This is part of the course is the most important part.
The later parts are implicit in it.
It's the intellectual heart of the course.
It is on Newtonian physics which is part of classical physics which also includes classical electromagnetism which is a subject for another course.
Newtonian physics includes
But the same thing about intellectual heart could be said about the concept of energy.
There are two intellectual hearts---but no matter.
First of all Newton's laws are pretty easy to write down: they are short and somewhat memorable.
Many people can even recite them in one form or another to some accuracy.
But they are NOT at all obvious. A matter of opinion, of course---but it's almost a universal opinion.
NO ONE ever finds those laws for themselves.
Even Isaac Newton (1642/1643--1727) did not find those laws from scratch: he was at the end of a long debate going back to Aristotle (384--322 BCE) and probably before.
Well they are the basic most elementary laws
But most motions we see in everyday life and in the wide world are at least somewhat complicated and often immensely complex.
Many forces and accelerations go on and continual feedback from the senses.
It's an immensely complex series of motions.
Each little element obeys Newton's laws, but it's hard to see the trees for the forest.
It's taken a long time for roboticists to make a robot walk even passingly well (see Wikipedia: Human Robot).
Even much more simple motions have their complications.
Taking dropping objects.
They all accelerate down with g=9.8 m/s**2 or so.
Except for air resistance.
But his critics argued that he was talking nonsense: there was no perfect vacuum according to Aristotle.
Galileo was thinking of precise underlying mathematical laws in ``a universe of precision'' (A. Koyre) that could be found by a combination of thought experiments on ideal systems and well controlled real experimentation that would take one beyond the complications that were everywhere.
His critics occupied a ``world of more or less'' (A. Koyre) in which there were no such precise laws---but only more-or-less laws.
One of these more-or-less laws was that a more-or-less constant velocity motion needed a more-or-less constant force.
Because in many cases that is more or less what you see.
They and Aristotle had a tough time with projectile motion and didn't think about perfect frictionless ice.
But even when you start looking for such precise mathematical laws, it is hard to find them.
Galileo and Rene Descartes only got part way to Newton's laws.
I think to find Newton's 2nd law for instance you have to have some clear idea of force and some clear force law: Newton had the law of gravity which he also discovered.
To have the idea of mass as a quantifiable resistance to acceleration by a force you need some idea of Newton's 2nd law and the role force plays in it.
In fact, the concepts utilized in Newton's laws (e.g., force, mass, even acceleration) cannot---in my opinion anyway---be completely defined without specifying the relationships among them.
So one kind of has to see the whole picture of Newton's laws come into focus as whole.
The bits are not completely definable apart.
But once you get the whole picture then everything works.
Note I put acceleration in the list of concepts that needed a Newtonian setting to be fully understood.
This is because you have recognize that acceleration not velocity needs a force for a cause.
An acceleration requires a continuing net force to exist.
Both acceleration and velocity in the above discussion are relative to inertial frames which we elucidate below.
You need also to recognize that acceleration is a vector.
A change in direction is a change in acceleration even if speed is unchanged.
Other things you need in Newtonian physics are geometry (including the all important straight line) and the time parameter.
This geometry is the geometry of Newtonian physics.
Newton may never have wondered about which geometry his laws applied to: he probably never thought about anything, but Euclidean geometry and nothing showed he was wrong in his time.
But he did wonder a bit about time (see the video).
He wondered if it did flow the same everywhere and at all times, but came to the conclusion that the simplest hypothesis was that it did---and that hypothesis consistent with all he knew.
Fortunately, time as measured by Newtonian physics is the time of periodic motions of pendulums and the cycles of astronomical bodies.
Credit: T.A.Rector, I.P.Dell'Antonio /NOAO /AURA/NSF. NOAO gives open permission for educational use. I wonder if Dell'Antonio is my old friend Iain from Harvard 1991--1993.
So in regard to space and time, there was no shock of the new with Newtonian physics.
That would come with the special relativity and general relativity of Albert Einstein (1879--1955).
``When I was young I despised all authority---and I have been punished for it by having been made into an authority myself.''---from memory. This is my favorite Einstein quote.
Download site: NASA Astronomy Picture of the Day: 1995dec19.
This picture was obviously taken in the first decade of the 20th century and by U.S. copyright law is now out of copyright. According to the informative, but not authoritative, source WebMuseum, Paris copyright in all other jurisdictions would have expired if the holder died more than 70 years ago.
There are two meanings to FUNDAMENTAL LAWS in physics:
The absolutely true exact physical laws of nature are sometimes called the theory of everything (TOE).
