Question

From the dimensional consideration, which of the following equation is correct

• A. $T = 2\pi\sqrt{\frac{R^{3}}{GM}}$
• B. $T = 2\pi\sqrt{\frac{GM}{R^{3}}}$
• C. $T = 2\pi\sqrt{\frac{GM}{R^{2}}}$
• D. $T = 2\pi\sqrt{\frac{R^{2}}{GM}}$

Answer 1

Answer: (A)

$T = 2\pi\sqrt{\frac{R^{3}}{GM}}$

Answer 2

$T = 2\pi\sqrt{\frac{R^{3}}{GM}}$ $= 2\pi\sqrt{\frac{R^{3}}{gR^{2}}} = 2\pi\sqrt{\frac{R}{g}}$

Now by substituting the dimension of each quantity in both sides.

$\lbrack T\rbrack = \left\lbrack \frac{L}{LT^{- 2}} \right\rbrack^{1/2} = \lbrack T\rbrack$

L.H.S. = R.H.S. i.e., the above formula is Correct**.**