Question

Dimensions of Universal Gravitational Constant (G) are:

• A. $[M^{-1}L^{3}T^{-2}]$
• B. $[ML^{2}T^{-2}]$
• C. $[M^{-2}L^{3}T^{-1}]$
• D. $[M^{1}L^{3}T^{-2}]$

Hint:

Always derive G from $F = \frac{Gm^{2}}{r^{2}}$ to avoid memorization errors.

Answer 1

The Correct Option is A

Step 1: Concept
Newton's Law of Gravitation states $F = \frac{Gm_{1}m_{2}}{r^{2}}$. To find the dimensions of $G$, we rearrange the formula to $G = \frac{Fr^{2}}{m_{1}m_{2}}$.

Step 2: Meaning
Dimensions represent the physical nature of a quantity in terms of base units: Mass (M), Length (L), and Time (T).

Step 3: Analysis
The dimension of Force (F) is $[MLT^{-2}]$, distance ($r^{2}$) is $[L^{2}]$, and mass product ($m_{1}m_{2}$) is $[M^{2}]$. Substituting these: $[G] = \frac{[MLT^{-2}][L^{2}]}{[M^{2}]}$.

Step 4: Conclusion
Simplifying the expression gives $[M^{1-2}L^{1+2}T^{-2}]$, which results in $[M^{-1}L^{3}T^{-2}]$. Final Answer: (A)