Just as the immensity of the universe is crucial to astrophysics, so is the order of magnitude smaller than the human scale. The smallest and the largest scales of the universe are deeply connected. But as we zoom inward, all the way down to the fundamental particles, we also need to consider the most familiar of the four fundamental forces of nature: gravity.
Gravity and Newton’s Law of Gravity
Gravity is what keeps us pinned here to the surface of the Earth. It never gets cancelled. That’s why, when we zoom way out to astronomical scales, gravity is the dominant force. That’s why gravity is what sculpts the properties of planets, stars, and galaxies.
Every mass attracts every other mass, according to Newton’s law of gravity, which says that the force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them. The constant of proportionality is big G, Newton’s gravitational constant, which has a value of 6.7 times 10to the minus-11 Newton meter-squared or kilogram-squared.
Let’s suppose we have a big mass, big M, held fixed at the origin of our coordinate system. And little-m is free to move, like a planet, orbiting a star. Little-m will feel a pull toward the origin. To convey the direction of the force, we’ll use vector notation.
A vector is an arrow, with a magnitude and a direction. We put a little arrow over the F to remind us it’s a vector. And by convention a “hat” on top means it’s a unit vector—a vector with a magnitude of one, so all it’s doing is specifying a direction; r-hat points in the direction of increasing r, that is, away from the origin. But the force is toward the origin, which is why we have the minus sign. The acceleration vector points in the opposite direction as r-hat.
Gravitational Potential Energy
In addition to using Newton’s law of gravity, we also need the formula for the potential energy associated with the gravitational force, which varies as the product of masses but goes inversely with r, as opposed to r-squared. And again, it’s negative.
If we let little-m fall toward the origin, r shrinks, and according to the formula, the potential energy becomes more negative, which implies that positive energy must be showing up somewhere else, since the total energy is conserved. And that does make sense.
The kinetic energy, 1/2 mv-squared, is increasing, as the mass accelerates toward the origin. The gravitational potential energy is being converted into kinetic energy. So, that’s gravity.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Focusing on the Microns
Let’s start zooming in logarithmically where with each step, we’ll magnify by a factor of 10. We’ll start here, on the human scale, where things are measured in meters. Zooming in to a 10th of a meter, we’re now centered on the human face, and as we keep narrowing our field of view to a 100th of a meter, 10 to the minus 2, we’re staring into the eye.
Another factor of 10, to the millimeter scale, and we can fit right through the pupil of the eye and dive inside. At 10 to the minus-4 meters we can see the blood vessels in the retina, and by the time we hit 10 to the minus-5, we’re seeing individual blood cells.
Now, we’re at 10 to the minus 6 meters, a millionth of a meter, a unit that comes up often enough that we give it a special name: a micron. Here we can see individual bacteria, each one is a few microns across. And starting here, the lighting looks strange. Everything’s blurry, and there are fringe patterns all around. That’s because we’ve zoomed in to the size of the wavelength of light.
Diffraction and Light Waves
Light is a wave, an oscillating pattern of electric and magnetic fields, but it’s hard to tell way up on human scales, because the wavelength is only about half a micron. Down here, though, it’s obvious.
Light waves bend and spread, like water waves, and it’s impossible to focus them sharply. That’s the phenomenon of diffraction, which will play a big role in telescopes.
Molecules in Nanometers
After another few orders of magnitude, at 10 to the minus-8 meters, we start to see that the water that surrounds us is not a continuous fluid; it’s made of individual molecules.
When we zoom in to 10 to the minus-9 meters, that is, the scale of nanometers, molecules look fuzzy and they’re in constant motion, jiggling and vibrating. They’re getting knocked around by other molecules. The energy of all those random motions is what we perceive as heat, way up on human scales. The hotter the material, the more vigorously the molecules are bouncing around.
Atoms and Electromagnetism
Zooming in closer we see the individual atoms that make up molecules. Let’s look at a single water molecule, made of 2 hydrogen atoms and an oxygen. Like any atom, oxygen has a nucleus, which has a positive electrical charge, and is surrounded by orbiting electrons, which have negative electric charge.
Because of those opposite charges, the nucleus and the electrons are attracted to each other which also explains the second of the four fundamental forces of nature: electromagnetism.
Common Questions about Fundamental Particles and Newton’s Law of Gravity
A micron is a unit to describe 10 to the minus 6 meters, that is a millionth of a meter, a unit that comes up often in physics. Starting here, the lighting looks strange. Everything’s blurry, and there are fringe patterns all around. That’s because we’ve zoomed in to the size of the wavelength of light.
The unit we arrive at after a few orders of magnitude, at 10 to the minus-8 meters is called a nanometer. When we zoom in to 10 to the minus-9 meters, that is, the scale of nanometers, we can see the individual molecules that look fuzzy and are in constant motion, jiggling and vibrating.
Electromagnetism is another fundamental force of nature that explains the opposite charges of atoms.