A mass m is raised from the surface of the earth to a point distant ßR(ß > 1) from the centre of the earth and then put into a circular orbit to make it an artificial satellite. The total work done to complete this job i

The correct option is mgR(2ß - 1)/2ß.

To find the total work done, we can calculate the difference between the final mechanical energy of the satellite in its orbit and the initial mechanical energy of the mass on the surface of the Earth.

Step 1: Identify the initial energy of the mass
Initially, the mass $m$ is at rest on the surface of the Earth (at a distance $R$ from the center of the Earth). Its initial energy $E_i$ is purely gravitational potential energy:
$E_i=-\frac{GMm}{R}$
where $G$ is the universal gravitational constant and $M$ is the mass of the Earth.

Step 2: Identify the final energy of the satellite in orbit
Finally, the mass is put into a circular orbit at a distance $r=ßR$ from the center of the Earth. The total mechanical energy (kinetic energy + potential energy) of a satellite in a stable circular orbit of radius $r$ is:
$E_f=-\frac{GMm}{2r}$
Substituting $r=ßR$ into the equation gives:
$E_f=-\frac{GMm}{2ßR}$

Step 3: Calculate the work done
The work done $W$ required to complete this job is equal to the change in the total mechanical energy of the system:
$W=E_f-E_i$
Substituting the expressions for $E_i$ and $E_f$:
$W=-\frac{GMm}{2ßR}-\left(-\frac{GMm}{R}\right)$
Simplifying the equation:
$W=\frac{GMm}{R}\left(1-\frac{1}{2ß}\right)$
$W=\frac{GMm}{R}\left(\frac{2ß-1}{2ß}\right)$

Step 4: Relate the expression to acceleration due to gravity (g)
The acceleration due to gravity at the surface of the Earth is given by:
$g=\frac{GM}{R^2}$
which gives:
$GM=g{R}^2$
Substituting $GM$ back into our work formula:
$W=\frac{\left(g{R}^2\right)m}{R}\left(\frac{2ß-1}{2ß}\right)$
$W=mgR\left(\frac{2ß-1}{2ß}\right)$

Thus, the total work done to complete this process is $\frac{mgR\left(2ß-1\right)}{2ß}$.