Question

Find the dimensions of work.

Answer 1

Hint: Recall the definition of work, convert the physical variable into their dimensions and solve.

Complete step by step answer:

The magnitude of work done upon or by a body is defined as the product of the force applied on or by the body and the displacement in that axis.
The SI unit of work is Joule (J) or Newton-metre (N-m).
Therefore by the definition of work,
$\text{W=F }\!\!\times\!\!\text{ d}$
Where W is the work done, F is the magnitude of force applied and d is the magnitude of displacement.

Now, using the above equation and converting the physical variables into their dimensions we get,

$\text{ }\!\![\!\!\text{ W }\!\!]\!\!\text{ = }\!\![\!\!\text{ F }\!\!]\!\!\text{ }\!\!\times\!\!\text{ }\!\![\!\!\text{ d }\!\!]\!\!\text{ }$ ---- (1)

Where the notation [] stands for the dimension of the physical variable.
Now, we will try to find the dimensions of force.
We know that,

$\text{F=ma}$
$\text{ }\!\![\!\!\text{ F }\!\!]\!\!\text{ = }\!\![\!\!\text{ m }\!\!]\!\!\text{ }\!\!\times\!\!\text{ }\!\![\!\!\text{ a }\!\!]\!\!\text{ }$ ---(2)

Where m is the mass of a body and ‘a’ is the magnitude of acceleration of the body. The dimensions of mass are [M] and the dimensions of acceleration are [LT^-2]

$\therefore \text{ }\!\![\!\!\text{ F }\!\!]\!\!\text{ =M }\!\!\times\!\!\text{ L}{{\text{T}}^{\text{-2}}}\text{=ML}{{\text{T}}^{\text{-2}}}$ --(3)
The dimension of displacement is Length (L).
$\text{ }\!\![\!\!\text{ d }\!\!]\!\!\text{ =L}$ ----(4)
Putting (3) and (4) in (1), we get
$\text{ }\!\![\!\!\text{ W }\!\!]\!\!\text{ =ML}{{\text{T}}^{\text{-2}}}\text{ }\!\!\times\!\!\text{ L=M}{{\text{L}}^{\text{2}}}{{\text{T}}^{-2}}$
Therefore, the dimensions of work are ML²T^-2.
Additional information: Many different physical quantities can have the same physical dimensions. For example, the dimensions of work, energy and torque are all the same.
Energy is in fact the ability to do work. Thus, energy and work are closely related. Both of them are also measured in Joules.

Note: Students should be extra careful while equating dimensions as dimensions of some physical quantities can be quite complicated. A good way to solve such problems is to first convert all the units of the physical quantities to the MKS system units and then proceed further by converting the MKS units into their respective dimensions.