Fundamental Interactions
![[url=https://pixabay.com/en/milky-way-universe-person-stars-1023340/]"Milky Way Galaxy"[/url] by Pixabay is in the [url=http://creativecommons.org/publicdomain/zero/1.0/]Public Domain, CC0[/url]
Fundamental forces are responsible for every physical event in our universe - from the nuclear fusion in the sun to the gravitationally driven motions of the bodies in the heavens.](https://www.geogebra.org/resource/B9XmF37h/6VNqotpR4kiBBsOl/material-B9XmF37h.png)
By most people's count, there are four fundamental forces or interactions in nature. By doing some fancy math, or considering the interactions at higher energy levels, some of them may be viewed as being combined or unified. For now, however, we will speak of four forces. They are the:
- Gravitational force - a force responsible for an attraction between any two objects based on their energy content. Since energy for massive objects is equivalent to mass via Einstein's famous E=mc2 equation, we generally view this force as one that draws masses together. In future discussions, however, you'll see that the mass of ordinary matter itself is mostly made of the energy holding that matter together and does not really speak of how much matter is present.
- Electromagnetic force - a force responsible for interactions among objects based on their charge and relative motion.
- Strong force - a force which acts to hold the protons and neutrons together in the nucleus and also is responsible for, among other things, holding quarks together to form those protons and neutrons.
- Weak force - a force which is responsible for radioactivity and nuclear fission, or interactions among particles in nature that wear the label "fermion", something which we'll discuss in third semester.
Gravitational Force
Gravity is one of the fundamental forces in nature. There is a gravitational vector field (an invisible force field that acts on mass) at every point in our universe. Its magnitude is small at locations in interstellar space, and it is unbelievably large near a black hole. Its strength is moderately-valued near earth's surface - the only place we've experienced gravity. The force that is felt by a mass in a gravitational field is given by From this equation we can see that the units associated with the gravitational field are newtons per kilogram. A field of strength g=1N/kg means that a one kg mass will feel one newton of force when immersed in the field and that therefore a 2kg mass will feel 2N of force, etc.
While units of N/kg are perfectly suited to describe these gravitational fields, a newton can be written as This means the field can also be attributed units of meters per second squared. Either one is fine to use.
The force that gravity exerts on an object near earth's surface is often called weight. Many physics textbooks - perhaps even the one you studied in high school - define it as such. I am going to swim against the mainstream and tell you that equating weight and gravitational force is a bad idea. Let me tell you why: Because it will lead to all sorts of misunderstanding about terms like weightlessness and about how scales work and about how we can actually feel heavier or lighter (and measure a different weight) in circumstances where gravity has not changed such as riding in an elevator, or while on a ride at an amusement park, or even when we stand still on a rotating planet... and all of us do that.
Rather than calling the force of gravity weight (there is another force that is rightfully equated to weight), I'll just call it what it is - the force of gravity, or perhaps the gravitational force. I'll use the symbol to describe it. This force of gravity is easily quantifiable in terms of mass m and the constant term that we've used before for earth's gravitational field strength near its surface (or anywhere else in the universe) :
If you are curious and want to know what gravity really is, I'll reveal what might come as a surprise. In a physics sense we still do not know! We can certainly quantify it and speak intelligently about it in certain ways, but we still do not know how it works. This might change in the next few years, but for now it is the last of the fundamental forces in nature for which we don't have a proper theoretical understanding... or one that just does not fit with the rest of nature.
While we have recently measured gravitational waves, the proper theoretical understanding and the detection of the mediating particles called gravitons, is still lacking. We need to understand the mechanism by which earth knows to pull on you and you know to pull back. That part is still unknown. Rest assured, however, that what we do know about gravity still allows us to successfully launch rockets to space, place satellites in orbits, predict the orbits of comets and planets, etc. Notice that I did not mention things like the orbits of stars around a galactic core. That's because at this moment in history our understanding of gravity does not match with what we see in those situations. This mismatch has led us to propose other missing (or dark) matter and exotic forms of energy (dark energy).
