question_answer 57)

                  An object of mass m is raised from the surface of the earth to a height equal to the radius of the earth, that is, taken from a distance R to 2R from the centre of the earth. What is the gain in its potential energy?

Answer:

                  Gain in PE \[={{U}_{f}}-{{U}_{f}}=-\frac{GMm}{2R}-\left( -\frac{GMm}{R} \right)\]                 \[=\frac{GMm}{R}\left( -\frac{1}{2}+1 \right)=\frac{GMm}{2R}\]             \[=\frac{{{(gR)}^{2}}m}{2R}=\frac{1}{2}mgR\]           \[(as\,\,g=GM/{{R}^{2}})\]