# Lecture 14: Gravitation and Kepler's Laws ## Physics I — Mechanics & Thermodynamics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. State Newton's law of universal gravitation and apply it to calculate gravitational force 2. Relate gravitational field strength g to the universal law of gravitation 3. Calculate gravitational potential energy for the general (non-uniform field) case 4. Derive and apply orbital velocity and period for circular orbits 5. State and apply Kepler's three laws of planetary motion 6. Calculate escape velocity --- ## Lecture Content ### I. Newton's Law of Universal Gravitation Every particle in the universe attracts every other particle with a gravitational force given by F = G m_1 m_2 / r^2, where G = 6.674 x 10^-11 N m^2/kg^2 is the universal gravitational constant and r is the distance between the centers of the two masses. This force is always **attractive**, directed along the line joining the masses. It forms an **action-reaction pair** (F_{1 on 2} = -F_{2 on 1}) and obeys an **inverse-square law**: doubling the distance reduces the force by a factor of four. For a uniform sphere, the gravitational force acts as if all the mass were concentrated at the center, a result known as the shell theorem. Surface gravity on a planet is given by g = GM/R^2, where M and R are the planet's mass and radius. Substituting Earth's values yields g = (6.674 x 10^-11)(5.97 x 10^24)/(6.37 x 10^6)^2 = 9.80 m/s^2. ### II. Gravitational Field The **gravitational field** at a point in space is the gravitational force per unit mass: **g** = **F**/m = -GM/r^2 r-hat, directed toward the mass creating the field. Any test mass placed in this field experiences a weight **F** = m**g**. The gravitational field varies with altitude according to g(h) = GM/(R + h)^2 = g_0 [R/(R + h)]^2. At low altitudes where h is much less than R, g is approximately constant. At large distances, g falls off as 1/r^2. ### III. Gravitational Potential Energy (General) Near Earth's surface where g is uniform, the gravitational potential energy is simply U = mgy. For the general case where g varies with distance, the potential energy is U = -G m_1 m_2 / r, with the reference U = 0 set at r = infinity. This potential energy is always **negative** for bound systems. The change in potential energy is Delta U = -G m_1 m_2 (1/r_f - 1/r_i), and the force can be recovered as F = -dU/dr = -G m_1 m_2 / r^2. A graph of gravitational potential energy U(r) = -GMm/r vs. distance r from the center of a planet. The curve starts very negative near the planet surface and asymptotically approaches zero as r goes to infinity. A horizontal dashed line at a negative energy level E represents the total energy of a bound orbit. The turning points where E = U (closest and farthest orbital distances) are marked. A second horizontal line at E = 0 represents the escape condition. Annotations show that E < 0 means bound orbit and E >= 0 means escape. ### IV. Circular Orbits For an object in circular orbit, gravity provides the centripetal force: GMm/r^2 = mv^2/r. The **orbital velocity** is v = sqrt(GM/r), which is independent of the orbiting object's mass and shows that closer orbits are faster. The **orbital period** follows from v = 2 pi r / T, giving T = 2 pi r^(3/2) / sqrt(GM), or equivalently T^2 = (4 pi^2 / GM) r^3, which is Kepler's third law for circular orbits. The **energy of a circular orbit** has a clean structure: K = (1/2)mv^2 = GMm/(2r), U = -GMm/r, and the total energy E = K + U = -GMm/(2r). The total energy is negative, confirming that the orbit is bound. A notable relationship is |U| = 2K for any circular orbit. ### V. Kepler's Three Laws Kepler's **first law (law of ellipses)** states that each planet moves in an elliptical orbit with the Sun at one focus. A circle is a special case of an ellipse, and the semi-major axis a characterizes the overall size of the orbit. Kepler's **second law (law of equal areas)** states that a line from the Sun to a planet sweeps out equal areas in equal time intervals. This is a direct consequence of conservation of angular momentum. The planet moves faster at perihelion (closest approach) and slower at aphelion (farthest distance), with dA/dt = L/(2m) = constant. Kepler's **third law (harmonic law)** states that the square of the orbital period is proportional to the cube of the semi-major axis: T^2 = (4 pi^2 / GM) a^3. For any two planets orbiting the same star, T_1^2/a_1^3 = T_2^2/a_2^3. This law allows the determination of celestial body masses from orbital observations. Panel A: An elliptical orbit with the Sun at one focus. The semi-major axis a, semi-minor axis b, perihelion distance r_p, and aphelion distance r_a are labeled. Panel B: The same orbit with two shaded triangular areas swept in equal time intervals — one near perihelion (short, wide triangle) and one near aphelion (long, narrow triangle). Both areas are equal, illustrating Kepler's second law. The velocity vectors at perihelion (large) and aphelion (small) are shown. ### VI. Escape Velocity **Escape velocity** is the minimum launch speed needed to escape a gravitational field entirely, reaching r = infinity with v = 0. Using energy conservation, (1/2)mv_e^2 - GMm/R = 0, which gives v_e = sqrt(2GM/R) = sqrt(2gR). For Earth, this works out to v_e = sqrt(2 x 9.80 x 6.37 x 10^6) = 11.2 km/s. Escape velocity is independent of the direction of launch and the mass of the escaping object. It is related to the orbital velocity at the surface by v_e = sqrt(2) v_orbit. If an object is launched at less than escape velocity, it follows a bound elliptical orbit. At exactly escape velocity, the trajectory is parabolic. Above escape velocity, the trajectory is hyperbolic. ### VII. Satellites and Weightlessness Astronauts in orbit are in **free fall**, accelerating toward Earth along with their spacecraft. They are not outside the reach of gravity; on the contrary, gravity is what keeps them in orbit. The sensation of "weightlessness" arises because the apparent weight (normal force) is zero: both the astronaut and the spacecraft accelerate at g = GM/r^2 together. A geostationary orbit has a period of 24 hours and lies in the equatorial plane, at a radius r_geo = (GM T^2 / 4 pi^2)^(1/3) = 4.22 x 10^7 m, or about 35,800 km above Earth's surface. Low Earth orbit (LEO) has altitudes of roughly 200 to 2000 km, with orbital periods of approximately 90 minutes. Earth with three orbits drawn: a low Earth orbit (small circle close to the surface, labeled "LEO, T ~ 90 min"), a medium orbit, and a geostationary orbit (large circle, labeled "GEO, T = 24 h"). For the LEO, a zoomed inset shows an astronaut and spacecraft both falling toward Earth at the same rate, with the astronaut floating inside — illustrating apparent weightlessness. The gravitational force vectors are shown on both the astronaut and the spacecraft, pointing toward Earth's center.