question_answer

Using the principle of homogeneity of dimensions, find which of the following is correct: (i) \[{{T}^{2}}=4{{\pi }^{2}}{{r}^{2}}\]                       (ii) \[{{T}^{2}}=\frac{4{{\pi }^{2}}{{r}^{3}}}{G}\]  (iii) \[{{T}^{2}}=\frac{4{{\pi }^{2}}{{r}^{3}}}{GM}\] where T is time period, G is gravitational constant, M is mass and r is radius of orbit.

Answer:

                (i) \[{{T}^{2}}=4{{\pi }^{2}}{{r}^{2}}\] Dimensionally, \[{{M}^{0}}{{L}^{0}}{{T}^{2}}={{M}^{0}}{{L}^{2}}{{T}^{0}}\] As LHS \[\ne \]RHS, the formula is incorrect. (ii) \[{{T}^{2}}=\frac{4{{\pi }^{2}}{{r}^{3}}}{G}\] Dimensionally,  \[{{M}^{0}}{{L}^{0}}{{T}^{2}}=\frac{{{L}^{3}}}{{{M}^{-1}}{{L}^{3}}{{T}^{-2}}}={{M}^{1}}{{L}^{0}}{{T}^{2}}\] As LHS \[\ne \] RHS, the formula is incorrect (iii) \[{{T}^{2}}=\frac{4{{\pi }^{2}}{{r}^{3}}}{GM}\] Dimensionally, \[{{M}^{0}}{{L}^{0}}{{T}^{2}}=\frac{{{L}^{3}}}{{{M}^{-1}}{{L}^{3}}{{T}^{-2}}M}={{M}^{0}}{{L}^{0}}{{T}^{2}}\] As LHS \[=\] RHS, the formula is correct.