Chapter 13: Newton's Theory of Gravity

A Little History

• Newton recognized that the laws governing motion on Earth and in the heavens were interconnected, illustrating this with the example of an apple falling and the moon orbiting the Earth. He posited that the same force responsible for the apple's descent also keeps the moon in its orbit.

Newton’s Universal Gravitational Force

• Circular orbits indicate centripetal acceleration: $F = \frac{mv^2}{R}$. This force is proportional to the product of the masses involved.

  • The gravitational force acts along the line that connects the centers of the two masses.

  • The distance $R$ is calculated from the center of one mass to the center of the other.

Little g and big G

• The constant "g" represents the acceleration due to gravity near Earth's surface: $W = mg = G \frac{mM<em>E}{R</em>E^2}$

  • The value of $g$ can be expressed as: $g = G \frac{M<em>E}{R</em>E^2} = 9.8 \frac{m}{s^2}$

• $G = 6.67 \times 10^{-11} \frac{m^3}{kg \cdot s^2}$, $M<em>E = 5.98 \times 10^{24} kg$, $R</em>E = 6.37 \times 10^6 m$

• Measuring Big $G$ allows us to "weigh" the Earth, determining its mass using the formula: $M_E = \frac{gR^2}{G}$

Gravitational and Inertial Mass

• Gravitational mass: the property responsible for gravitational attractions between objects.

• Inertial mass: a measure of an object's resistance to acceleration when a force is applied.

• Equivalence Principle: asserts that these two types of masses are fundamentally the same. This is expressed as: $F<em>{12} = G \frac{m</em>{1g}m<em>{2g}}{r</em>{12}^2}$

  • The relationship between force and acceleration is given by: $F = m_i a$

• Newton’s law of gravity: $F<em>{1 \text{ on } 2} = F</em>{2 \text{ on } 1} = G \frac{m<em>1m</em>2}{r^2}$

Kepler’s Laws

• 1st Law: Planets move in elliptical orbits with the sun at one focus. This law defines key points in a planet's orbit:

  • Perihelion: the point in a planet's orbit where it is closest to the Sun.

  • Aphelion: the point in a planet's orbit where it is farthest from the Sun.

  • Semimajor axis: represents the mean distance of the planet from the Sun, effectively the average of the perihelion and aphelion distances.

  • 1 AU (Astronomical Unit) = $1.5 \times 10^{11} m$

• 2nd Law: A line joining a planet to the Sun sweeps out equal areas during equal intervals of time. This implies:

  • As a planet moves closer to the Sun, its speed increases, and as it moves farther away, its speed decreases. This is a consequence of the conservation of angular momentum.

  • Mathematically, this is expressed as: $\frac{dA}{dt} = \frac{L}{2M} = \text{constant!}$

• 3rd Law: $T^2 = Cr^3$, where $T$ is the orbital period and $r$ is the semi-major axis of the orbit.

  • The detailed form of the 3rd Law is: $T^2 = \frac{4\pi^2}{GM_S} r^3$, where $M_S$ is the mass of the Sun.

Gravitational Potential Energy

• The change in gravitational potential energy is given by: $\Delta U = U<em>B - U</em>A \equiv -W_{A \rightarrow B}$, where $W_{A \rightarrow B}$ is the work done by gravity as an object moves from point A to point B.

• The reference point for gravitational potential energy is set such that $U<em>G = 0$ when the objects are infinitely far apart. Therefore, the gravitational potential energy at a distance $r$ is: $U</em>G = -G \frac{m<em>1m</em>2}{r}$

• Close to Earth's surface, the gravitational potential energy can be approximated as: $U_G = mgy$

• Escape speed: the minimum speed required for an object to escape the gravitational pull of a planet (ignoring atmospheric friction) is given by: $v = \sqrt{\frac{2GM<em>E}{R</em>E}}$