- 3372 - PHYSICS - the Theory of Everything? The fact that Newton could have easily written F = ma instead of F = (dp/dt), perhaps Newton actually anticipated “special relativity“. Insight into the workings of our Universe, along with the development of invaluable tools for problem solving, embedded in the seemingly simple equation behind Newton’s second law: F = ma.
--------------------- 3372 - PHYSICS - the Theory of Everything?
- You thought math was easy. Once defined it was straight forward and consistent. Velocity is a simple example. You take the distance and divide by the time it takes to travel that distance. That ratio is the average speed. You would not thing that this simple ratio would lead us to a “Theory of Everything”
-
- There are few ideas in science that are as captivating as a Theory of Everything. This theory would presumably brings together all our knowledge of fundamental physics. Even as we understand more, our view of nature is always incomplete, unable to grasp what lies beyond the limited reach of our instruments and our imagination.
-
- This Theory of Everything does not mean that it will be able to predict all things that will happen. This Theory, despite its grandiose name, means a theory that describes how the particles of matter, the most fundamental building blocks of the stuff that makes the Universe (us included), interact with one another. It is how atoms are built and where they came from.
-
- The theory’s goal is to show that the four fundamental forces of nature, gravity, electromagnetism, and the strong and weak nuclear forces, are, in fact, manifestations of a single, ultimate force, hidden beneath the appearance of difference.
-
- The Theory rests on the assumption that if we could only perceive the depths of physical reality with enough clarity, we would see all forces as “oneZ“. We are almost there.
-
- We do not see this unity today because it is only manifest at extremely high energies, well beyond what we can perceive even with our most powerful machines. The four forces are like four rivers that join upstream to become three, then join again further upstream to become two, and finally, join once more to become a single river at the source of it all.
-
- The problem is that the very notion of a Theory of Everything, even if restricted to the world of subatomic particles and their interactions, rests on an incorrect understanding of how science works and what it can do.
-
- Physical theories are data-driven, based on a painstaking process of empirical validation; any hypothesis must be vetted by experiments before it is accepted. And even when it is accepted, and the hypothesis is sometimes elevated to the level of a “theory,” such as in the general theory of relativity or the theory of evolution, this acceptance is always temporary.
-
- A physical theory can only be proved wrong, never right, at least in any permanent sense. This is because every theory is necessarily incomplete, always ready for updates as we learn more about the physical world.
-
- What makes science exciting is less the success of a theory and more the moment when a theory fails. That’s when “the new” happens.
-
- As the physicist Werner Heisenberg, of Uncertainty Principle fame, once wrote, “What we observe is not Nature itself but Nature exposed to our methods of questioning.” What we can say about Nature depends on how we measure it, with the precision and reach of our instruments dictating how “far” we can see.
-
- Therefore, no theory that attempts to unify current knowledge can seriously be considered a “final” theory given that we cannot ever be sure that we aren’t missing a huge piece of evidence.
-
- How are we to know that there isn’t a fifth or sixth force lurking out there in the depths? We cannot know, and quite often, hints of a “new force” are announced in the media. To put it differently, our perennially myopic view of nature precludes any theory from being complete. Nature doesn’t care how compelling we think our ideas are.
-
- The best that we can do is to keep searching for more encompassing explanations of natural phenomena, possibly even achieving some level of unification as we go along.
-
- The history of physics has a few of these, such as Newton’s theory of gravity bringing together terrestrial and celestial motions, and electromagnetism that, in the absence of sources such as electric charges and currents, shows a beautiful unification between electricity and magnetism.
-
- Einstein spent the last two decades of his life searching for a unified theory of gravity and electromagnetism and failed.
-
- The “Grand Unified Theory” proposed in 1974 by Sheldon Glashow and Howard Georgi to unify electromagnetism with the two nuclear forces also failed to be vindicated, including many of its more recent extensions. This does not mean that such unifications are impossible or wrong. They may even work, although current evidence is slim.
-
- The moral of the history here is not that unification ideas are useless or impossible but that the notion of achieving a final unification is. Science is an ongoing process of discovery that is fueled by our lack of answers.
-
- The very process of discovery leads to more unknowns, not fewer. As science advances, it creates new lines of questioning that feeds our curiosity and creativity. How awfully boring if, one day, we arrived at a complete fundamental understanding of matter and its interactions. It is much better to look at the world through our myopic eyes, always wondering what lies beyond what we can see.
