Question

Using the principle of homogeneity of dimensions, which of the following is correct?

• A. $T^{2} = \frac{4\pi^{2}r^{3}}{GM}$
• B. $T^{2} = 4\pi^{2}r^{2}$
• C. $T^{2} = \frac{4\pi^{2}r^{3}}{G}$
• D. $T = \frac{4\pi^{2}r^{3}}{G}$

Answer 1

Answer: (C)

$T^{2} = \frac{4\pi^{2}r^{3}}{G}$

Answer 2

$T^{2} = \frac{4\pi^{2}r^{3}}{GM}$

Taking dimensions on both sides, we get

$\lbrack T\rbrack^{2} = \frac{\lbrack L\rbrack^{3}}{\lbrack M^{- 1}L^{3}T^{- 2}M\rbrack} = \lbrack M^{0}L^{0}T^{2}\rbrack$

$\therefore$LHS = RHS

Now, $T^{2} = 4\pi^{2}r^{2}$

Takings dimension on both sides

$\lbrack T\rbrack^{2} = \lbrack L^{2}\rbrack$

$\therefore$LHS $\neq$RHS

Now, $T^{2} = \frac{4\pi^{2}r^{3}}{G}$

$\lbrack T\rbrack = \frac{\lbrack L^{3}\rbrack}{\lbrack M^{- 1}L^{3}T^{- 2}\rbrack} = \lbrack ML^{0}T^{2}\rbrack$

LHS $\neq$ RHS.