Question

Applying the principle of homogeneity of dimensions, determine which one is correct. Where $T$ is the time period, $G$ is the gravitational constant, $M$ is the mass, and $r$ is the radius of the orbit.

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

Answer 1

Answer: (C)

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

Answer 2

According to the principle of homogeneity of dimensions, the dimensions on the left-hand side (LHS) must match those on the right-hand side (RHS).
1. Check Dimensions of Each Term in Option (3):
Consider:
$T^2 = \frac{4\pi^2 r^3}{GM}.$ - The dimensions of $T^2$ are $[T^2]$.
- The dimensions of $G$ (gravitational constant) are $[M^{-1}L^3T^{-2}]$.
- The dimensions of $M$ are $[M]$.
- The dimensions of $r$ (radius) are $[L]$.
2. Dimensional Analysis:
Substitute the dimensions into RHS:
$\left[\frac{L^3}{M \times M^{-1}L^3T^{-2}}\right] = [T^2].$ Since both sides have the dimension of $[T^2]$, option (3) is dimensionally correct.
Answer: $\frac{4\pi^2 r^3}{GM}$