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# 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

1. A dimensionless quantity

Question 1 of 5

2. The dimensions of impulse are equal to that of

Question 2 of 5

3. The dimensions of universal gravitational constant are

Question 3 of 5

4. A unitless quantity

Question 4 of 5

5. Choose the correct statement(s)

Question 5 of 5

#### Important resources and other dimensional formulas

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