Gravitation & Orbits Cheat Sheet

A printable reference covering Newton's law of gravitation, gravitational field, potential energy, circular orbits, orbital speed, and Kepler's laws for grades 10-12.

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Gravitation explains how masses attract, how planets move, and why satellites stay in orbit. This cheat sheet helps students connect force, field strength, energy, and orbital motion in one organized reference. It is especially useful for solving multi-step problems involving planets, moons, satellites, and escape speed. The most important idea is that gravity is an inverse-square force, so distance from the center of mass matters. Circular orbit problems usually combine Newton's law of gravitation with centripetal force to find speed or period. Energy formulas use gravitational potential energy, which is negative when zero is defined at infinity.

Key Facts

Newton's law of universal gravitation is $F_g = G\frac{m_1m_2}{r^2}$ , where $r$ is the center-to-center distance between the two masses.
The gravitational field strength around a spherical mass is $g = \frac{GM}{r^2}$ , and the force on a small mass is $F_g = mg$ .
Gravitational potential energy for two masses is $U_g = -\frac{GMm}{r}$ when the zero of potential energy is chosen at infinity.
For a circular orbit, the orbital speed is $v = \sqrt{\frac{GM}{r}}$ and the period is $T = 2\pi\sqrt{\frac{r^3}{GM}}$ .
Escape speed from distance $r$ is $v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$ , which is $\sqrt{2}$ times the circular orbital speed at the same radius.
Kepler's third law for a small object orbiting a much larger mass is $T^2 = \frac{4\pi^2}{GM}a^3$ , where $a$ is the semimajor axis.
For a circular orbit, the kinetic energy is $K = \frac{GMm}{2r}$ and the total mechanical energy is $E = -\frac{GMm}{2r}$ .
Because gravity follows an inverse-square law, doubling $r$ makes the gravitational force and field strength become $\frac{1}{4}$ as large.

Vocabulary

Universal gravitational constant: The constant $G = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$ that sets the strength of gravitational attraction.
Gravitational field: A region around a mass where another mass experiences gravitational force, measured by $g = \frac{F_g}{m}$ .
Orbital radius: The distance $r$ from the center of the central body to the orbiting object, not just the height above the surface.
Centripetal force: The net inward force needed for circular motion, given by $F_c = \frac{mv^2}{r}$ .
Escape velocity: The minimum launch speed needed to reach infinite distance with zero final speed, ignoring air resistance and propulsion after launch.
Semimajor axis: Half the longest width of an elliptical orbit, represented by $a$ in Kepler's third law.

Common Mistakes to Avoid

Using altitude instead of orbital radius, which is wrong because gravity depends on distance from the center of the planet. Use $r = R + h$ , not just $h$ .
Assuming the satellite's mass changes its circular orbital speed, which is wrong because $m$ cancels when $G\frac{Mm}{r^2} = \frac{mv^2}{r}$ . For a given central mass and radius, $v = \sqrt{\frac{GM}{r}}$ .
Forgetting that gravitational potential energy is negative, which gives incorrect energy comparisons. With zero at infinity, use $U_g = -\frac{GMm}{r}$ .
Using $g = 9.8\ \text{m/s}^2$ far from Earth's surface, which is wrong because $g$ decreases with distance. Use $g = \frac{GM}{r^2}$ for satellites and high-altitude motion.
Confusing orbital speed with escape speed, which is wrong because escape requires enough energy to leave the gravitational well. At the same radius, $v_{\text{esc}} = \sqrt{2}v_{\text{orbit}}$ .

Practice Questions

1 Calculate the gravitational force on a $1000\ \text{kg}$ satellite at an altitude of $400\ \text{km}$ above Earth. Use $G = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$ , $M_E = 5.97 \times 10^{24}\ \text{kg}$ , and $R_E = 6.37 \times 10^6\ \text{m}$ .
2 Find the circular orbital speed of a satellite orbiting Earth at $r = 7.00 \times 10^6\ \text{m}$ from Earth's center. Use $G = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$ and $M_E = 5.97 \times 10^{24}\ \text{kg}$ .
3 A planet orbits a star of mass $2.00 \times 10^{30}\ \text{kg}$ with semimajor axis $1.50 \times 10^{11}\ \text{m}$ . Use $T^2 = \frac{4\pi^2}{GM}a^3$ to estimate the orbital period.
4 Explain why astronauts in orbit feel weightless even though Earth's gravitational force is still acting on them.