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Q.
In a system of units if force $(F)$, acceleration $(A)$, and time $(T)$ are taken as fundamental units, then the dimensional formula of energy is:
JIPMERJIPMER 2006Physical World, Units and Measurements
Solution:
Let energy $E=F^{x} A^{y} T^{3}$
Writing the dimensions on both sides
$\left[ ML ^{2} T ^{-2}\right]=\left[ MLT ^{-2}\right]^{x}\left[ LT ^{-2}\right]^{y}[ T ]^{2}$
$\left[ ML ^{2} T ^{-2}\right]=\left[ M ^{ x } L ^{x +y} T ^{-2 x -2 y +z}\right]$
On comparing both sides
$x =1$
$x +y =2$
$\Rightarrow y=2-x=2-1=1$
and $-2 x-2 y +z=-2$
$\Rightarrow -2 \times 1-2 \times 1+z=-2$
$\Rightarrow z=-2+2+2$
$=2$
$\therefore $ Dimensional formula of energy $= FAT^{2}$