Answer
The work done on the object is $~~\frac{G~M_E~m}{12R_E}$
Work Step by Step
We can find the work done on the object:
$W = \int^{4R_E}_{3R_E}~F~dr$
$W = \int^{4R_E}_{3R_E}~\frac{GM_Em}{r^2}~dr$
$W = -\frac{GM_Em}{r}~ \Big\vert^{4R_E}_{3R_E}$
$W = -\frac{GM_Em}{4R_E}-(-\frac{GM_Em}{3R_E})$
$W = \frac{GM_E~m}{3R_E}-\frac{GM_E~m}{4R_E}$
$W = \frac{4G~M_E~m}{12R_E}-\frac{3G~M_E~m}{12R_E}$
$W = \frac{G~M_E~m}{12R_E}$
The work done on the object is $~~\frac{G~M_E~m}{12R_E}$.
