Understanding the World: Newtonian Mechanics and Dimensional Analysis
Personal Reflections on Physics and Learning
Note: This essay was prepared with the research assistance and ghostwriting of ChatGPT 4.0. No LLM AI were harmed in the process, although I felt inclined to threaten them from time to time. I apologize for any errors in the document; they are mine, not the AI’s.
Author's Preface
I studied physics in high school up to the 13th grade (yes, we had a 13th grade), and later I took a second-year course in physics at community college. I may have been a standout in that course, but mainly in skipping class. My interest in further math and science waned when I attended university a few years later. However, as I got into practical matters later in life, I found that I could grasp physics concepts fairly well, as long as I was interested and willing to put in the necessary hours.
Over the years, I haven't met that many physics graduates—just a few, perhaps. Interestingly, I didn't notice any exceptional intelligence among them, but I did recognize that there is a specialized sort of intelligence required for this field. More importantly, what stood out to me was their work ethic—a level of dedication that often made me feel ashamed of my own lack of perseverance. IQ might be a good measure of general intelligence, but I think it's a simplistic concept that doesn't fully capture the complexities of human ability.
Getting through an undergraduate program in physics certainly requires specialized intelligence and determination, but to be a true physicist—one who pushes the boundaries of the field—requires rare and quite astonishing gifts. I have read about such individuals and even worked with a former professor of physics, who is still a friend. While he is highly knowledgeable in physics, he isn't one of those incredibly gifted physicists you read about. I've never met one of those, and I imagine that, like everyone else, they put their pants on one leg at a time. They may not be exceptional outside of their specialty—but then, I wouldn't know, having never met one.
When I first studied physics, I didn’t put much effort into it. It wasn’t until later in life, when I started trying to understand practical things, that I began to see the value in these concepts. I may have encountered dimensional analysis during my schooling, but if so, the idea didn’t take root. Years later, I encountered—or perhaps re-encountered—this concept and realized its significance. I now think of it as an ontological idea, something that deals with the very nature of being.
Initially, I saw physics as a collection of disconnected formulas to be memorized and used as needed. While that’s partly true, I’ve since discovered that there is a deeper organizational principle: these formulas are all connected through dimensional analysis. Simpler formulas define more dimensionally complex concepts, which in turn can be used to define even more complex ones.
I once used a trial version of the Maple software to try to get this all straight, before turning to ChatGPT years later.
This document represents my attempt to put everything together into a coherent and integrated framework. It may contain errors and oversimplifications. I realize it might be too nerdish for most people and not technical enough for the real nerds in physics and engineering. But for those who are curious and want to start making sense of how the physical world works, I hope this serves as a helpful introduction and is not misleading.
Introduction
Newtonian mechanics, also known as classical mechanics, is a branch of physics that deals with the motion of objects and the forces acting upon them. Developed by Sir Isaac Newton in the 17th century, this framework provides a set of mathematical laws and principles that explain how objects move in space and time, how forces affect their motion, and how energy is transferred and conserved. It's called "classical" because it forms the foundation upon which much of modern physics is built, although it has since been supplemented by more advanced theories like quantum mechanics and relativity.
Why Should You Care? Understanding Newtonian mechanics is crucial because it applies to the everyday world around us. Whether you're driving a car, throwing a ball, or even just walking, the principles of Newtonian mechanics are at work. By understanding these principles, you can better grasp how things move and interact, which is essential in fields ranging from engineering and architecture to sports and medicine.
This essay will look nto the fundamental dimensions used in physics—space, time, and mass—and explore how they combine to form the building blocks of Newtonian mechanics. We’ll examine key equations that describe different aspects of the physical world, using dimensional analysis to show how these concepts are interconnected.
Defining the Dimensions
In the context of physics, dimensions are the fundamental building blocks that describe physical quantities. The three primary dimensions we will work with are space, time, and mass. These dimensions form the foundation for all physical quantities in Newtonian mechanics.
Space: Space represents the extent of an object or the distance between objects. It is measured in units like meters (m) and can be expressed in terms of length, width, height, area (length * width), or volume (length * width * height).
Time: Time is the progression of events from the past to the present and into the future. It is measured in seconds (s) and is used to describe how quickly or slowly something happens, such as in the measurement of speed or acceleration.
Mass: Mass is the measure of the amount of matter in an object. It is measured in kilograms (kg) and is a fundamental quantity that describes an object's resistance to acceleration when a force is applied. Mass is also related to energy, as expressed in Einstein's equation
E = M * C^2, but in everyday life, we typically think of mass as the "weight" of an object (though technically, weight is a force).Caveat: The note on Einstein's equation
E = M * C^2being related to energy and mass is accurate for contextual understanding, although it's not directly applicable to Newtonian mechanics, which is acknowledged.
These three dimensions—space, time, and mass—are combined in various ways to describe more complex physical phenomena, such as velocity, acceleration, force, and energy. Understanding how these dimensions interact is key to understanding the principles of Newtonian mechanics.
Key Equations: Connecting the Dots
Now that we understand our basic building blocks—space, time, and mass—let's explore how they connect. Each of the following equations helps us describe a different aspect of the physical world. Before we dive into the explanations, we’ll first look at the fundamental dimensions involved in each calculation.
1. Area
Dimensions: Space * Space
Formula:
area = length * widthExplanation: Area is like the size of the surface of a table. To find the area, you multiply the length of the table by its width. Both length and width are measurements of space, so area tells us how much space is covered.