-
- Here is a “simple” example of how science expands our understanding. “ F = m*a“. Force equal mass time acceleration. What seems like a simple, three-letter equation contains an enormous amount of information about our Universe. The physics within it is vital for understanding all of motion, while the mathematics is the most important application of calculus to our reality.
-
- This equation can even lead us to ‘relativity“, and remains eternally useful to physicists of all levels. If there’s one equation that people learn about physics, not Einstein’s “E = m * c2” , it is Newton’s “F = m * a“.
-
- In isolation, any system, whether at rest or in motion, including angular motion, will be unable to change that motion without an outside force. In space, your options are limited, but even in the International Space Station, one component (like an astronaut) can push against another (like another astronaut) to change the individual component’s motion.
-
- Newton put forth three laws of motion describing these bodies:
-
---------------------- An object at rest remains at rest and an object in motion remains in constant motion, unless acted upon by an outside force.
-
---------------------- An object will accelerate in the direction of whatever net force is applied to it, and will accelerate with a magnitude of that force divided by the object’s mass. a = F / m
-
---------------------- Any action must have an equal and opposite reaction. If anything exerts a force on any object, that object exerts an equal and opposite force on the “thing” that pushes or pulls it.
-
----------------------- An object at rest will remain at rest, unless acted upon by an outside force.
-
- If you can measure the mass of your object and how it’s accelerating, you can use F = ma to determine the net force acting on the object.
-
- If you can measure the mass of your object and you know (or can measure) the net force being applied to it, you can determine how that object will accelerate. This is particularly useful when wanting to determine how an object will accelerate under the influence of gravity.
-
- If you can measure or know both the net force on an object and how it’s accelerating, you can use that information to determine your object’s mass.
-
- Other famous examples include Hubble’s law for the expanding Universe, which is
v = H * r . The recession speed equals the Hubble constant multiplied by distance
-
- Ohm’s Law, which is V = I * R . Voltage equals current multiplied by resistance.
-
- The way to take F = ma to the next level uses acceleration is a change in velocity (v) over a time (t) interval.
-
- This can either be an average acceleration, such as taking your car from 0 to 60 mph , or an instantaneous acceleration, which asks about your acceleration at one particular moment in time.
-
- We normally express this as a = Δv/Δt, where the “Δ” symbol stands for a change between a final and an initial value, or as a = dv/dt, where the “d” denotes an instantaneous change.
-
- Velocity itself is a change in position (x) over time, so we can write v = Δx/Δt for an average velocity, and v = dx/dt for an instantaneous velocity.
-
- The relationship between position, velocity, acceleration, force, mass, and time is profound.
-
- Given that we live in a three-dimensional Universe, every one of these equations is actually three equations: one for each of the three dimensions ( x, y, and z directions) present in our Universe.
-
- The fact that F = ma is a three-dimensional equation. One of the remarkable things about these sets of equations is that they’re all independent of one another.
-
- What happens in the x-direction — in terms of force, position, velocity, and acceleration — only affects the other components in the x-direction. The same applies for the -y- and -z- directions as well: What happens in those directions only affects those directions.
-
- This explains why when you hit a golf ball on the Moon, gravity only affects its motion in the “up and down” direction, not in the side to side direction. The ball will continue on, constantly, with its motion unchanged; it’s an object in motion with no external forces in that direction.
-
- Instead of treating objects as though they’re idealized point masses, we can consider masses that are extended objects. Instead of treating objects that only move in lines, accelerating at a constant rate in one or more directions, we can treat objects that orbit and rotate.
-
- Through this procedure, we can begin to discuss concepts like “torque” and moment of “inertia“, as well as angular position, angular velocity, and angular acceleration. Newton’s laws and equations of motion all still apply here, as everything in this discussion can be derived from that same core equation: F = ma.
-
- Calculus and Rates: Velocity is the rate at which your position changes. It’s a distance over a time, or a change in distance over a change in time, and that’s why it has units like meters per second or miles per hour.
-
- Similarly, acceleration is the rate at which your velocity changes. It’s a change in velocity over a change in time, and that’s why it has units like meters per second^2: because it’s a velocity (meters per second) over a time (per second).
-
- If you know:
-
----------------- Where something is right now
-
----------------- What time it is right now
-
----------------- How quickly it’s moving right now
-
----------------- What forces are and will be acting on it
-
- Then you can predict what it will do in the future. That means we can predict where it will be at any point in time, including arbitrarily far into the future.
