Gravitation potential energy (10-4)

Review

The expression in Equation 10-4 looks nothing like the familiar formula $U_{\mathrm{grav}} = mgy$. To understand the difference, let’s consider two different cases in which gravitational potential energy changes (Figure 10-8).

In Figure 10-8a a woman on Earth’s surface raises a book of mass $m_{\mathrm{book}}$ a vertical distance $d$. To do this she must work against a gravitational force on the book of magnitude $m_{\mathrm{book}}g$, and so she must exert an upward force of the same magnitude. So she does an amount of work $m_{\mathrm{book}}gd$ in the process. The book begins and ends at rest, so the work $m_{\mathrm{book}}gd$ that she does goes into increasing the gravitational potential energy associated with the book: $\Delta{U_{\mathrm{grav}}} = m_{\mathrm{book}}gd$. If the woman stands on a ladder and once again lifts the book by a distance $d$, the gravitational force she works against is still $m_{\mathrm{book}}g$ since the gravitational force exerted by Earth changes hardly at all near Earth’s surface. So she again does work $m_{\mathrm{book}}gd$, and the gravitational potential energy increases by the same amount $m_{\mathrm{book}}gd$ as before. As Figure 10-8b shows, this means that the gravitational potential energy increases in direct proportion to the object’s height $y$, and so the graph of $U_{\mathrm{grav}}$ versus height is a straight line: $U_{\mathrm{grav}} = m_{\mathrm{book}}gy$.

Now imagine the same person has donned a spacesuit and has brought the book with her to a point in space a distance $r_A$ from the center of Earth (Figure 10-8c). At this point the gravitational force that Earth exerts on the book has magnitude $F_A = Gm_{\mathrm{Earth}}m_{\mathrm{book}}/r_A^2$. If she moves the book an additional distance $d$ away from Earth, she has to exert a force of the same magnitude on the book and so does an amount of work $F_{A}d = (Gm_{\mathrm{book}}/r_A^2)d$. Just as on Earth, the work that she does goes into increasing the gravitational potential energy. If she now moves to a greater distance $r_B$ from the center of Earth and repeats the process, Earth's gravitational force has a smaller magnitude $F_B = Gm_{\mathrm{Earth}}m_{\mathrm{book}}/r_B^2$d, and so adds a smaller amount of gravitational potential energy. This means that at great distances, the gravitational potential energy does $\textit{not}$ increase in direct proportion to the distance $r$ and the graph of $U_{\mathrm{grav}}$ versus distance is not a straight line. Instead, the graph becomes shallower with increasing values of $r$ (Figure 10-8d). This is just what Equation 10-4 tells us, which helps to justify this expression for gravitational potential energy.