3.3 Newton’s Universal Law of Gravitation

Newton's law of gravitation describes the force of attraction between any two objects that have mass. It connects mass and distance to the strength of gravity, which makes it possible to predict orbits, calculate the masses of distant objects, and understand motion throughout the universe.

Newton's Universal Law of Gravitation

Gravity's dependence on mass and distance

Gravitational force depends on two things: how massive the objects are and how far apart they are. The more massive the objects, the stronger the pull between them. The farther apart they are, the weaker the pull becomes.

Newton captured this relationship in a single equation:

$F = G \frac{m_1 m_2}{r^2}$

Here's what each piece means:

• $F$ is the gravitational force between the two objects
• $m_1$ and $m_2$ are the masses of the two objects
• $r$ is the distance between their centers
• $G$ is the gravitational constant: $G = 6.67 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}$

The force is directly proportional to the product of the masses. Double one object's mass, and the force doubles. The force is inversely proportional to the square of the distance. Double the distance, and the force drops to one-quarter of what it was. That square in the denominator is why distance matters so much.

A few other details worth knowing:

• Gravity is a vector quantity, meaning it has both magnitude and direction. The force always points along the line connecting the two objects' centers.
• $G$ is extremely small, which is why you don't feel gravitational attraction toward your desk. You need at least one very massive object (like a planet) for the force to be noticeable.
• Newton described gravity as a force that acts across empty space, sometimes called "action at a distance."

Newton's law for orbital predictions

An orbit happens when an object is moving fast enough sideways that it keeps "falling around" the central body instead of crashing into it. Two things are at work:

• Gravitational force pulls the orbiting object inward (toward the central body)
• Inertia keeps the object moving forward in a straight line

A stable orbit occurs when the gravitational force provides exactly the centripetal force needed to bend the object's path into a curve. Too slow, and the object spirals inward. Too fast, and it flies away.

Kepler's laws describe the patterns of orbital motion, and Newton's gravity explains why they work:

1. Law of Ellipses: Orbits are ellipses, with the central object (like the Sun) at one focus of the ellipse, not at the center.
2. Law of Equal Areas: A line connecting the orbiting body to the central body sweeps out equal areas in equal time intervals. This means a planet moves faster when it's closer to the Sun and slower when it's farther away.
3. Harmonic Law: The square of the orbital period is proportional to the cube of the semi-major axis: $P^2 \propto a^3$

The speed needed to maintain a circular orbit at a given distance is:

$v = \sqrt{\frac{GM}{r}}$

where $M$ is the mass of the central object and $r$ is the orbital radius. Notice that larger $r$ means slower orbital speed. Mercury zips around the Sun at about 47 km/s, while Neptune crawls at roughly 5.4 km/s. Similarly, satellites in low Earth orbit move faster than GPS satellites in higher orbits.

Escape velocity is the minimum speed an object needs to leave a body's gravitational pull entirely and never come back. It's related to orbital velocity but higher, because the object needs enough energy to coast to an infinite distance.

Mass calculations from orbital data

One of the most powerful applications of Newton's gravity is figuring out the mass of an object you can't put on a scale. If you can observe something orbiting that object, you can calculate its mass.

Newton's form of Kepler's third law relates the orbital period $P$, the semi-major axis $a$, and the mass of the central object $M$:

$\frac{P^2}{a^3} = \frac{4\pi^2}{GM}$

Rearranging to solve for mass:

$M = \frac{4\pi^2 a^3}{GP^2}$

To use this, you need to measure two things about the orbiting object: its orbital period and the size of its orbit (semi-major axis). From those two measurements, the mass of the central body falls right out.

This technique is used throughout astronomy:

• Planet masses come from tracking the orbits of their moons or artificial satellites
• Star masses come from observing binary star systems where two stars orbit each other
• Galaxy masses are estimated from the orbital speeds of stars within them

Gravitational effects and energy

Acceleration due to gravity (often written $g$) isn't the same everywhere. It depends on your distance from the center of mass. On Earth's surface, $g \approx 9.8 \text{ m/s}^2$, but at higher altitudes or on less massive bodies, it's weaker.

Gravitational potential energy is the energy stored in a system because of the positions of its objects relative to each other. The farther apart two objects are, the more potential energy the system has (think of it as the energy you'd get back if you let them fall toward each other). This concept is what makes escape velocity meaningful: an object needs enough kinetic energy to overcome the gravitational potential energy binding it to the central body.