Question

The quantity having negative dimensions in mass is:

A Gravitational Potential B Gravitational constant C Acceleration due to gravity D None of the above

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Answer

To determine which quantity has a negative dimension in mass, let's analyze the given options based on their respective dimensional formulas.

  1. Gravitational Potential:
    The formula for gravitational potential (φ) is given by: $$ \phi = \frac{GM}{r} $$ Here, $G$ is the gravitational constant, $M$ is mass, and $r$ is the distance. The dimensional formula is: $$ [\phi] = [M]^0[L]^2[T]^{-2} $$ This shows it has no mass ($M$) involved in its dimensions.

  2. Gravitational Constant:
    The gravitational constant (G) is given by: $$ G = \frac{F \cdot r^2}{M^2} $$ Where $F$ is force, $r$ is distance, and $M$ is mass. The dimensional formula is: $$ [G] = [M]^{-1}[L]^3[T]^{-2} $$ This clearly shows a negative dimension in mass ($M^{-1}$).

  3. Acceleration due to Gravity:
    The formula for acceleration due to gravity (g) is: $$ g = \frac{GM}{r^2} $$ The dimensional formula is: $$ [g] = [M]^0[L][T]^{-2} $$ Similar to gravitational potential, it has no mass ($M$) involved in its dimensions.

Conclusion: Based on the dimensional analysis, the quantity that has a negative dimension in mass is the gravitational constant. Therefore, the correct option is B (Gravitational constant).


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