PHSX 220 Physics I (w/ Calculus) · Unit 3: Momentum, rotation, gravitation · on Exam 3
Ch 13 Gravitation
Study notes for Gravitation in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Newton's law of gravitation: $F = \dfrac{Gm_1m_2}{r^2}$, attractive, along the line joining the centers, with $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. $r$ is measured from center to center.
- Surface gravity $g = GM/R^2$ (Earth: $M = 5.97\times10^{24}$ kg, $R = 6.37\times10^6$ m gives 9.8). At altitude $h$: $g = g_0\left(\dfrac{R}{R+h}\right)^2$.
- Circular orbits: gravity supplies the centripetal force, $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, so $v = \sqrt{GM/r}$ and $T = 2\pi\sqrt{r^3/GM}$ (Kepler's third law). Faster orbits are lower.
- Gravitational potential energy with the zero at infinity: $U = -\dfrac{GMm}{r}$. Orbital energy $E = -\dfrac{GMm}{2r}$ (negative: bound). Escape speed from a surface: $v_{esc} = \sqrt{2GM/R}$ (11.2 km/s for Earth).
- Kepler: orbits are ellipses with the Sun at a focus; equal areas in equal times (angular momentum conservation); $T^2 \propto a^3$.
Key formulas
Force and field
$$F = \frac{Gm_1m_2}{r^2},\qquad g = \frac{GM}{R^2}$$
Orbits
$$v = \sqrt{\frac{GM}{r}},\qquad T^2 = \frac{4\pi^2}{GM}r^3,\qquad E = -\frac{GMm}{2r}$$
Energy
$$U = -\frac{GMm}{r},\qquad v_{esc} = \sqrt{\frac{2GM}{R}}$$
Worked example
Find the speed and period of a satellite 400 km above Earth's surface.
Show the solution
$r = 6.37\times10^6 + 4.0\times10^5 = 6.77\times10^6$ m. $v = \sqrt{6.67\times10^{-11}(5.97\times10^{24})/6.77\times10^6} = 7.67\times10^3$ m/s. $T = 2\pi r/v = 5550$ s $\approx 92$ min.
Common mistakes
- Using altitude instead of distance from Earth’s center.
- Using U = mgh far from the surface.
- Forgetting that G and g are different things.
Exam tipAlmost every orbit problem is "gravity = centripetal force". Write that one line and solve for whatever is asked.
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