Question

If force $F$, work $W$ and velocity $v$ are taken as fundamental quantities, then the dimensional formula of time is:

Answer 1

Hint We are asked to find the dimensional formula of time taking the fundamental quantities as force, work and velocity. Thus, we will consider an equation which relates all these parameters. Then, we will form an equation having the final parameter as time.
Formulae used:
$\Rightarrow$ $W = F.v.t$
Where, $W$ stands for work done, $F$ stands for force applied, $v$ stands for the velocity of the body and $t$ stands for the time.

Complete Step By Step Solution
Here,
We will first rearrange the former equation to an equation which formulates time to be the final quantity.
Thus, we get
$\Rightarrow$ $t = \dfrac{W}{{Fv}}$
Now,
Let us take the dimensions of the fundamental quantities as:
$[F]$ for force, $[v]$ for velocity, $[W]$ for work and $[T]$ for time.
Thus,
From the above equation, we can say
$\Rightarrow$ $[T] = \dfrac{{[W]}}{{[F][V]}}$
In order to have a proper dimensional formula, we have to have the dimensions in a single line.
Thus,
$\Rightarrow$ $[T] = [W{F^{ - 1}}{V^{ - 1}}]$

Additional Information The concept of dimensional analysis stands very necessary to understand and analyze a formula in a very basic manner. There are mainly two ways of bringing dimensions to our hope. First is the case where we need to check whether a formula is fundamentally correct or not. Here, we substitute the dimensions of the parameter on either side of the equations and check whether they come out to be equal or not.
Secondly we use dimensional analysis in situations where we are in need to formulate a basic equation in order to proceed for a solution. In this particular case, we take into account the dependence of the final parameter on the basic parameters and then put in the dimensions to proceed.


Note We got the dimensional formula of time with respect to work, force and velocity. But for any other case, we have to consider the given basic quantities as fundamental dimensions. Then we will take up an equation connecting all the parameters together.