Question

If time $\left(t\right)$, velocity $\left(v\right)$, and angular momentum $\left(l\right)$ are taken as the fundamental units. Then the dimension of mass $\left(m\right)$ in terms of $t,v$ and $l$ is :

• A. $\left[t^{-2}{v}^{-1}{l}^1\right]$
• B. $\left[t^{-1}{v}^1{l}^{-2}\right]$
• C. $\left[t^{-1}{v}^{-2}{l}^1\right]$
• D. $\left[t^1{v}^2{l}^{-1}\right]$

Answer 1

The correct option is C $\left[t^{-1}{v}^{-2}{l}^1\right]$

Given:
$t,v$ and $l$ are the fundamental units.

$m\propto t^a{v}^b{l}^c$

Using dimensions

$\left[M^1{L}^0{T}^0\right]=\left[T{]}^a\right[L{T}^{-1}{]}^b[M{L}^2{T}^{-1}{]}^c$

$M^1{L}^0{T}^0=M^c{L}^{b+2c}{T}^{a-b-c}$

After comparing powers, we get

$c=1,b=-2,a=-1$

$\therefore m=\left[t^{-1}{v}^{-2}{l}^1\right]$

Hence, option $\left(\text{D}\right)$ is correct.