I happen to think that TOE is a misnomer.
There are principles which one believes (me among a huge herd I think) that should operate in alternate universes where different laws of physics apply: e.g., entropy (which has a special status in physics), consciousness, ethics, evolution.
But we are stuck with TOE for now.
Sometimes some of those laws are considered candidates for being FUNDAMENTAL LAWS in the first meaning and sometimes we know they can't be---or there is a debate.
For a long time, up to circa 1900, Newtonian physics was thought to be FUNDAMENTAL LAW or at least a candidate for that status.
We now know that it is not FUNDAMENTAL LAW.
As you go small to the atomic realm, Newtonian physics progressively fails and must be replaced by quantum mechanics.
As you go toward the vacuum speed of light, Newtonian physics progressively fails and must be replaced by special relativity.
As you go toward strong gravity cases like black holes or toward the whole of the observable universe, Newtonian physics progressively fails and must be replaced by general relativity.
This is about 1/4 of the original HST deep field image (1995dec18--28): a random empty speck of sky: size here is about 1/60 of a degree: the whole deep field was 2.7 X 2.7 arcminutes.
It is true color (as much as possible) and the blue galaxies must be very blue to still be blue after high redshifts???. There are hundreds of galaxies visible: some the vary faintest are as they were probably only a billion years after the Big Bang. Most may be from 6--9 billion light years or 6--9 billion years in look-back time.
Credit: NASA/HST. This image is the public domain as it was
and is not otherwise noted as copyrighted.
This is ``a'' download site, but not where I originally got the
image which was who knows.
Download site: Image:HubbleDeepField.800px.jpg. This is ``a'' download site, but not where I originally got the image which was who knows.
So there is still some work to do with our current FUNDAMENTAL LAW.
But within its realm of validity which stretches from above the atom level to large structures in space and from zero to very high velocities it is almost entirely adequate for describing what one observes.
Often so adequate that no reference to more general theories needs be made to explain any discrepancies which are usually due to observational error or limited control over the variables of any system.
Why not use the general theories for these cases always?
They are harder---immensely harder usually---to apply directly.
So when one doesn't have to, one doesn't.
One merely says---or actually doesn't bother to say---that Newtonian physics is the correct approximation for those more general theories for the case at hand and procedes calculating with Newtonian physics.
Newtonian physics is what I call a TRUE APPROXIMATE THEORY.
We know it's not FUNDAMENTAL LAW in either sense.
Perhaps, we will never know TOE.
But we can know TRUE APPROXIMATE THEORIES---and Newtonian physics is one of those.
Newtonian physics consists of Newton's 3 laws of motion, some definitions, inertial frames, non-inertial frames, force laws, and lots more---the list goes on and on as one goes into details and applications.
For the moment let us just say an inertial frame is a pretty commonplace frame of reference like the Earth's surface.
But an accelerating car (relative to the Earth's surface) is NOT an inertial frame.
Newton's 3 laws of motion are:
Newton's 1st law.
Alternatively, one could say a body is unaccelerated relative to an inertial frame unless acted on by a NET FORCE.
The two statements mean the same thing:
at rest or in straight-line, constant-velocity motion = unaccelerated
F_net is the net force,
m is the mass,
a is the acceleration,
HTML LINKED indicates a vector quantity in this case (though the links are real too).
The equation gives us the unit of force:
kg*m/s**2 is defined to be a newton (N). One newton = 0.22481 pounds approximately 1/5 pounds. It is not a big lot of force.Frequently, one just says `` F=ma'' when one means the 2nd law.
Force and acceleration are in fact VECTORS: they are quantities with both magnitude and direction.
Force is a physical relation between bodies or between a body and field of a force (e.g., a gravity or electric field) that can cause an acceleration and/or can balance (i.e., cancel) other forces and/or cause a deformation of a body.
Acceleration is a kinematic quantity evaluated using distances and times.
Mass is a scalar (i.e., it only has a magnitude) and is a measure of the resistance of a body to acceleration.
It is sometimes defined as the quantity of matter which is a secondary definition and in one perspective means no more than the first definition since one can quantify matter by its resistance to acceleration.
This is another way in which mass tells you the quantity of matter: i.e., the number of each class of particle.
Newton's 2nd law ( F=ma) illustrated.
Note that because acceleration occurs in F=ma that the 2nd law specifies the connection between forces and kinematics.
The 1st law is actually logically redundant: it is a special case of the 2nd law.
But for historical and conceptual reasons, we retain 1st law as a seperate law.
It's answer 1.
Newton's 3rd law illustrated.
