Question

In a new system of units, the fundamental quantities mass, length and time are replaced by acceleration $a$, density $\rho$ and frequency $f$. The dimensional formula for force in this system is

• A. $\left[\rho a^3{f}^{-6}\right]$
• B. $\left[\rho a^4{f}^{-6}\right]$
• C. $\left[\rho a^4{f}^6\right]$
• D. $\left[{\rho }^{-1}{a}^4{f}^{-6}\right]$

Answer 1

The correct option is B $\left[\rho a^4{f}^{-6}\right]$

As we know that force is given as $F=ma=\rho va$, where $v$ is the volume.
So, in this system we have to write volume in the terms of $a$ and $f$
$\Rightarrow$ volume, $\left[v\right]=\left[L^3\right]={\left[\frac{L}{T^2}\right]}^3\left[T^6\right]$
$\Rightarrow v=a^3{f}^{-6}$
Thus, we have dimensional formula for force in this system as
$F=\rho a^4{f}^{-6}$