Question

The dimensions for the gravitational constant $G$ are ____.

• A. $M^{-1}L^2T^{-2}$
• B. $M^0L^0T^0$
• C. $MT^{-2}$
• D. $ML^2T^{-2}$
• E. $M^{-1}L^3T^{-2}$

Hint:

If you forget the dimensions of Force, remember $F = ma$, so Force is Mass ($M$) $\times$ Acceleration ($LT^{-2}$).

Answer 1

Answer: (E)

Step 1: Understanding the Concept
Dimensional analysis involves expressing physical quantities in terms of fundamental dimensions: Mass ($M$), Length ($L$), and Time ($T$). We derive the dimensions of $G$ from Newton's Law of Universal Gravitation.

Step 2: Key Formula or Approach
1. Newton's Law: $F = G \frac{m_1 m_2}{r^2}$.
2. Rearranging for $G$: $G = \frac{F r^2}{m_1 m_2}$.

Step 3: Detailed Explanation
1. Dimensions of Force ($F$) = $[MLT^{-2}]$.
2. Dimensions of distance squared ($r^2$) = $[L^2]$.
3. Dimensions of mass product ($m_1 m_2$) = $[M^2]$.
4. Substitute into the formula for $G$: \[ [G] = \frac{[MLT^{-2}] [L^2]}{[M^2]} \]
5. Simplify the expression: \[ [G] = M^{1-2} L^{1+2} T^{-2} = M^{-1} L^3 T^{-2} \]

Step 4: Final Answer
The dimensions of the gravitational constant $G$ are $M^{-1}L^3T^{-2}$.