The oceans were sailed west by Columbus, east by the Polynesians, and pyramids and cathedrals were build and all without knowing Newtonian physics.
The Giza Pyramids.
Khufu, c. 2520-2494 BC pyramid (Great Pyramid; Cheops' Pyramid); Khafre, c. 2520-2494 BC pyramid; Menkaure c. 2490-2472.
Credit: Digital Imaging Project of Mary Ann Sullivan, Bluffton College; download site Digital Imaging Project's Pyramid Gallery. The download site gives more information.
But you couldn't build a suspension bridge or a Moon rocket without Newtonian physics---and a lot of other cool stuff too, of course.
At some point in scientific and technological development you must give up on the purely empirical approach and look for fundamental laws or at least very general, basic laws.
You can then build up complex systems by understanding them in terms of these general, basic laws---plus a lot of empirical knowledge too.
Also Newtonian physics is truth---well approximate truth about the basics of motion---and truth is good to know---it enlightens our darkness.
Inertial frames are a special kind of frame.
They are NOT hard to understand and everyone really knows something about them---they know when they are NOT in one for example: e.g., a car making too sharp a turn and you feel that you are being thrown sideways: really you are just trying to move in a straight line.
To our best modern understanding, primary inertial frames are the frames that participate in the mean expansion of the universe (see raisinbread animation).
The CMB pervades space is quite measurable especially above the atmosphere.
Measurements of cosmologically distant objects can also be used to determine inertial frames.
Probably almost NO FRAME attached to a physical body with rest mass is truly an inertial frame.
Almost every body with rest mass is being accelerated by gravity and/or some other force.
To be explicit almost every large material body is rotating: clusters of galaxies, galaxies, stars, planets in orbit and on their axes, all things on planets.
Frames rotating relative to inertial frames are always NON-INERTIAL strictly speaking.
The frame of reference attached a merry-go-round is non-inertial.
Just rotating platforms with handels.
Great for physics demonstrations.
They used to be all over the place, but then a lot of them got removed as being too dangerous.
Of course, there are also amusement-park merry-go-rounds with horses and the like.
But some non-inertial frames are more non-inertial than others---to paraphrase George Orwell (1903--1950).
If a merry-go-round rotates once in 10 seconds, it is MORE non-inertial than if it went around in 20 seconds.
The Earth goes around once in about 24 hours = 86400 seconds.
Although radius counts too as we'll see later, the much longer period of Earth's rotation suggests it is more inertial than merry-go-rounds.
When this is NOT done it is for pedagogical (or sloppiness) reasons, not logical ones.
Newton's laws are just as valid in NON-INERTIAL FRAMES, but they are NOT referenced to them.
Objects thrown in the frame of the merry-go-round do NOT travel in straight lines in the horizontal even though no force acts in the horizontal.
Thus, they are accelerated relative to the non-inertial frame of the merry-go-round with NO force acting.
But they travel in straight lines in the relatively inertial ground frame, and so are unaccelerated in that frame and obey Newton's laws.
It's not just Newtonian physics that needs inertial frames.
Einstein and all modern physics still needs them.
It seems that physics is NOT just about the interactions objects in a neutral of background of space, but that space has active properties.
In relativistic physics, one would say spacetime has active properties.
It does seem weird though to have one of our basics physical concepts---inertial frames---being something we have difficulty---but not impossibility---in finding exact examples of.
But it's not really weird.
It's part of the procedure of formal science particularly physics.
Envision the ideal case and then add on the complications as perturbations.
In fact we can find approximately inertial frames adequate for most purposes.
The Earth is sufficiently INERTIAL for many purposes.
There are clever ways of doing this.
In discussing weather systems, one has to consider a NON-INERTIAL FRAME EFFECT called the Coriolis force.
They are not real forces.
They are non-inertial-frame effects that we treat by introducing them as forces.
Actually you are just trying to move in a straight line from an inertial frame perspective.
The ``gravity-like force'' you feel in rectilinear accelerated non-inertial frames is also a inertial force.
See the rotating parabolic disk animation which unfortunately is complicated by the parabolic shape of the disk for the bouncing particle---so four effects are going on at the same time centrifugal force, Coriolis force, gravitational force accelerating the particle, and collisions.
On the left, one sees the system from a rest frame.
On the right, one sees the system from the rotating frame.
The particle actually collides with the same point on the surrounding wall all the time actually.
Newton took the fixed stars as defining the primary inertial frame.
The fixed stars are just the stars which seemed for most of human history to be fixed---and that is what Newton hypothesized too.