2. Volume
Dimensions: Space * Space * Space
Formula:
volume = length * width * heightExplanation: Volume is how much space something takes up in three dimensions. Imagine filling a box with water. The volume tells you how much water the box can hold. You find it by multiplying the length, width, and height of the box.
3. Displacement
Dimensions: Space
Formula:
displacement = final_position - initial_positionExplanation: Displacement is the straight-line distance between where something starts and where it ends up. To understand displacement, we first need to understand position:
Position is a point in space where something is located. You can describe position using coordinates, like saying a ball is at (2, 3, 4) in three-dimensional space.
Final Position is where the object ends up, and Initial Position is where it starts.
So, if you start at position (2, 3, 4) and end up at position (5, 7, 6), your displacement is the straight-line distance between these two points. Displacement cares only about the change in position, not the path taken.
4. Velocity
Dimensions: Space / Time
Formula:
velocity = displacement / timeExplanation: Velocity is a way of saying how fast something is moving and in which direction. It’s calculated by dividing how far you’ve gone (displacement) by how long it took (time). So, if you run 10 meters in 5 seconds, your velocity is 2 meters per second.
5. Acceleration
Dimensions: Space / Time^2
Formula:
acceleration = change in velocity / timeExplanation: Acceleration is how quickly something speeds up or slows down. If you’re riding a bike and start pedaling faster, your velocity increases, and so does your acceleration. It’s like how much you’re changing your speed every second. If your speed doesn’t change, your acceleration is zero.
Caveat: The formula acceleration = change in velocity / time is correct, but it might be better clarified as acceleration = (final_velocity - initial_velocity) / time for complete precision. The current explanation is not incorrect but could be more explicit.
6. Energy
Dimensions: Mass * (Space / Time)^2
Formula:
energy = 0.5 * mass * velocity^2Explanation: Energy is what makes things move or change. When you kick a ball, you give it energy. The amount of energy depends on the ball's mass (how heavy it is) and how fast it’s moving (velocity). The faster or heavier the ball, the more energy it has. This formula specifically describes kinetic energy, which is the energy of motion.
7. Momentum
Dimensions: Mass * Space / Time
Formula:
momentum = mass * velocityExplanation: Momentum is the quantity of motion an object has. The more mass an object has and the faster it’s going, the more momentum it has. For example, a heavy truck moving at a high speed has a lot more momentum than a light bicycle moving at the same speed. Momentum is important because it tells us how difficult it is to stop a moving object.
8. Density
Dimensions: Mass / Space^3
Formula:
density = mass / volumeExplanation: Density tells us how much stuff (mass) is packed into a certain amount of space (volume). Imagine two boxes of the same size: one filled with feathers and the other with rocks. The box with rocks is denser because it has more mass in the same amount of space.
9. Force
Dimensions: Mass * Space / Time^2
Formula:
force = mass * accelerationExplanation: Force is what makes an object move or change direction. When you push or pull something, you’re applying a force. The amount of force depends on how much mass the object has and how quickly you want it to change its velocity (which is acceleration). For example, it takes more force to push a heavy object than a light one to get them moving at the same speed.
10. Pressure
Dimensions: Mass / (Space * Time^2)
Formula:
pressure = force / areaExplanation: Pressure measures how much force is applied over a certain area. If you press your finger on a table, the force you apply is spread over the area of your finger. The smaller the area, the more pressure you exert. This is why stepping on someone’s foot with a high heel hurts more than with a flat shoe—the force is concentrated on a smaller area, increasing the pressure.
11. Work
Dimensions: Mass * Space^2 / Time^2
Formula:
work = force * displacementExplanation: Work is what happens when you apply a force to move something over a distance. If you push a box across the floor, you’re doing work on the box. The amount of work depends on how much force you use and how far you move the box. If you push harder or move the box further, you do more work.
Note on the Dot Product: When force and displacement are in the same direction, you can simply multiply them to find the work. However, if the force is not in the same direction as the displacement (for example, if you're pushing at an angle), you only count the part of the force that acts in the direction of the displacement. This is where the concept of the dot product comes in. For simplicity, think of it as multiplying the force by the displacement and then by the cosine of the angle between them to get the correct amount of work done in the direction of the movement.
12. Power
Dimensions: Mass * Space^2 / Time^3
Formula:
power = work / timeExplanation: Power is the rate at which work is done. It tells us how quickly energy is being used or transferred. For example, if two people do the same amount of work but one does it faster, that person is using more power. Power is important because it helps us understand how efficiently energy is being used.
13. Final Velocity
Dimensions: Space / Time
Formula:
final_velocity = initial_velocity + (acceleration * time)Explanation: Final velocity is the speed and direction an object is moving after a certain amount of time has passed, considering any acceleration. If you start from a standstill and start accelerating, this formula helps you figure out how fast you’ll be going after a certain period of time. It’s like knowing how fast you’ll be driving if you keep pressing the gas pedal.
Summary
In this essay, we explored the basics of Newtonian mechanics and how dimensional analysis provides a deeper understanding of the connections between various physical quantities. By breaking down complex concepts into their fundamental dimensions—space, time, and mass—we can see how these building blocks combine to form the foundation of classical physics.
While my initial experiences with physics were marked by a lack of effort and enthusiasm, I’ve come to appreciate the elegance and interconnectedness of these ideas. Although the material presented here may contain errors and oversimplifications, it represents my attempt to make sense of the physical world and share that understanding with others. For those who, like me, have found themselves grappling with these concepts later in life, I hope this essay offers a helpful starting point on the journey to deeper understanding.