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)

The gravitational force is  

Explanation:

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

from the sketch 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

Where is the mass o the earth and

          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

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

Where is the mass density  which is mathematically represented as

Now substituting these equation 3

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

From equation 2 we see that the

So substituting for g we have

see earlier answer