Question

A tunnel is dug along the diameter of the earth. If \[{M_e}\]​ and \[{R_e}\]​ are the mass and radius, respectively, of the earth, then the force on a particle of mass m placed in the tunnel at a distance r from the center is:
A) $\dfrac{{G{M_e}mr}}{{R_e^3}}$
B) $\dfrac{{G{M_e}m}}{{R_e^3r}}$
C) $\dfrac{{G{M_e}mR_e^3}}{r}$
D) $\dfrac{{G{M_e}mr}}{{R_e^2}}$

Answer 1

Hint: Mass is defined as the amount of matter in a body. The SI unit of mass is kg. At any time, the mass of the body will not change. It will remain constant all the time and does not change from place to place.

Complete step by step solution:
Given data:
Mass of the earth \[ = {M_e}\]
The radius of the earth \[ = {R_e}\]
Force=?
Let us calculate the mass of the inner solid sphere.
Let the radius of the inner solid sphere = r
 Thus the mass of the inner solid sphere, \[{M'} = \dfrac{{{M_e}}}{{\dfrac{4}{3}\pi R_e^3}} \times \dfrac{4}{3}\pi {r^3} = \dfrac{{{M_e}}}{{R_e^3}}{r^3}\]
 Then, \[g = \dfrac{{G{M_e}{r^3}}}{{R_e^3}} \times \dfrac{1}{{{r^2}}} = \dfrac{{G{M_e}r}}{{R_e^3}}\]
Hence force on the particle $ = mg = \dfrac{{G{M_e}mr}}{{R_e^3}}$

Hence the correct option is A.

Note: 1. Force is a vector quantity that has both magnitude and direction. It is defined as the push or pulls on an object causing a change in its velocity. Tension force, spring force, Gravitational force, Frictional force are some of the types of forces. Distance is a scalar quantity. It indicates how far the object has been covered. It is the length of the path taken by the object. Displacement is the distance between the initial point and the endpoint.
 2. Earth is the third planet from the sun. The distance from the center of the earth to a point on its surface is called an Earth radius. The value ranges from 6,378 km at the equator to 6,37 km at the pole.
 3. Mass is a property of the body which determines the body’s acceleration under the action of the force. Thus mass can be measured either by the acceleration produced or by the amount of force required in a given time.