The more physicsy definition offered was given above:
At least, I can't see any way it can't be.
There must be force laws or force formulae that exist independently of Newton's laws of motion.
It takes more laws than Newton's 3 laws of motion to make up Newtonian physics.
Of course, force laws do exist.
There are only 4 fundamental forces:
Einstein famously tried to find a UNIFIED FIELD THEORY that would show that gravity and the electromagnetic force are really one force---but he failed at least in finding a useful theory.
Still he set the agenda for physics: i.e., to find a UNIFIED FIELD THEORY---but for all the forces.
This search is now subsumed under the search for TOE.
In some sense the last 3 forces have been united in a very abstract way, but gravity has resisted unification---but there are speculative theories that may do the trick.
I'm sort of old-fashioned, but I'd rather stick to saying four fundamental forces.
When TOE comes along---and ``when come the Saints marching in''---I'll talk of the one fundamental force in its four basic manifestations.
The strong and weak nuclear forces are intrinsically quantum mechanical and can't be treated by Newton's laws.
Here we will just say that the strong nuclear force binds nuclei together.
The weak force turns up in some radioactive decay proceses.
The weak force has actually been united with the electromagnetic force to form the electroweak force.
But this unification is at such a deep level that it is not too relevant for many purposes.
The electromagnetic force is an immensely complex force.
It has many complex manifestations which are often given their own special names.
Although there are LONG-RANGE MANIFESTATIONS of the electromagnetic force (the macroscopic Coulomb [i.e., electrostatic] and magnetic forces)---which in the modern textbook jargon are called FIELD FORCES---the forces important for giving structure to solids are primarily contact forces.
If cut a board in half you just break (ideally) one layer of bonds, but that is enough: the two halves don't stick together.
Jump off the floor and the forces between feet and floor vanish.
The CONTACT FORCE NATURE of the most manifestations of the electromagnetic force are a consequence of the TWO SOURCES of the Coulomb force (electrostatic force).
Actually, Ben Franklin (1706--1790) assigned the names positive and negative to charge (HRW-507).
Caption: Portrait of Benjamin Franklin by Jean-Baptiste Greuse.
Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Benjamin_Franklin_by_Jean-Baptiste_Greuze.jpg.
Actually, Ben got it wrong.
Electrons, which are negative, are the most mobile charge carriers and it would have been mentally useful to have them positive since conventional current in circuits (which was decided on before anyone knew about electrons) assumes positive carriers.
This wasn't the only time Ben had a choice out of two options.
The unlike charges attract so strongly that they usually keep the world mainly neutral over macroscopic distances.
And neutral matter will NOT exhibit a strong, long-range (macroscopic) Coulomb force.
The positive and negative charges cancel macroscopically.
But at the microscopic level, positive and negative patches are everywhere.
At the microscopic level, quantum mechanical laws prevent exact cancellation.
But why is it a good thing that the electromagnetic force has complex manifestations?
Answer 1, I'd say. But you could argue for answer 3.
It has only one ``charge'': MASS.
This double function is just a coincidence in Newtonian physics.
general relativity the coincidence is explained---but we will never go into that esoteric point.
And mass always attracts mass: gravity is never canceled by having two flavors of mass.
Now gravity on Earth was always known. Newton didn't discover that.
Isaac Alien discovers gravity.
What Newton discovered was that gravity is universal: both on Earth and in the astronomical realm there was gravity obeying the same law.
The UNIVERSAL LAW OF GRAVITY.
The gravity force law (or Newton's law of gravity) which holds between ideal point masses is:
G M_1 M_2 F_12 = ------------------- R_12**2 where G = 6.6742*10**(-11) in MKS units (circa 2002) is the universal constant of gravity. M_1 is the mass of point mass 1. M_2 is the mass of point mass 2. R_12 is the distance between the masses. Notice this distance comes in as an inverse-square. We say that the formula is an inverse-square law and gravity is an inverse-square law force. F_12 is the force that 1 exerts on 2 and the force that 2 exerts on 1. The forces are directly on the line between the two objects and point in opposite directions. The gravitational forces are attractive always. No anti-gravity exists in the ordinary realm of physics, but there may be a cosmological anti-gravity that is discussed in IAWL Lecture 31: Cosmology. By the way, the MKS unit of force is the newton: N=kg*m/s**2: 1 N = about 1/5 lb. The gravity force law gives force in newtons when MKS units are used consistently.
Now POINT MASSES are one of those idealizations that physicists love.
They may not exist in any sense---but then again they may: i.e., black hole singularities, but black holes can't be treated by Newtonian physics in any case.
