Dimensional Formula - Units and Dimensions

Dimensional Formula: Every physical quantity in science can be written using seven basic units like length, mass, time, temperature, and more. The dimensional formula helps students understand the connection between these quantities.

Learning about the dimensional formula makes it easier to remember important science rules and also builds a strong base in the topic of units and dimension formulas. Keep reading to understand what is dimensional formula and how it is used with easy-to-understand examples.

Read More: Work Done Formula

What is Dimensional Formula?

The dimensional formula shows how a physical quantity is made using basic units like mass, length, and time. It tells us which base units are used and how many times they are used in a physical quantity.

We can simply explain the dimensional formula as "It is a way to write the units inside a physical quantity." It helps students understand how different quantities like speed, force, and energy are connected to basic units.

The dimension formula of any physical quantity is written like this: Q = MᵃLᵇTᶜ

• Here M, L, and T stand for Mass, Length, and Time, respectively.

• a, b, and c are the powers (or exponents) that tell how many times each unit is used.

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Units and Dimensions Formulas for Different Physical Quantities

In science, every physical quantity has a unit and a dimensional formula. The unit tells us how large or small the quantity is, and the dimension formula tells us which basic units (like mass, length, and time) are used to make that quantity. Learning these units and dimension formulas helps students remember measurements easily and solve problems correctly.

Find out some common physical quantities with their units and dimensional formulas in the table below:

Read More: How to Find the Angle of a Triangle?

Relation Between Dimensional Formula and Dimensional Equations

As we learned above, the dimensional formula shows how a physical quantity is written in terms of the basic units like mass (M), length (L), time (T), electric current (I), temperature (θ), amount of substance (N), and luminous intensity (J). A dimensional equation is the equation that uses these dimensional formulas to relate the physical quantity with its base units. It helps us understand how different quantities are connected through their dimensions.

For example, let’s see the dimensional equation for Area:

• Area = Length × Breadth

• Both length and breadth have the dimension [L].

• So, Area = [L] × [L] = [L²]

This shows that the dimensional formula of Area is [L² M⁰ T⁰].

Hence, the dimensional formula tells us how to write a quantity using basic units, and the dimensional equation shows the step-by-step relation using these dimensional formulas.

Read More: Measurement

When To Use Dimensional Formula?

The dimensional formula is very helpful in solving and checking physics problems. Moreover, we can use the dimensional formulas:

• To Check if an Equation is Correct: Students can use the dimensional formula to see if a physics formula is right or wrong by comparing the dimensions on both sides of the equation.

• To Find Relations Between Quantities: When students don’t know the formula, the dimensional formula can help them find how two or more physical quantities are related.

• To Convert Units: If students need to change units from one system to another, like from centimeters to meters, the dimensional formula makes this easy.

• To Write Quantities in Base Units: The dimensional formula helps to express any big or small physical quantity using simple base units like length, mass, time, etc.

Read More: Integers

Limitations of Dimension Formula

While the dimensional formula is useful in many ways, it also has some limits. Let us understand the limitations of the dimension formula here:

• Does Not Work with Special Functions: Dimensional formulas cannot be used when the equation has trigonometric (like sin, cos), logarithmic, or exponential terms.

• Covers Only Some Quantities: We can use dimensional formulas for many physical quantities, but not for all. Some quantities cannot be written using dimensional formulas.

• Cannot Find Constants: The dimensional formula does not help to find special numbers or constants used in formulas, like 2, ½, or π.

• Only Simple Math is Allowed: We can only add or subtract terms with the same dimensions in a dimensional formula. Complex operations are not allowed.

Read More: What is Factorization Formula? 

Dimensional Formula Examples With Solutions

Example 1: Find the dimensional formula of Force.

Solution: We know that Force = mass × acceleration.

• Dimensional formula of mass = [M]

• Dimensional formula of acceleration = [LT⁻²]

• So, Force = [M] × [LT⁻²] = [MLT⁻²]

Hence, Dimensional formula of Force is [MLT⁻²].

Example 2: Using the formula Q = MᵃLᵇTᶜ, find the values of a, b, and c for acceleration.

Solution: We know that Acceleration = (velocity) ÷ (time) = (displacement ÷ time) ÷ time = L ÷ T² = LT⁻²

So, Acceleration = M⁰L¹T⁻².

By comparing with Q = MᵃLᵇTᶜ, we get: a = 0, b = 1, and c = -2.

Hence, the values of a, b, and c for acceleration are: a = 0, b = 1, c = -2.

Example 3: Check if the formula Work = Force × Displacement is correct using dimensional analysis.

Solution:

Left Side:

• Dimensional formula of Work = [ML²T⁻²]

Right Side:

• Force = [MLT⁻²]

• Displacement = [L]

So, Force × Displacement = [MLT⁻²] × [L] = [ML²T⁻²].

Since LHS = RHS, the formula is dimensionally correct.

Hence, the formula Work = Force × Displacement is verified dimensionally.

Example 4: Find the angle dimensional formula.

Solution:

We know that Angle = Arc Length / Radius. Both arc length and radius are measured in metres (m).

• So, Angle = metre / metre = m / m = 1 (no units).

• This means the angle has no units. It is a dimensionless quantity.

So, the Angle Dimensional Formula = M⁰L⁰T⁰.

Also Read: Imperial system

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