A) \[\sqrt{\frac{Gh}{{{c}^{5}}}}\]
B) \[\sqrt{\frac{h{{c}^{5}}}{G}}\]
C) \[\sqrt{\frac{Gh}{{{c}^{3}}}}\]
D) \[\sqrt{\frac{{{c}^{3}}}{Gh}}\]
Correct Answer: A
Solution :
\[T\,\,=\,\,k{{G}^{a}}{{h}^{b}}{{c}^{c}}\] \[[T]={{[{{M}^{-1}}\,{{L}^{3}}{{T}^{-2}}]}^{a}}\,\,{{[M{{L}^{2}}{{T}^{-1}}]}^{b}}\,{{[L{{T}^{-1}}]}^{c}}\] \[=\,\,{{M}^{-a+b}}\,L{{\,}^{3a+2b+c}}\,{{T}^{-2a-b-c}}\] \[~-a+b=0\] \[\Rightarrow \,\,\,a=b~~~~~~~~...\left( 1 \right)\] \[3a+2b+c=0~~~~~~...\left( 2 \right)\] \[-2a-b-c=1~~~~~~~~~~~...\left( 3 \right)\] \[a+b=1\] \[a=b\,\text{=}\,\frac{1}{2}\] \[C=-2\,\,\times \,\,\frac{1}{2}-\frac{1}{2}-1=-\frac{5}{2}\] \[T\,\propto \,\frac{{{G}^{\frac{1}{2}}}{{h}^{\frac{1}{2}}}}{{{C}^{\frac{5}{2}}}}\,=\,\sqrt{\frac{Gh}{{{C}^{5}}}}\] Option (a)
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