{"id":5696,"date":"2019-11-08T09:49:54","date_gmt":"2019-11-08T04:19:54","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=5696"},"modified":"2022-11-04T12:06:29","modified_gmt":"2022-11-04T06:36:29","slug":"dimensional-formula-of-work","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/dimensional-formula-of-work\/","title":{"rendered":"What is the Dimensional Formula of Work and its derivation?"},"content":{"rendered":"\n

In this article, we\u2019ll talk about the Dimensional Formula of Work<\/strong>. Here, we will look at the dimension of work and how to find the work dimensional formula.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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,<\/p>\n\n\n\n

$W=\\vec F \\cdot \\vec d$<\/p>\n\n\n\n

here both force and displacement are vector quantities and work done being the scalar product of two vectors is a scalar quantity.<\/p>\n\n\n\n

Dimensional Formula of Work<\/h2>\n\n\n\n

The Dimensional formula for Work done is<\/p>\n\n\n\n

$[M^1L^2T^{-2}]$<\/p>\n\n\n\n

Where
M<\/strong> -> Mass<\/p>\n\n\n\n

L<\/strong> -> Length<\/p>\n\n\n\n

T<\/strong> -> Time
We would now derive this dimensional formula.<\/p>\n\n\n\n

dimension of work done derivation<\/h3>\n\n\n\n

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

W-> Work done by the Force<\/p>\n\n\n\n

So,<\/p>\n\n\n\n

Now the dimension of displacement= $[L^1]$<\/p>\n\n\n\n

Let’s derive the dimension of Force<\/p>\n\n\n\n

$F= ma$
Now<\/p>\n\n\n\n

Where $m\\rightarrow$ mass<\/p>\n\n\n\n

$a$ -> Acceleration<\/p>\n\n\n\n

Dimension of Mass = $[M^1]$<\/p>\n\n\n\n

Now acceleration<\/p>\n\n\n\n

$a = \\frac {\\Delta v}{t}$<\/p>\n\n\n\n

Now dimension of Velocity= $[M^0 L^1T^{-1}]$<\/p>\n\n\n\n

dimension of Time = $[M^0 T^1]$<\/p>\n\n\n\n

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

So, Dimension of force is given by<\/p>\n\n\n\n

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

Now we know both the displacement and Force dimension , we can calculate the dimension of Work easily as<\/p>\n\n\n\n

$\\text {dimension of Work} = \\text {dimension of Force} \\times   \\text {dimension of displacement}$<\/p>\n\n\n\n

$= [M^1L^1T^{-2}] \\times [L^1] = [M^1L^2T^{-2}]$<\/p>\n\n\n\n

Unit of Work Done is Joule.<\/p>\n\n\n\n

Try the free Quiz given below to check your knowledge of Dimension Analysis:-<\/p>\n\n\n\n

Quiz on Dimensional Analysis<\/h4>\n\n\n