Question

The work done to raise a mass m from the surface of the earth to a height h, which is equal to the radius of the earth, is :
(1) mgR
(2) 2 mgR
(3) (1/2) mgR
(4) (3/2) mgR

Answer 1

The work done is $\bold{\frac{1}{2}mgR}$

Explanation:

Work done can be calculated by the following formula.

                   $\bold{W= U_{f}-U_{i}}$.........................(1)

          $U_{i}=initial\,potential\,energy\,at\,the\,earth\,surface=\frac{-GMm}{R}$

         $U_{f}=final\,potential\,energy\,at\,the\,earth\,surface=\frac{-GMm}{2R}$

Substitute the initial and final potential energies of earth in equation(1)

                   $\Rightarrow \frac{-GMm}{2R}-\frac{-GMm}{R}$

                  $\Rightarrow \frac{GMm}{2R}$

The acceleration due to gravity   of surface of the earth is

        $\bold{g=\frac{GM}{R^{2}}or GM=gR^{2}}$

                $\Rightarrow \frac{gR^{2}m}{2R}$ ($GM=gR^{2}$)

               $\Rightarrow \frac{mgR}{2}$

               $\Rightarrow \frac{1}{2}mgR$

Therefore, The work done is $\bold{\frac{1}{2}mgR}$

Hence, correct option is 3