What is the Dimensional Formula of Work and its derivation?

In this article, we’ll talk about the Dimensional Formula of Work. Here, we will look at the dimension of work and how to find the work dimensional formula.

When a force displaces an object, work is said to be done in science. The work done by a force is defined as the product of the force’s component in the direction of displacement and the magnitude of displacement.

Work can be computed by multiplying Force by displacement that occurs in the direction of the applied force. Mathematically work done is given by the equation,

$W=\vec F \cdot \vec d$

here both force and displacement are vector quantities and work done being the scalar product of two vectors is a scalar quantity.

Dimensional Formula of Work

The Dimensional formula for Work done is

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

Where
M -> Mass

L -> Length

T -> Time
We would now derive this dimensional formula.

dimension of work done derivation

Work is defined as the cross product for Force  and displacement
$W= F.d$
Where d -> displacement
F -> Force applied

W-> Work done by the Force

So,

Now the dimension of displacement= $[L^1]$

Let’s derive the dimension of Force

$F= ma$
Now

Where $m\rightarrow$ mass

$a$ -> Acceleration

Dimension of Mass = $[M^1]$

Now acceleration

$a = \frac {\Delta v}{t}$

Now dimension of Velocity= $[M^0 L^1T^{-1}]$

dimension of Time = $[M^0 T^1]$

So dimension of Acceleration = $ \frac {[M^0 L^1T^{-1}]}{ [M^0 T^1]}= [M^0 L^1T^{-2}]$

So, Dimension of force is given by

$\text {Dimension of Force} =[M^1] \times [M^0 L^1T^{-2}] = [M^1L^1T^{-2}]$

Now we know both the displacement and Force dimension , we can calculate the dimension of Work easily as

$\text {dimension of Work} = \text {dimension of Force} \times   \text {dimension of displacement}$

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

Unit of Work Done is Joule.

Try the free Quiz given below to check your knowledge of Dimension Analysis:-

Quiz on Dimensional Analysis

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1. Which of the following pair does not have similar dimensions

angle and strain

tension and surface tension

stress and pressure

Planck's Constant and angular momentum

Question 1 of 5

2. Choose the correct statement(s)

A dimensionally incorrect equation may be correct

A dimensionally incorrect equation may be incorrect

A dimensionally correct equation may be correct

A dimensionally correct equation may be incorrect

Question 2 of 5

3. A dimensionless quantity

always has a unit

may have a unit

never has a unit

does not exist

Question 3 of 5

4. Which of the following has the dimensions of pressure?

\([ML^{-2}T^{-2}]\)

\([ML^{-1}T^{-2}]\)

\([M^{-1}L^{-1}]\)

\([MLT^{-2}]\)

Question 4 of 5

5. The dimension of angular velocity is

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

\([MLT^{-2}]\)

\([ML^2T^{-2}]\)

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

Question 5 of 5

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Important resources and other dimensional formulas