Question

A man of mass m starts falling towards a planet of mass M and radius R. As he reaches near to the surface, he realizes that he will pass through a small hole in the planet. As he enters the hole, he sees that the planet is really made of two pieces: a spherical shell of negligible thickness of mass $2M/3$ and a point mass $M/3$ at the centre. Change in the force of gravity experienced by the man is

Answer 1

The gravitational force, when the person or the object is near to another larger object (planet).
$F=\dfrac{GmM}{{{R}^{2}}}$
where $F$ is the force experience by the man or object when coming in contact with a larger body which has its own gravitational force, $G$ is the gravitational force constant, $R$ is the distance from the man to the planet and $M$ is the mass of the planet.

Complete step by step solution:
When the person started flying towards the planet, the person experiences the gravitational force due to the gravity of the planet, the force experienced by the man is $F=\dfrac{GmM}{{{R}^{2}}}$.
When the person heads inside the planet through the hole, the gravitational force that he was once experiencing due to the shell of the planet is now gone and due to this the gravitational force on him becomes zero.
Now, when the person enters through the hole, he sees two shells of mass $2M/3$ and point mass $M/3$. Thus the gravitational force acting on the person due to the point of mass of the shell is given as:
$F=\dfrac{GmM}{{{R}^{2}}}$
Placing the value of the point of mass in the above formula we get the value of the force acting is:
$F=\dfrac{GmM}{3{{R}^{2}}}$
Now that we have gained the two energies that are one outside the planet and another inside the planets due to another shell-like structure, the net gravitational force acting on the person is the value of the second shell gravitational force subtracted from the first shell gravitational force.
The net force is $\dfrac{GmM}{{{R}^{2}}}-\dfrac{GmM}{3{{R}^{2}}}$
$\Rightarrow \dfrac{GmM}{{{R}^{2}}}-\dfrac{GmM}{3{{R}^{2}}}$
$\Rightarrow \dfrac{2GmM}{3{{R}^{2}}}$
Therefore, the net force acting on the man after he reaches inside the planet is $\dfrac{2GmM}{3{{R}^{2}}}$.

Note:
The gravitational force is dependent on the length of the object and the force radiates from the core to the exterior surface that is why shell surfaces have gravity on the outside of the surface and not inside or until the shell’s length making the gravity inside zero but in case of series of shells, the gravity will be influenced continuously as long as the object passes through the shells meaning, the force experienced by the person once he passes one shell will be lost by the previous shell but have gained a new one by the shell in front of it and so on if the objects continuously moves inside.