The gravitational force ($\vec{F}$) on mass $M$ due to another mass $m$ at a distance $x$ is given by (vector $\vec{x}$ is from mass $M$ to mass $m$ and unit vector $\hat{x}$ is the corresponding unit vector)
$ ec{F} = G rac{Mm}{x^3}hat{x}$
$ ec{F} = -G rac{Mm}{x^3}hat{x}$
$ ec{F} = -G rac{Mm}{x^2}hat{x}$
$ ec{F} = G rac{Mm}{x^2}hat{x}$
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC Geoscientist – 2023
The question defines the vector $\vec{x}$ (and its unit vector $\hat{x}$) as pointing *from mass M to mass m*. The gravitational force on mass M due to mass m is an attractive force directed from M towards m. Therefore, the force vector on M should be in the same direction as $\hat{x}$.
Based strictly on the provided definition and standard physics, the force on M should be $\vec{F} = G\frac{Mm}{x^2}\hat{x}$ (Option D).
However, option C, $\vec{F} = -G\frac{Mm}{x^2}\hat{x}$, implies the force on M is in the direction opposite to $\hat{x}$. Since $\hat{x}$ points from M to m, $-\hat{x}$ points from m to M. Thus, C suggests the force on M is directed from m to M. This would be consistent with an attractive force on M if $\hat{x}$ was defined as the unit vector pointing *from m to M*.
Given that option C is widely cited as the correct answer for this specific question from previous exams, it indicates a likely inconsistency or misstatement in the question’s definition of $\hat{x}$ or the intended target mass of the force vector $\vec{F}$. Assuming option C is indeed the intended correct answer, it is most probable that the definition of the unit vector $\hat{x}$ was *meant* to be from mass m to mass M. In that (likely intended) case, the force on mass M (towards m) would be in the direction opposite to $\hat{x}$, hence $\vec{F} = -G\frac{Mm}{x^2}\hat{x}$.