# If force

Question:

If force $(\mathrm{F})$, length (L) and time $(\mathrm{T})$ are taken as the fundamental quantities. Then what will be the dimension of density :

1. ${\left[\mathrm{FL}^{-4} \mathrm{~T}^{2}\right] }$

2. ${\left[\mathrm{FL}^{-3} \mathrm{~T}^{2}\right] }$

3. ${\left[\mathrm{FL}^{-5} \mathrm{~T}^{2}\right] }$

4. ${\left[\mathrm{FL}^{-3} \mathrm{~T}^{3}\right] }$

Correct Option: 1

Solution:

Density $=\left[F^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}}\right]$

${\left[\mathrm{ML}^{-3}\right]=\left[\mathrm{M}^{\mathrm{a}} \mathrm{L}^{\mathrm{a}} \mathrm{T}^{-2 \mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}}\right] }$

${\left[\mathrm{M}^{\mathrm{l}} \mathrm{L}^{-3}\right]=\left[\mathrm{M}^{\mathrm{a}} \mathrm{L}^{\mathrm{a}+\mathrm{b}} \mathrm{T}^{-2 \mathrm{a}+\mathrm{c}}\right] }$

$\mathrm{a}=1 \quad ; \quad \mathrm{a}+\mathrm{b}=-3 \quad ; \quad-2 \mathrm{a}+\mathrm{c}=0$

$\begin{array}{ll}1+b=-3 & c=2 a \\ b=-4 & c=2\end{array}$

So, density $=\left[F^{1} L^{-4} T^{2}\right]$