Topic 6.6 Motion of Orbiting Satellites: analyze circular and elliptical orbits using gravity as the centripetal force, gravitational potential energy, and conservation of energy and angular momentum.

What this topic is asking

The College Board (Topic 6.6) wants you to analyze the motion of orbiting satellites, treating gravity as the centripetal force for circular orbits, and using conservation of mechanical energy and angular momentum for elliptical orbits. This topic ties the gravitational force (Topic 2.6), circular motion (Topic 2.7), and the conservation laws of Unit 6 into the dynamics of orbits.

Gravity as the centripetal force

The whole circular-orbit analysis flows from setting gravity equal to the centripetal requirement. An orbit needs no engine: the satellite is in continuous free-fall, accelerating toward the planet, but its sideways velocity is large enough that it keeps "falling around" the planet rather than into it. This is Newton's cannonball idea made precise.

Orbital speed and radius

This result, $v = \sqrt{GM/r}$, is the workhorse of the circular-orbit part of the topic. Its mass independence (the satellite's own mass cancels) parallels the way mass cancels in free-fall: all satellites at a given radius orbit at the same speed. The inverse-root dependence on radius explains why the International Space Station (low orbit) races around the Earth in about 90 minutes while the Moon (far out) takes a month.

Conservation laws in elliptical orbits

Most orbits are ellipses, not perfect circles, and there the speed changes around the path. Two conservation laws govern the motion:

• Mechanical energy is conserved: $E = K + U_g = \tfrac{1}{2}mv^2 - \dfrac{GMm}{r}$ is constant. As the satellite moves closer (smaller $r$), the gravitational potential energy $U_g = -\dfrac{GMm}{r}$ becomes more negative, so the kinetic energy, and hence the speed, increases.
• Angular momentum is conserved: gravity always points along the line to the planet, so it exerts no torque about the planet, and $L = mvr_\perp$ is constant.

Together these explain Kepler's qualitative picture: the satellite is fastest at perihelion (closest approach) and slowest at aphelion (farthest point), because shrinking $r$ forces $v$ up to keep $L$ fixed, while the energy bookkeeping says the kinetic energy rises as the potential energy falls. The exam reward here is recognizing that an orbit is the meeting point of the unit's two great conservation laws, energy and angular momentum, applied to a gravitational system. The same reasoning extends to comets, which whip through perihelion at enormous speed and crawl through the outer solar system, all while conserving both energy and angular momentum.

Try this

Q1. State what provides the centripetal force for a satellite in a circular orbit. [1 point]

• Cue. The gravitational force from the body it orbits.

Q2. A satellite moves to a larger circular orbit. State whether its orbital speed increases or decreases. [1 point]

• Cue. It decreases, because $v = \sqrt{GM/r}$ falls as $r$ rises.