But back to the practical part: Since is the product of a mass in kilograms (kg) and a field strength that may be written with units of acceleration ( ) , or as force per unit mass (N/kg), this product is in units of Newtons. By definition, There's that funny triple equal sign denoting equivalence again.
Please note that while is certainly a vector that acts in the direction of , that often it is defined as a magnitude with the reader expected to understand in problems dealing with terrestrial scenarios that its direction is that of gravity - downward on earth's surface. We will be using this force in problems along with other forces, and in such instances it must be written as a vector.
We are to understand when we see that the field is always present near earth's surface, but to be experienced, a mass must be present. The "experience" that the mass has is one of being tugged downward toward earth's center. What's important is to recognize that the field is actually there even when the mass is not there to have the experience. This is the tree in the forest making a sound apart from anyone hearing it.
GRAMMATICAL ASIDE:
Please note the correct grammatical rules for units: The man Isaac Newton obviously capitalized his name. The unit that commemorates his life's work is written as a word in lower case (40 newtons of force), and the unit symbol is capitalized (40 N of force) since it was named after the man. This should be contrasted with other units we use such as meters or seconds, which do not come from associated names of scientists, and which therefore have unit symbols that are written in the lower case such as m, s, etc. One exception to this rule is the liter. It is generally written with an 'L', even though there was no Mr or Mrs Liter. But that's because a lower case L can easily be mistaken for a number one (1). Imagine writing 245 liters and writing 245L versus 245l. I actually typed a lower case L there, but can you tell?
The Unfortunate Naming of Forces
We have a case of unfortunate naming of forces used in physics. Why I call it unfortunate is that forces in nature are always between two objects. It'd be nice to call forces by names which indicate the two objects involved. We don't. When a box rests on a table, the common name for the force exerted on the box (to support it) by the table is the normal force. It would be easier to understand if we called it something like force on the underside of the object by the surface on which it rests, but you can see the inefficiency in doing this. Maybe "force on box by surface" would work. Regardless, that's what we should do, but don't tend to in practice.
Naturally the names of forces often come from ordinary language - words like gravity or friction force, so we don't have the liberty of just changing them. Therefore it'd be useful at least when we're thinking of these forces to think clearly of forces using the on/by notation. So the normal force is force on the block by the tabletop, and the gravitational force on the block is the force on the block by planet earth.
When we discuss Newton's third law pairs, you'll see that finding the paired force just requires reversing these labels.
What we call the third law pairing of the upward normal force is the downward force on the tabletop by the block and the pairing for the downward force of gravity on an object by earth is the upward force of gravity on planet earth by the object. Whether easy to measure like the first case, or not, as in the second case, these forces are always present.
If it's hard to convince you that the block pulls gravitationally on the planet, consider something like a much bigger "block" such as the moon in orbit around our planet. The earth pulls the moon toward itself via gravity, and the moon pulls back on the earth equally and oppositely. That is a third law pair. The force on earth by the moon can be easily demonstrated just by taking note that the earth's ocean tides are a consequence of this pull. The tides always face the moon with a slight delay due to the viscous flow of the water. There are actually two high and two low tides around the planet. Both the side closest to the moon and the one farthest from the moon are high, the orthogonal sides are low tides, but to understand this we will need to have discussions later about non-inertial reference frames.
Universal Gravitation
Newton is famous for having first shown - along with the calculus that he invented - that the gravitational force in general (not just near earth's surface) must be an inverse square law. Such forces drop off as the square of the distance separating the objects. Newton realized gravity needed such a form in order for calculated planetary orbits to follow the paths that Johannes Kepler had experimentally measured and described just a few decades before Newton's time. The law describing this inverse square gravitational force is called Newton's law of universal gravitation, and looks like this:
This equation allows us to calculate the force of gravity on object A due to object B's gravitational pull. The force depends on a constant G called the universal gravitational constant. In the expression, the unit vector's whole purpose is to indicate the direction of the force of gravity that object A experiences. The unit vector is one pointing toward object A from object B, but the minus sign in the beginning of the expression reverses that, so that we are to understand that the force is directed toward object B from object A, which is exactly what we expect for an attractive gravitational force. The rest of the expression indicates the magnitude of the force. The r2 in the denominator is the magnitude squared of the vector that goes to A from B, so it's a scalar quantity.
