To solve this problem, we need to calculate the work done to raise a mass $m$ from the surface of the Earth to a height $h$ equal to the radius of the Earth $R. \\$The gravitational potential energy at a distance $r$ from the center of the Earth is given by:$\\ U = -\frac{G M m}{r} \\$where $G$ is the gravitational constant and $M$ is the mass of the Earth.$\\$Initially, the mass is at the surface of the Earth, so the initial potential energy $U_i$ is:$\\ U_i = -\frac{G M m}{R} \\$When the mass is raised to a height $h = R,$ the final distance from the center of the Earth is $2R. \\$The final potential energy $U_f$ is:$\\ U_f = -\frac{G M m}{2R} \\$The work done $W$ is the change in potential energy:$\\ W = U_f - U_i \\$Substitute the expressions for $U_f$ and $U_i: \\ W = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{R}\right) \\$Simplify the expression:$\\ W = -\frac{G M m}{2R} + \frac{G M m}{R} \\ W = \frac{G M m}{R} - \frac{G M m}{2R} \\ W = \frac{G M m}{2R} \\$We know that the gravitational force at the surface of the Earth is $mg = \frac{G M m}{R^2}. \\$Thus, $G M m = mg R^2. \\$Substitute $G M m = mg R^2$ into the expression for $W: \\ W = \frac{mg R^2}{2R} \\ W = \frac{mg R}{2} \\$Therefore, the work done to raise the mass is $\frac{1}{2} mg R. \\$This corresponds to Option 3.