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

The Correct Option is C

Dimensional homogeneity requires that the dimensions on the left-hand side (LHS) equal those on the right-hand side (RHS).
1. Dimension Verification for Option (3):
Examine:
\[ T^2 = \frac{4\pi^2 r^3}{GM}. \] - 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 Consistency Check:
Substitute dimensions into the RHS:
\[ \left[\frac{L^3}{M \times M^{-1}L^3T^{-2}}\right] = [T^2]. \] Both sides exhibit the dimension of \([T^2]\), confirming option (3) is dimensionally sound.
Result: \( \frac{4\pi^2 r^3}{GM} \)