In the expression $P = El^2m^{-5}G^{-2}$, $E$, $l$, $m$ and $G$ denote energy, mass, angular momentum and gravitational constant respectively. Show that $P$ is a dimensionless quantity.
Given, expression is, $\quad P=E L^{2} m^{-5} G^{-2}$
where $\mathrm{E}$ is energy
$[\mathrm{E}]=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]$
$\mathrm{m}$ is mass
$[\mathrm{m}]=\left[\mathrm{M}^{1}\right]$
$\mathrm{L}$ is angular momentum $[\mathrm{L}]=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]$
$\mathrm{G}$ is gravitational constant $[\mathrm{G}]=\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]$
Substituting dimensions of each term in the given expression,
$[\mathrm{P}] =\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right] \times\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]^{2} \times\left[\mathrm{M}^{1}\right]^{-5} \times\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]^{-2}$
$=\left[\mathrm{M}^{1+2-5+2} \mathrm{~L}^{2+4-6} \mathrm{~T}^{-2-2+4}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]$
Hence, $P$ is a dimensionless quantity.
If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express length in terms of dimensions of these quantities.
Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck's constant $h$ and the speed of light $c$, as $Y=c^\alpha h^\beta G^\gamma$. Which of the following is the correct option?
Applying the principle of homogeneity of dimensions, determine which one is correct. where $\mathrm{T}$ is time period, $\mathrm{G}$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.