Topic: Physics 101 at Ornery U- mechanics with minimal algebra.

Everard
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Chapter 1- Kinematics- Describing motion.

Physics is the study of matter and energy and the interaction between the two. When matter interacts with other matter there is some change in the state of the interacting matter. One of the most obvious ways that matter can change its state is the state of its motion. So, the first task is to describe motion.

The terms we use to describe motion are 1) Distance. The path length from point A to point B is the distance between point A and B.

2) Displacement. The "straight line" distance from point A to point B.

You're on an outdoor track. You run around the 400m track once, so you start at the point you end at. Your "distance" traveled is 400m, and the "displacement" of your travels is 0m, because the straight line distance from where you started to where you ended is zero.

3) Speed. The distance traveled, divided by the time it takes to travel that distance. How fast an object is covering ground.

4) Velocity. The Displacement traveled, divided by the time it takes to travel that displacement. How fast an object is "closing" from its start point to end point. This is sometimes called the "state of motion," as an objects current state of motion can be completely defined by how fast its moving, and which direction it is moving in.

5) Acceleration. How rapidly the velocity is changing.

Displacement, velocity, and acceleration have positive and negative values. A negative value does not mean "slow," though. The positive and negative signs indicate direction. Velocity is sometimes called the "state of motion," as an objects current state of motion can be completely defined by how fast its moving, and which direction it is moving in. velocity can change in one of two ways: Either the speed can change, or the direction can change. Thus, on your car, the steering wheel could properly be termed an accelerator, as it accelerates your car by changing its velocity by changing its direction.

For example, a common type of problem is the projectile problem. A cannonball on its way from the cannon to the target is an example of a projectile. Ignoring friction for now. The cannonball is going to be traveling up, and then down, at different points along its path. Gravity is going to "accelerate" the object downwards, but its initial velocity out of the muzzle will be upwards. We choose one of those directions as positive, and the other as negative, so, for example, gravity could be negative, meaning that our downwards direction is negative. Everything in that direction will now be negative, until I start to examine another situation.

Its important to understand the relationships between velocity, displacement, and acceleration.

Displacement is defined as the change in position. If I move from point 3 to point 7, my position has changed 4 units. This is my displacement. It doesn't matter HOW I moved from point 3 to point 7, what my path was, it just matters that I did.

Velocity is defined as the change in position, over time. Movin from point 3meters to point 7meters, my position changed 4 meters. If this took me two seconds, my velocity is 2meters/second.

Acceleration is defined as the change in velocity, over time. If my velocity goes from 1 m/s to 3 m/s, in 1 second, my acceleration is 2m/s/s, or two meters per second squared.

These definitions give us certain relationships.

The relationships are called the kinematics equations, and you can look them up if you are interested. They tell us how to solve for displacement, velocity, acceleration, or time, if we know other of those parameters.

Motion is usually not in one dimension, so its important to realize that motion in one direction does not influence motion in other directions. That is, gravity points towards the surface of the earth, so the acceleration due to gravity, for our everyday life, is "down." But motion horizontally is unaffected by gravity.

You can prove this to yourself if you do the following simple experiment. Take out a meter stick, or some other long flat object. Put a quarter on the corner of a desk, and another quarter on top of the meter stick, at the end (98cm mark or so). Line up the quarters. Then flick the meterstick into the quarter resting on the desk, such that the quarter on the desk goes flying, and the quarter on top of the meterstick drops to the ground. Listen for the clinks. The two quarters will hit the ground at virtually the same time.

Similarly, if you took several guns with different muzzle velocities, pointed them all horizontally, and fired them all while simultaneously dropping a bullet from the same height as the guns, all the bullets would hit the ground at the same time.

This is because the horizontal speed of the bullets does not determine how long it takes for the bullets to hit the ground... only the vertical components of motion, since the direction of travel towards the ground is vertical.

In other words, motion along one axis is independent of motion along another axis. (Take home point).

