Question

If the speed of light [c], acceleration due to gravity [g] and pressure [P] are taken as the fundamental quantities, then the dimension of gravitational constant is

• A. ${{c}^{2}}{{g}^{0}}{{p}^{-2}}$
• B. ${{c}^{0}}{{g}^{2}}{{p}^{-1}}$
• C. $c{{g}^{3}}{{p}^{-2}}$
• D. $c{{g}^{0}}{{p}^{-1}}$

Answer 1

Answer: (B)

${{c}^{0}}{{g}^{2}}{{p}^{-1}}$

Answer 2

: $[c]=[L{{T}^{-1}}]$ $[g]=[L{{T}^{-2}}]$ $[P]=[M{{L}^{-1}}{{T}^{-2}}]$ $\frac{c}{g}=\frac{L{{T}^{-1}}}{L{{T}^{-1}}}=T$ $\therefore$ $T=\frac{c}{g}$ ?..(i) Again $\frac{{{c}^{2}}}{g}=\frac{{{L}^{2}}{{T}^{-2}}}{L{{T}^{-2}}}=L$ $\therefore$ $L=\frac{{{c}^{2}}}{g}$ ?.(ii) $P=M{{L}^{-1}}{{T}^{-2}}\to P=M{{\left[ \frac{{{c}^{2}}}{g} \right]}^{-1}}{{\left[ \frac{c}{g} \right]}^{-2}}$ $P\left[ \frac{{{c}^{2}}}{g} \right]{{\left[ \frac{c}{g} \right]}^{-2}}=M$ Or $M=\frac{P{{c}^{4}}}{{{g}^{3}}}$ ?..(iii) Now $G={{M}^{-1}}{{L}^{3}}{{T}^{-2}}$ $\therefore$ $[G]={{\left[ \frac{P{{c}^{4}}}{{{g}^{3}}} \right]}^{-1}}{{\left[ \frac{{{c}^{2}}}{g} \right]}^{3}}{{\left[ \frac{c}{g} \right]}^{-2}}$ $[G]={{P}^{-1}}{{c}^{0}}{{g}^{2}}$ $[G]={{c}^{0}}{{g}^{2}}{{P}^{-1}}$