The amount of work done to raise a mass ‘m’ from the surface of the Earth to a height equal to the radius of the Earth ‘R’, will be: ____.

We are asked for the amount of work required to lift a mass $m$ from the Earth’s surface (distance $R$ from the Earth’s centre) to a height equal to the Earth’s radius, i.e., to a distance $2R$ from the centre.

The gravitational force on the mass at a distance $r$ from the centre is

$F(r)=\backslash frac\{GMm\}\{r^\{2\}\}$

Here $G$ is the gravitational constant and $M$ is the Earth’s mass. At the surface ($r=R$) this force equals the weight $mg$, so

$mg=\backslash frac\{GMm\}\{R^\{2\}\}$

The work done in moving the mass from $R$ to $2R$ is the integral of the force over the displacement:

$W=\backslash int\_\{R\}^\{2R\}\backslash frac\{GMm\}\{r^\{2\}\}\backslash ,dr$

Integrating gives

$W=GMm\backslash Bigl[-\backslash frac\{1\}\{r\}\backslash Bigr]\_\{R\}^\{2R\}\; =GMm\backslash left(-\backslash frac\{1\}\{2R\}+\backslash frac\{1\}\{R\}\backslash right)\; =GMm\backslash left(\backslash frac\{1\}\{2R\}\backslash right)$

Using the relation $mg=\backslash frac\{GMm\}\{R^\{2\}\}$, we substitute $GMm=mgR^\{2\}$:

$W=\backslash frac\{mgR^\{2\}\}\{2R\}=\backslash frac\{mgR\}\{2\}$

Hence the work required to raise the mass to a height equal to the Earth’s radius is

$W=\backslash frac\{mgR\}\{2\}$

The correct answer is mgR/2.