AP Physics C: Unit 7 Study Guide - Orbital Mechanics

Newton's Law of Universal Gravitation

Describes the gravitational force between two masses; proportional to the product of their masses and inversely proportional to the square of the distance between them.

Centripetal Force

A force that acts on an object moving in a circular path, directed toward the center of rotation.

Orbital Speed

The speed required for an object to maintain a stable orbit around a celestial body, given by the formula v = √(GM/r).

Orbital Period ($T$)

The time taken to complete one full orbit, calculated using T = 2π√(r³/GM).

Kepler’s First Law

All planets move in elliptical orbits with the Sun at one focus.

Eccentricity ($e$)

A measure of how much an orbit deviates from being circular; a circle has e=0.

Kepler’s Second Law

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Conservation of Angular Momentum

The principle stating that if no external torque acts on a system, the angular momentum of the system remains constant.

Kepler’s Third Law

States that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Gravitational Potential Energy ($U_g$)

The energy possessed by an object due to its position in a gravitational field; U_g = -GMm/r.

Total Mechanical Energy ($E$)

The sum of kinetic energy and gravitational potential energy in a system; for circular orbits, E = -GMm/(2r).

Escape Speed

The minimum speed needed for an object to break free from a planet's gravitational pull without further propulsion, given by v_{esc} = √(2GM/r).

Centripetal force equation

F_g = mv²/r, relating gravitational force to circular motion of an orbiting body.

Negative sign in potential energy

Indicates that the gravitational potential energy is negative in a bound system, and approaches zero as the distance increases.

Linear Momentum vs Angular Momentum

Linear momentum is not conserved in orbit due to external forces, while angular momentum is conserved.

Weightlessness in Orbit

The sensation of weightlessness experienced by astronauts is due to continuous freefall, not the absence of gravity.

Radius ($r$) in orbital calculations

The distance from the center of the celestial body to the orbiting object; must include the body's radius and altitude.

Forces in circular motion

Gravity provides the necessary centripetal force that keeps a satellite in a stable orbit.

Orbital Mechanics

The study of the motion of celestial objects under the influence of gravitational forces.

Equation for gravitational force

F_g = GMm/r², where G is the gravitational constant.

Fundamental derivation for orbital velocity

Equates gravitational force to centripetal force to derive velocity.

Ratio of potential to kinetic energy

For a satellite in a circular orbit, the potential energy is typically -2 times the kinetic energy (U = -2K).

Torque in orbital motion

In orbital motion, torque is zero because the gravitational force acts along the radius vector.

Importance of conservation of energy

Energy analysis often provides a clearer understanding of orbital dynamics than force analysis.

Gravitational constant (G)

A proportionality constant used in the calculation of gravitational force, approximately 6.67 × 10^-11 N·m²/kg².

Elliptical Orbits

Orbits that are not circular but have varying distances between the orbiting body and the center mass.

Foci of an ellipse

The two points located in an ellipse; the mass of the celestial body resides at one of these foci.