Question

The gravitational force on an object of mass m, located inside the earth a distance r < Re, from the center (Re is the radius of the earth), is due only to the mass of the earth that lies within a solid sphere of radius r. What is the gravitational force as a function of the distance r from the center? Do not look up or use any values yet! (5 pts)

Answer 1

Answer:

The gravitational force is  $\r F_{g} = \frac{mg}{R_e} \r rr$

Explanation:

The sketch describing this question is shown on the first uploaded image

from the sketch $\r r$ represent a unit vector pointing outward (This is shown the distance and the direction of the object  

Now the gravitational force of a object on the surface earth with mass m is

Mathematically represented as

                    $\u F = - \frac{Gmm_e}{R_e^2} =- mg \r r ----(1)$

Where $m_e$ is the mass o the earth and

          $R_e$ is the radius of the earth

The  minus sign shows that it is acting in the negative x-axis

Making the acceleration  due to gravity a subject of the formula we have

                     $g = \frac{Gm_e}{R_e^2} ----(2)$

Considering when  the distance of the object to the center of the earth is r

Note: Any mass of the earth that is outside the distance covered by radius r as shown in the diagram does not affect the gravitational force

 The mass of the enclosed earth can be mathematically represented as

                     $M_{en} = \rho \frac{4}{3} \pi r^3 ----(3)$

Where $\rho$ is the mass density  which is mathematically represented as

                         $\rho = \frac{m_e}{(\frac{4}{3} )\pi R_e^3} ----(4)$

Now substituting these equation 3

        $M_{en} = \frac{,m_e}{(\frac{4}{3} )\pi R_e^3} (\frac{4}{3} ) \pi r^3 = \frac{m_er^3}{R_e^3} ---(5)$

Now substituting equation 5 into equation 1 to obtain the gravitational force as a function of distance r

        $\r F = -\frac{GmM_{en}}{r^2} \r r = \frac{Gmm_e r^3}{r^2R_e^3} \r r = \frac{Gmm_e}{R_e^3} r ----(6)$

From equation 2 we see that the $g = \frac{Gm_e}{R_e^2}$

So substituting for g we have

                 $\r F_{g} = \frac{mg}{R_e} \r rr$