As you may have noticed, I haven't talked about mass yet, either. The mass of the bullets does not determine how long it takes for the bullets to hit the ground, only how high up they are, and other initial vertical conditions. (If you point a gun up, it takes longer for the bullet to hit the ground).

To prove this to yourself, here's a dramatic 20 second demonstration.

Take a sheet of paper, and a ball. Hold them in seperate hands, and drop them from the same height at the same time. The ball hits the ground first. Now, pick the objects up, crumple the paper until its ball-shaped. Drop the two objects again, again from the same height and at the same time.

They should hit the ground at basically the same time.

But obviously, the mass of the paper didn't change... only its shape. Shape can change how long it takes for an object to fall, but mass won't. We'll see exactly why later on.

quote:As you may have noticed, I haven't talked about mass yet, either. The mass of the bullets does not determine how long it takes for the bullets to hit the ground, only how high up they are, and other initial vertical conditions

Do we know why inertial and gravitic mass are identical?
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Everard
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This skips us ahead a little bit, but...

quote: Do we know why inertial and gravitic mass are identical?

Nope! I always tell my students, when we come to the concept of inertia "If you want to win a noble prize, figure out the answer to the question "Why inertia?"

In principle, we can determine the "inertial mass" of an object by slamming two bodies into each other, and measuring the relative accelerations, because inertial mass is the measure of resistence to acceleration. When we collide objects, they experience identical forces. Objects with more inertia will acccelerate more slowly under the same force then objects with less inertia.

Gravitational mass is the mass of an object measured by the effect of the gravitational field on the object. This is what we read off of a bathroom scale... the scale "reads" how much weight (force) is put upon it, divides by the gravitational field, and gives us the gravitational mass.

If we take gravitational mass, divide by inertial mass, and multiply by the gravitational field, we should get the acceleration. Iff all objects fall at the same rate in a given gravitational field, then, using apprpriate units, m/M (gravitational divided by inertial mass)=1.

Using better and better experiments, so far this equivalence has held true. That is, a=mg/M down to about 12 decimal places.

Einstein postulated that within a sufficiently small region of space time, a and g are indistinguishable from each other.

But no one has ever proved that, theoretically, they are always equal to each other. And we don't know why they are. Just that they are.

The upshot of this- Inertial and gravitational mass are, conceivably, distinct quantities. But if they are, the person who figures out that they are, or why they should be different, gets to be Big Man in physics for a while.

That took me a little far afield for chapter 1, but thats ok
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Well, that was the only question I had on the actual content of 'Chapter 1'

I was going to niggle about explicitly stating some assumptions (no air resistance) while, well, assuming others (flat Earth society ) but decided against it.
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Everard
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You're right. "Until further notice, we're living in a world where sheep sleep with the wolves. I call this world 'room 124, physics room!"

The time to edit the initial post is over, but I'm basically gearing this towards people who don't know much if any physics, and would like to learn. I don't think thats very many of our posters, but its probably a few.

Over time, i hope to move this into more advanced topics in physics, but I think the best way to do that is to build from the ground up. Obviously, we will come to a point where someone like vulture is a better poster to run this then someone like me, who deals with trying to beat the rudiments of physics into unwilling teens

posted
No questions, Everard. This is empty chatter so you don't think there's vacuum on this end.

(I apparently had a faulty definition of acceleration in my head, but I imagine that's the status quo for the physics-ignorant.)
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Feels like the day I had to tell my second grade teacher that I cheated my way through first grade by memorizing all the storybooks when the teacher read them aloud.
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posted
I never took physics eiher, Jesse, so don't feel alone there. And believe me, I felt its lack when I was trying to do some research into acoustics..
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posted
If you think of a car moving at 5kph in a straight line (northward, say), it has zero velocity in the east/west axis.

If you make a 90 degree right-hand turn (by simply coasting and turning the steering wheel) the end result (assuming zero energy loss due to friction) is your car travelling eastward at 5 kph with zero velocity in the north/south axis.

