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Q.
Planck's constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are)
JEE AdvancedJEE Advanced 2015Physical World, Units and Measurements
Solution:
$E = \frac{hc}{\lambda}$
$E = \frac{-Gm_{1}m_{2}}{r}$
$\frac{hc}{\lambda} = \frac{Gm^{2}}{r}$
unit of $λ$ and r is L
So $M = \left(\frac{hc}{G}\right)^{1/2}\quad\quad\dots\left(1\right)$
$M \propto \sqrt{c}\quad\quad M \propto \frac{1}{\sqrt{G}}\quad\quad Ans. \left(A\right)$
$E = mc^{2}$
$E = \frac{hc}{\lambda}$
$\frac{hc}{\lambda} = mc^{2}$
$\lambda = \frac{h}{m}$
$\therefore \,L = \frac{h}{M}$
$L = \frac{h}{\left(hc\right)^{1/2}}G^{1/2}\quad\quad$ from $\left(1\right)$
$\therefore \,L \propto \sqrt{G} \quad L \propto \sqrt{h}$