But the gravity force law is actually of the greatest use.
Gravity between objects of general shape.
Gravity between objects with simplifying conditions.
What is the gravitational force between two, 1 kilogram spherically symmetric masses held 1 meter apart?
Gravity is a very weak force.
The last figure illustrates that the gravitational force between human size and even much larger objects is usually unnoticeable.
G M_1 M_2 F_12 = ------------------- R_12**2
In a sense, gravity drops off rapidly with distance because of the 1/R**2 factor which makes it an INVERSE-SQUARE LAW FORCE.
This behavior is shown in cartoon in the figure below.
Some simple function behaviors.
But it is still called a long-range force or a BODY FORCE because it interacts with the whole body not just the surface as CONTACT FORCES do.
Gravity is much more long range than any contact force.
Question: If we double one mass, the force:
Answer 2 is right.
Question: If we double both masses, the force:
Answer 3 is right.
Recall all masses attract.
Question: Why don't we in this room feel mutually attracted?
Answer 3 is right.
Gravity can actually be measured for human-sized objects, but it takes very sensitive apparatus.
The acceleration g due to gravity on the Earth's surface.
Accelerating downward under the force of gravity alone.
In fact the whole kinematics of falling objects should be the same regardless of mass---if you can neglect air resistance.
Drop a chalk brush and a coin: air resistance relatively small.
Then drop a brush and a sheet of paper: air resistance not relatively small for the sheet of paper.
Air resistance causes falling to reach a terminal velocity.
Examples of terminal velocities.
In free fall you feel weightless, but this is not because gravity has turned off.
Gravity is just pulling you down atom by atom and you arn't resisting, and so there is no internal stress or pressure to resistance.
Standing up and resisting gravity is a different matter.
Standing up and resisting gravity.
Now not only you, but the atmosphere, the oceans, and the solid Earth must stand up under gravity pulling it down.
Only PRESSURE FORCES can withstand the self-gravity of dense, massive bodies like planets and stars.
In normal gases (but not degenerate gases) they are caused by atoms and molecules bouncing off of one and another: the electromagnetic force is the actual interaction.
In liquids and solids the atoms and molecules are pressed into contact.
The electromagnetic forces which give atoms their structure strongly resist compression of atoms from their unbound size.]
The pressure force will not provide strength for complex structures.
For example, water has a strong pressure force: you CANNOT compress it easily.
But water cannot resist shearing forces very well: drops can hold a shape, but nothing much bigger.
Even solids will not resist a shearing force if their mass is too big for a shape to be sustained by inter-atomic bonds: i.e., they will act like fluids.
Inter-atomic bonds make a boulder keep its shape under planet-size gravity.
But a boulder as big as a mountain on a planet is flattened into a mountain: i.e., a small protuberance on the face of a planet.
The pressure force can hold up the super-big boulder's mass, but it will push sideways causing the boulder to ``flow'' sideways and slump done to being a mountain.
A boulder as big as a planet in space would be pulled into spherical shape.
The solid pressure resists collapse, but not shearing that leads to spherical symmetry.
We see the combined effect of self-gravity and pressure is to produce a body with nearly exact spherical symmetry.
There will be a few low protuberances (i.e., mountains, continents, etc.) and relatively small interior asymmetries due inter-atomic bonds strong enough to resist the relatively low pressures at the base of the protuberances.
There are TWO QUALIFICATIONS:
The centrifugal force is not a real force, but the tendency of bodies to move in a straight line. It is the thing that tends to throw you off playground merry-go-rounds. It increases with rotation rate.
You note that Saturn is obviously oblate with equatorial diameter (which is parallel to the bands and rings) is about 10 % larger than the polar axial diameter.
The defined oblateness is
(R_equator-R_polar)/R_polar=0.0979624 , where R_equatorial is the equatorial radius and R_polar is the polar radius. (Cox-295).
The oblateness is caused by the centrifugal force which is high for Saturn because of its fast rotation.
Saturn's deep interior rotation period relative to the fixed stars ( sidereal rotation period) is 0.44401 days or 10.656 hours (Cox-295).
Pressures at various depths in the Earth.
Besides pressure, MOTION can withstand strong gravity. This is what holds up planetary and galactic systems.
The strong self-gravity of these systems is countered by motion.
Usually rotational motion quantified as ANGULAR MOMENTUM or KINETIC ENERGY (i.e., energy of motion which we discuss this below).
ANGULAR MOMENTUM is, loosely speaking, the tendency of rotating bodies to keep rotating.
Let us now move on to gravity in space.