-
- Newton’s equations are entirely deterministic, so if we can measure or know what an object’s initial conditions are at some time, and we know how that object will experience forces over time, we can predict precisely where it will wind up.
-
- While planetary motion may look simple, it’s governed by a second-order differential equation relating force to acceleration. The difficulty in solving this equation should not be underestimated.
-
- This is how we predict planetary motion and comet arrivals, assess asteroids for their potential to strike Earth, and plan missions to the Moon. At its core, F = ma is what we call a differential equation, and a second-order differential equation at that. Why?
-
- Because “second-order” means it has a second time-derivative in there: Acceleration is a change in velocity over a change in time, while velocity is a change in position over a change in time. Differential equations are their own branch of mathematics.
-
- A differential equation is an equation that tells you, assuming you know what your object is doing right now, what it will be doing at the very next moment. Then, when that next moment has elapsed, that very same equation tells you what will occur at the subsequent moment, and so on, forward to infinity.
-
----------------------- F = ma is one of those very hard differential equations.
-
- Spacetime can only be described if we include not only the position of the massive object, but where that mass is located throughout time. Both instantaneous location and the past history of where that object was located determine the forces experienced by objects moving through the Universe, making General Relativity’s set of differential equations even more complicated than Newton’s.
-
- Newton never said, “force equals mass times acceleration.” Instead, he said, “force is the time rate of change of momentum,” where momentum is the product of mass times velocity.
-
- These two statements are not the same. F = ma tells you that force, which occurs in some direction, leads to an acceleration of masses: a changing velocity over time for every mass that experiences a force. Momentum represent with the letter p, is the product of mass times velocity:
-
--------------------------- (p = m * v).
-
- Can you see the difference? If we change momentum over time, whether it’s with average momentum (Δp/Δt) or with instantaneous momentum (dp/dt), we run into an issue. Writing down F = ma makes the assumption that mass does not change; only velocity changes. This isn’t universally true, however, and the two big exceptions have been hallmarks of 20th-century advances.
-
- Rockets convert fuel into energy and thrust, expelling it and losing mass as they accelerate. As a result, F = ma is too oversimplified to be used to calculate a rocket’s acceleration.
-
- One is the science of rocketry, since rockets actively lose their mass (burning it and expelling it as exhaust) as they actively accelerate. In fact, the “changing mass, also” version of the equation, where both velocity and mass are allowed to vary over time, is known by many as simply “the rocket equation.”
-
- When a loss or gain in mass is occurring, it affects your objects motion, and how that motion changes over time as well. Without the mathematics of calculus and differential equations, and without the physics of how objects such as this behave in real life, calculating the behavior of a spacecraft powered by propellant would be impossible.
-
- The other is the science of “special relativity“, which becomes important when objects move close to the speed of light. If you use Newton’s equations of motion, and the equation F = ma to calculate how an object’s position and speed change when you apply a force to it, you can incorrectly calculate conditions that lead to your object exceeding the speed of light.
-
- If, however, you instead use F = (dp/dt) as your force law, the way Newton himself wrote it, then so long as you remember to use relativistic momentum (where you add in a factor of the relativistic γ: p = mγv), you’ll find that the laws of special relativity, including time dilation and length contraction, all naturally appear.
-
- When you’re at rest , a photon travels up-and-down between two mirrors at the speed of light. When you’re boosted by a rocket, the photon also moves at the speed of light, but takes longer to oscillate between the bottom and the top mirror.
-
- As a result, time is “dilated” for objects in relative motion compared to stationary ones.
-
- Based on this observation and the fact that Newton could have easily written F = ma instead of F = (dp/dt), that perhaps Newton actually anticipated special relativity. Insight into the workings of our Universe, along with the development of invaluable tools for problem solving, embedded in the seemingly simple equation behind Newton’s second law: F = ma.
-
- The idea of forces and accelerations will come into play every time a particle moves through curved spacetime; every time an object experiences a push, pull, or forceful interaction with another entity; and every time a system does anything other than remain at rest or in constant, unchanging motion.
-
- If you’re only going to teach one physics equation to someone, make it this one. With enough effort, you can use it to decode the workings of almost the entire Universe.
-
December 12, 2021 PHYSICS - the Theory of Everything? 3372
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ---
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
--------------------- --- Saturday, December 18, 2021 ---------------------------
No comments:
Post a Comment