Example 2.74. Calculate the work done by a body of mass $m$ when falling to the surface of the Earth from a height $h$.
Solution. The force $F$ of attraction of a mass $m$, located at a distance $x$ from the Earth's surface is $F=\frac{\gamma m M}{(R+x)^{2}}$, where $\gamma$ is the gravitational constant, $R$ is the radius of the Earth. The elementary work $d A=F(x) d x$ (the $O X$ axis is directed to the center of the Earth along the radius, the origin of coordinates is at a height $h$ from the Earth), hence it follows:

$$
\begin{aligned}
& A=\int_{h}^{0} \frac{\gamma m M}{(R+x)^{2}} d x=-\left.\frac{\gamma m M}{R+x}\right|_{h} ^{0}= \\
& =\gamma m M\left(\frac{1}{R}-\frac{1}{R+h}\right)=\frac{\gamma m M h}{R(R+h)}
\end{aligned}
$$

Since on the Earth's surface $F=m g=\frac{\gamma m M}{R^{2}}$, then $A=\frac{m g R h}{R+h}$.