Question

If we draw a potential energy (U) vs distance (r) curve for a mass 'm' in the gravitational field of the Earth 'M' then why the graph does not exist for r=0 to r=R (where R is the radius of the Earth)but starts from r=R to r=infinity?

Answer 1

Gravitational potential function inside Earth.

The gravitational field strength due to uniform solid sphere within it can be shown to decrease linearly with `r` and `=0` as we reach the center of the sphere. This is due to fact that the force of gravitational attraction exists between part of the sphere below the point of location of another mass. Force between the remaining outside spherical shell aggregates to zero.
Figure below shows the gravitational field strength for both regions inside and outside the sphere.
`r` is the distance from the center of sphere and `a` is the radius of sphere.

As such Gravitational field function for values of distance`=0` and `a` between the the body of mass `m` and sphere reduces to

`E=G((4/3pir^3rho)m)/r^2`
Substituting value of density `rho` in terms of Mass of the planet `M=4/3pia^3rho`
`E=G(4/3pir^3(M/(4/3pia^3))m)/r^2`
`E=G(Mmr)/a^3`

Using steps similar to used in above derivation, Gravitational potential function for values of `r< a` inside the sphrical body will be

`U(r)=-G(m_rm)/r`
`=>U(r)=-G(Mmr^2)/a^3` .......(1)
where `m_r` is mass of smaller sphere of radius `r`.

Gravitational potential function outside Earth.

We know that Gravitational potential energy function outside the spherical body is given by the expression
`U(r)=-G(Mm)/r` ......(2)
which has a value on the surface of planet
`U(r)=-G(Mm)/a`

We know that gravitational potential of a point is defined as work done on a unit mass in moving it to that point from `oo` (a point remote from all other masses).

Therefore, total Gravitational potential of a body of mass `m` can be found by sum of integral of equation (1) from `lim r=oo` to `r=a` and integral of equation (2) from `lim r=a" to "r=r`

We also note that even though Gravitational potential function exists there is no physical significance attributed to Potential inside the Earth calculated with the help of equation (1) as it is physically not possible to take unit mass inside the solid earth, perform actual measurements and compare results. This remains theoretical exercise.

Therefore, from practical point graph is not drawn for values of distances lower than radius of planet as shown below.

For the purpose of calculations of escape velocity from earth and calculations regarding satellite orbits etc, potential at the surface of earth only is required.