertion that addition and subtraction can be performed as needed to determine the dimensions of each term in an equation.

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