Remember Galileo’s observation that free-falling objects (near the earth’s surface) all accelerate at roughly the same rate? Newton’s laws explain why this is so! Consider the famous experiment at the Leaning Tower of Pisa:
When Galileo drops a big rock and a little rock from the top of the Leaning Tower, each rock is pulled to the ground by the gravitational force acting between the rock and the earth. The strength of that force is given by Newton’s law of universal gravitation:
force =| G × mass of Earth × mass of rock |
| d2 |
In this case, d is the distance between the center of the earth and the center of the rock. Since the distance to the center of the earth is very large compared to the height of the Leaning Tower, d remains approximately constant throughout the fall, and hence the force on the rock remains constant too. Now, according to Newton’s second law of motion, the net force on the rock is equal to its mass times its acceleration: net force on rock = mass of rock × acceleration of rock Since there are no other forces acting on the rock (assuming air resistance is negligible), the net force is the same as the gravitational force; so we can simply equate them:
mass of rock × acceleration of rock =| G × mass of Earth × mass of rock |
| d2 |
Notice that the mass of the rock appears on both sides of this equation. So, if we divide both sides of the equation by the mass of the rock, it cancels out, and we get:
acceleration of rock =| G × mass of Earth |
| d2 |
Therefore, the rock’s acceleration doesn’t depend on its mass! That’s why big rocks and little rocks fall to the ground at the same rate.
Newton also explained Kepler’s discoveries about planetary motion. (As discussed in the previous chapter, Kepler was the mathematician and astronomer who figured out that planets move in elliptical orbits, not circles, around the sun. He also determined how the velocities of the planets change throughout an orbit. Kepler summarized his findings in three principles known today as Kepler’s laws of planetary motion.) Using the calculus techniques that he had invented, Newton was able to prove mathematically that Kepler’s laws could be derived—to a close approximation—from the second law of motion and the law of universal gravitation. In other words, Newton’s laws explained why Kepler’s laws were true!
This interactive simulation of orbital systems is based on Newton’s laws. Click the “Intro” simulation, then click the play button and notice that both the planet and the star follow elliptical orbits around their mutual center of mass. Try adjusting the masses, initial positions, and velocities of the star and/or planet to see how their orbits vary. (After making adjustments, a button labeled “Follow Center of Mass” will appear near the bottom of the simulation window. Click that button to keep the system from drifting out of the field of view.) In the “Lab” simulation, you can experiment with more complex systems, such as a three-body system of sun, planet, and moon.
Newton’s laws likewise accounted for the motions of other celestial objects, like moons and comets. In addition, Newton was able to explain how the ebb and flow of the ocean’s tides were due to the gravitational pull of the moon on the earth. All of these were tremendous successes for Newton’s theory of gravity, and removed all doubt about the correctness of Newton’s laws—until the twentieth century, when Einstein entered the scene. We’ll get to that story in chapter 6. Meanwhile, let’s take a look at the other half of classical physics: electromagnetic theory.