Newton’s Law of Universal Gravitation is a fundamental physical law. We experience its effects everywhere on this planet, and it is the prime mover in the vast world of astronomy. It can also be expressed in a relatively simple mathematical formula on which SAT II Physics is almost certain to test you.

In 1687, Isaac Newton published his Law of Gravitation in Philosophiae Naturalis Principia Mathematica. Newton proposed that every body in the universe is attracted to every other body with a force that is directly proportional to the product of the bodies’ masses and inversely proportional to the square of the bodies’ separation. In terms of mathematical relationships, Newton’s Law of Gravitation states that the force of gravity, , between two particles of mass and has a magnitude of:

where r is the distance between the center of the two masses and G is the gravitational constant. The value of G was determined experimentally by Henry Cavendish in 1798:

The force of gravity is a vector quantity. Particle attracts particle with a force that is directed toward , as illustrated in the figure below. Similarly, particle attracts particle with a force that is directed toward .

Note that the gravitational force, , acting on particle is equal and opposite to the gravitational force acting on particle , –. This is a consequence of Newton’s Third Law.

Let’s consider two examples to give you a more intuitive feel for the strength of the gravitational force. The force of gravity between two oranges on opposite sides of a table is quite tiny, roughly 10^–13 N. On the other hand, the gravitational force between two galaxies separated by 10⁶ light years is something in the neighborhood of 10²⁷ N!

Newton’s Law of Gravitation was an enormous achievement, precisely because it synthesized the laws that govern motion on Earth and in the heavens. Additionally, Newton’s work had a profound effect on philosophical thought. His research implied that the universe was a rational place that could be described by universal, scientific laws. But this is knowledge for another course. If you are interested in learning more about it, make sure to take a class on the history of science in college.

Previously, we noted that the acceleration due to gravity on Earth is 9.8 m/s² toward the center of the Earth. We can derive this result using Newton’s Law of Gravitation.

Consider the general case of a mass accelerating toward the center of a planet. Applying Newton’s Second Law, we find:

Note that this equation tells us that acceleration is directly proportional to the mass of the planet and inversely proportional to the square of the radius. The mass of the object under the influence of the planet’s gravitational pull doesn’t factor into the equation. This is now pretty common knowledge, but it still trips up students on SAT II Physics: all objects under the influence of gravity, regardless of mass, fall with the same acceleration.

To find the acceleration due to gravity on the surface of the Earth, we must substitute values for the gravitational constant, the mass of the Earth, and the radius of the Earth into the equation above:

Not coincidentally, this is the same number we’ve been using in all those kinematic equations.

If you were to burrow deep into the bowels of the Earth, the acceleration due to gravity would be different. This difference would be due not only to the fact that the value of r would have decreased. It would also be due to the fact that not all of the Earth’s mass would be under you. The mass above your head wouldn’t draw you toward the center of the Earth—quite the opposite—and so the value of would also decrease as you burrowed. It turns out that there is a linear relationship between the acceleration due to gravity and one’s distance from the Earth’s center when you are beneath the surface of the Earth. Burrow halfway to the center of the Earth and the acceleration due to gravity will be ¹/₂ g. Burrow three-quarters of the way to the center of the Earth and the acceleration due to gravity will be ¹ /₄ g.

The orbit of satellites—whether of artificial satellites or natural ones like moons and planets—is a common way in which SAT II Physics will test your knowledge of both uniform circular motion and gravitation in a single question.

Imagine a baseball pitcher with a very strong arm. If he just tosses the ball lightly, it will fall to the ground right in front of him. If he pitches the ball at 100 miles per hour in a line horizontal with the Earth, it will fly somewhere in the neighborhood of 80 feet before it hits the ground. By the same token, if he were to pitch the ball at 100,000 miles per hour in a line horizontal with the Earth, it will fly somewhere in the neighborhood of 16 miles before it hits the ground. Now remember: the Earth is round, so if the ball flies far enough, the ball’s downward trajectory will simply follow the curvature of the Earth until it makes a full circle of the Earth and hits the pitcher in the back of the head. A satellite in orbit is an object in free fall moving at a high enough velocity that it falls around the Earth rather than back down to the Earth.

Let’s take the example of a satellite of mass orbiting the Earth with a velocity v. The satellite is a distance R from the center of the Earth, and the Earth has a mass of .

The centripetal force acting on the satellite is the gravitational force of the Earth. Equating the formulas for gravitational force and centripetal force we can solve for v:

As you can see, for a planet of a given mass, each radius of orbit corresponds with a certain velocity. That is, any object orbiting at radius R must be orbiting with a velocity of . If the satellite’s speed is too slow, then the satellite will fall back down to Earth. If the satellite’s speed is too fast, then the satellite will fly out into space.