Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 402

\[1)\ Сила\ тяжести:\]

\[F(x) = \frac{Gm \bullet M}{x^{2}} = \frac{R^{2}m}{x^{2}} \bullet \frac{\text{GM}}{R^{2}} =\]

\[= \frac{R^{2}\text{mg}}{x^{2}}.\]

\[2)\ A = \int_{R}^{R + h}{F(x)\text{\ dx}} =\]

\[= \int_{R}^{R + h}{\frac{R^{2}\text{mg}}{x^{2}}\text{dx}} = \left. \ - \frac{R^{2}\text{mg}}{x} \right|_{R}^{R + h} =\]

\[= - \frac{R^{2}\text{mg}}{R + h} + \frac{R^{2}\text{mg}}{R} =\]

\[= \frac{(R + h)R^{2}mg - R \bullet R^{2}\text{mg}}{R(R + h)} =\]

\[= \frac{R^{3}mg + R^{2}mgh - R^{3}\text{mg}}{R(R + h)} =\]

\[= \frac{R^{2}\text{mgh}}{R(R + h)} = \frac{\text{Rmgh}}{R + h}.\]

\[Ответ:\ \ \frac{\text{Rmgh}}{R + h}.\]