Question

A hole is drilled from one end to the other end of Earth and an object of mass $m$ is dropped down the hole. The gravitational force acting on the object as a function of distance $r$ from the center of Earth is (Assume mass of Earth = $M$, radius = $R$, and uniform density)

• A. $\vec{F} = \dfrac{GMm}{R^2} \hat{r}$
• B. $\vec{F} = \dfrac{GMm}{R^2} \vec{r}$
• C. $\vec{F} = \dfrac{GMm}{R^3} \vec{r}$
• D. $\vec{F} = \dfrac{GMm}{R^3} \hat{r}$

Hint:

Inside a uniform solid sphere like Earth, the gravitational force is proportional to the distance from the center.

Answer 1

The Correct Option is D

Inside the Earth, the gravitational force varies linearly with distance $r$ from the center due to the shell theorem.
The force is given by: $\vec{F} = \dfrac{GMm}{R^3} r \ \hat{r}$
Here, $M$ is the mass of Earth, $R$ is the radius of Earth, and $r$ is the distance from the center. The direction is radial, hence $\hat{r}$.