Question

From the dimensional consideration, which of the following equations is correct?
(Here, $T =$ Time period, $R =$ Radius, $M =$ Mass and $G =$ Gravitational constant)

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

Answer 1

Answer: (A)

$\text{T}=2\pi\sqrt{\frac{R^{3}}{GM}}=2\pi\sqrt{\frac{R^{3}}{gR^{2}}}=2\pi\sqrt{\frac{R}{g}}$ [As, $GM=gR^2]$
Now, by substituting the dimensions of each quantity on both sides, we get
$\left[T\right]=\left[\frac{L}{LT^{-2}}\right]^{1/2}=\left[T\right]$