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Physics C: Mechanics Unit 9: Gravitation and Orbits

Gravity is a universal inverse-square force. This unit applies Newton's law, energy and angular momentum to surface gravity, orbits, escape and Kepler's laws.

Covers: Newton's law of universal gravitation, Gravitational field and the shell theorem, Gravitational potential energy (−GMm/r), Circular orbits and orbital energy, Escape speed, Kepler's laws and elliptical orbits

Study guide

Newton's law of universal gravitation says any two point masses attract with F = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N·m²/kg² and r is the distance between centers. The shell theorem extends this to spheres: a uniform spherical shell acts on outside masses as if all its mass were at its center, and exerts zero net force on masses inside it. So outside a planet, g = GM/r², and at the surface g = GM/R². Inside a uniform-density sphere, only the mass within radius r pulls, so g grows linearly with r, from zero at the center to GM/R² at the surface.

Because gravity varies with distance, mgh only works for small height changes. The general potential energy is U = −GMm/r, with U = 0 at infinite separation. It is negative because the masses are bound: you must add energy to separate them. For several masses, add −Gm₁m₂/r for each distinct pair. Energy conservation with this U finds how high a launched projectile rises or how fast something falls from far away. Escape speed is the launch speed that makes the total energy zero: ½mv² − GMm/R = 0, so v_esc = √(2GM/R), independent of the escaping object's mass.

For a circular orbit, gravity supplies the centripetal force: GMm/r² = mv²/r, giving v = √(GM/r). Writing v = 2πr/T gives T² = (4π²/GM)r³, Kepler's third law, which lets you find a central mass from any satellite's orbit. In a circular orbit, K = GMm/(2r), U = −GMm/r and E = −GMm/(2r). So K = −U/2 and E = U/2. A higher orbit has more total energy but a lower speed. That is why drag makes a satellite spiral inward and speed up: it loses energy and drops to a faster, lower orbit.

Kepler's three laws describe elliptical orbits: planets move in ellipses with the Sun at one focus; a line from the Sun sweeps equal areas in equal times; and T² ∝ a³, where a is the semimajor axis. The second law follows from conservation of angular momentum, because gravity is a central force. At perigee and apogee, the velocity is perpendicular to r, so v_p r_p = v_a r_a. Combining that with energy conservation gives the speed at either end. The semimajor axis is the average of the perigee and apogee distances, and the orbit's total energy is E = −GMm/(2a).

Orbit changes follow from energy. Firing an engine forward at one point raises the energy and turns that point into the perigee of a larger ellipse; firing backward makes it the apogee. Increasing speed by a factor of √2 reaches escape. Two stars of similar mass both orbit their common center of mass: the force depends on their separation d, while each star's orbital radius is its distance from the center of mass. A satellite orbiting just above a planet's surface has a period that depends only on the planet's average density, T = √(3π/(Gρ)).

Key ideas

• Gravity between point masses is proportional to the product of masses and to 1/r².
• A uniform shell acts like a point mass outside and exerts no force inside.
• U = −GMm/r replaces mgh for large distances.
• Escape speed is √(2GM/R), independent of the escaping mass.
• Circular orbits have v = √(GM/r), K = −U/2 and E = −GMm/(2r).
• Kepler's third law links the period to the semimajor axis and the central mass.
• Elliptical orbit problems combine energy and angular momentum conservation.

Formulas and rules

Common mistakes

• Measuring r from the surface instead of from the planet's center.
• Using mgh for large changes in altitude.
• Forgetting that U is negative and that bound orbits have negative total energy.
• Using perigee or apogee distance instead of the semimajor axis in Kepler's third law.
• Thinking a satellite slowed by drag moves to a higher or slower orbit.

Before the test, make sure you can

• Calculate gravitational forces and surface or altitude gravity.
• Apply the shell theorem, including g inside a uniform sphere.
• Use U = −GMm/r with energy conservation for launches and falls.
• Derive and use the circular-orbit speed, period and energy.
• Calculate escape speed from a surface or from an orbit.
• Apply Kepler's third law to find periods, radii or central masses.
• Find perigee and apogee speeds using energy and angular momentum.
• Predict how burns and drag change an orbit.

Words to know

Universal gravitation

Every pair of masses attracts with F = Gm₁m₂/r².

Gravitational constant G

6.67 × 10⁻¹¹ N·m²/kg², the constant in Newton's law of gravitation.

Gravitational field

Gravitational force per unit mass, g = GM/r² outside a spherical body.

Shell theorem

A uniform shell acts as a point mass at its center for outside masses and exerts no force inside.

Gravitational potential energy

U = −GMm/r, taking U = 0 at infinite separation.

Escape speed

The minimum launch speed to reach infinity with zero speed, √(2GM/R).

Orbital speed

The speed for a circular orbit of radius r, √(GM/r).

Kepler's first law

Planets move in ellipses with the Sun at one focus.

Kepler's second law

A line from the Sun to a planet sweeps equal areas in equal times.

Kepler's third law

T² ∝ a³ for bodies orbiting the same central mass.

Semimajor axis

Half the longest diameter of an ellipse; the average of perigee and apogee distances.

Perigee and apogee

The closest and farthest points of an orbit around Earth.

Geosynchronous orbit

A circular orbit with a period of one sidereal day, radius about 4.2 × 10⁷ m.

Bound orbit

An orbit with negative total energy, such as a circle or ellipse.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.