If you think about those subscripted vectors they really are representing the difference vectors of the two objects' positions. It is for this reason that the gravitational force law is often written just in terms of the individual objects' position vectors. Writing it in this alternate way makes understanding the individual terms easier, but at the loss of the "look" of an inverse square law. To rewrite the equation we use . That means , and the original term in the denominator is just .
Making those substitutions leads to the alternate form of Newton's law of universal gravitation:
This is the form that I will emphasize with you in class because it only contains position vectors of each object involved in the attraction. Keep in mind, however, that while it looks like an inverse cube law, we saw that written in more compact (but sometimes confusing) notation, it really is an inverse square law. We will see many inverse square laws through the semesters we spend together that all owe their form to the fact that the universe has three spatial dimensions.
Interactive Graphic of Newton's Law of Universal Gravitation
How to find the gravitational field
We know that gravitational fields affect masses by How does this relate to the universal law of gravitation? Maybe a good place to start is to recognize that the mass in the equation here is the one experiencing the force, but the gravitational field is due to other nearby or distant masses. So there must be a mass responsible for and a different mass (the one in the equation) experiences a force. If we divide out the mass that is experiencing the force, we can get the field on its own, or
If you look back at the law of universal gravitation, it has both the mass that's experiencing the force (m1) and the one responsible for it (m2), in the same expression. If we follow the same line of reasoning and divide out the one experiencing the force (m1), what we should get is an expression for the field due to mass 2 at the location of mass 1. Let's do this by finding the gravitational field at a point A, where we can imagine the first mass to be located, due to the other mass or masses at locations Given multiple masses we need a sum of expressions that look like this:
The gravitational field above with its subscripts should be read "the gravitational field at A due to mass i".
An easy simplification is to assume we want to find the gravitational field at a point A, which we can define as the origin if we haven't been told to do otherwise. This will simplify the above expression in the sense that
If we pay careful attention to the signs (the one in the magnitude doesn't matter, but the one in the numerator does), we get for the gravitational field at point A due to masses i:
Applying the Law of Universal Gravitation
Gravity is such a weak force that we only become aware of it when extremely large masses are involved. In reality, every object and person on the planet is attracted toward every other object and person via gravity, but it's never noticed. In fact, a light gust of wind is many orders of magnitude larger than the gravitational force pulling two people in conversation together.
The law of universal gravitation, like Coulomb's law which will be mentioned below, really only applies to spherically symmetric objects. In order to correctly calculate the gravitational force between two objects that aren't spherical (like the two people in conversation), we'd really need to consider the shape of the objects so we know how the mass is distributed in space and then we'd need to perform a volume integral to find the force. Luckily for us, since any significant gravitation tends to be due to planets or stars, and since gravity itself crushes such objects into spherical shapes, at least one of the involved objects is generally spherical. But what if I wanted to know the force between you and the planet? You are not spherical. It turns out that since you are so small as compared to the distance from earth's center (average location of mass) to your own center, it doesn't really matter that you aren't spherical. So calculating the force magnitude between you and the planet would be where the radius of the earth in the denominator represents the distance from your center to earth's center.
We may approximate gravitational forces between non-spherical objects, but it will always only be an approximation. The magnitude of the force between two people in conversation could be approximated using where r would be measured from the center of mass of one person to the center of mass of the other.
We will learn to correctly account for the shape of objects involved in such inverse squared force laws as we tackle such problems during second semester in our studies of electrostatics, which we will only briefly discuss right now.
Electrostatic Forces
There are two forces that tell us how charged objects behave in the vicinity of other charged objects or due to the fields created by other charged objects. These will be extensively discussed in the second semester of physics. For now I just want to mention the force between spherical distributions of charges (how fundamental particles behave). The law looks mathematically identical to the law of universal gravitation. The law is called Coulomb's force law and looks like this:
The terms q are the net charges residing on each object, and k is a different constant. The only difference otherwise is the missing minus sign that tells us that charges repel when they have the same sign - rather unlike gravity which attracts when both masses are positive (the only type in nature). The constant in this equation is where the unit C stands for Coulombs of charge.