Thus you accelerated by 5 kph eastward and -5 kph northward, simply by turning the steering wheel.

Therefore, the steering wheel (or your hands) caused your car to accelerate (change velocity).
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Everard
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Sure.

Velocity is made up of two different components, how fast you're going, and what direction you are going. You can drive from your house to the grocery store and back, and it can take the same amount of time each direction. But the trip isn't the same. Because the trip isn't the same, we need to account for that in our description of how you are moving.

Traveling at 30 miles per hour west, is different then traveling at 30 miles per hour east. The way we account for that, in one dimension, is the positive and negative sign I mentioned above. We could label west as positive and east as negative.

So, if we construct a completely imaginary world where you are traveling 30 miles an hour west, realize you need to turn around, and go back to get your wallet, one way you could do so is to grasp your steering wheel firmly, and spin! Over the course of a few seconds, you'll go from traveling 30 mph west to 30 mph east. Your velocity changed from positive to negative, even though the magnitude is the same. Since your velocity changed, by definition, there must have been an acceleration, since acceleration is the change in velocity over time.

The acceleration occured because you spun your steering wheel.
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So, aside from being an accelerator, is it also a decelerator (negative acceleration)?
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Everard
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Deceleration is one of those english words that doesn't really mean anything in physics.

In common parlance, it means "to slow down." But slowing down could be either a negative or positive acceleration, since positive and negative direction are arbitrary.

Negative acceleration, therefore, could be either speeding up or slowing down.

A positive acceleration tells us that the velocity is becoming larger in the positive direction, and a negative acceleration tells us the velocity is getting smaller in the positive direction.

In physics, we tend not to use "deceleration" in a technical way, because it creates more communication problems then it solves.

Here's an example.

Jump on a trampoline. You go up, you come down, you bounce and go up again. When you leave the trampoline, your speed is decreasing. When you come down, your speed is increasing. When you hit the trampoline, your speed is decreasing, and then increasing until the spring in the trampoline pushes you back into the air.

So, if we choose downwards as positive, there are four distinct phases.

1) The initial upwards flight, where you have negative velocity that is going from, say, 10 mph to 0 zer0 mph. This is negative velocity, positive acceleration.

2) The descent. Velocity is going from 0 to 10 mph, positive velocity, positive acceleration.

3) Hitting the trampoline, velocity goes from 10 mph to 0 mph. Positive velocity, negative acceleration.

4) The "spring." Launching us back into the air. Going from 0 mph to 10 mph, negative velocity, negative acceleration.

We have situations where the speed is increasing, and both positive and negative acceleration.

We also have situations where the speed is decreasing, both positive and negative accelerations.

Because of this overlap, we describe everything as acceleration, since slowing down can mean either positive or negative acceleration.

quote:Originally posted by flydye45: Displacement is the only term I've been unfamiliar with so far.

I've never completely understood centripital force, however.

Whenever you get to that, I'd be obliged.

The Wikipedia entry on it is good. Ignore the math stuff (it's hard), and just check that diagram at the top of the article.

The general gist of it is that there's two things you have to remember when things are going around in circles.

1) An object in motion will remain in motion at a constant velocity until a force acts upon it. In other words, once you start something moving, it'll keep going in a straight line at the same speed forever, until something comes along and acts upon it to either change its speed or change its direction.

2) Circles are definitely not straight lines. If something's going around in circles, its constantly changing its direction of motion, which means that its constantly changing its velocity and acceleration. Which means that a force is acting upon it.

And that force is centripetal force. Basically if something's moving in a circle, it's constantly being acted on by a force that's at right angles to its direction of travel, trying to pull it into the middle of the circle. If you've got a ball on a string, the tension in the string is trying to pull it in. If you've got a satellite in orbit, gravity is forever trying to make it fall down. If that centripetal force suddenly ceased to exist (eg the string holding your ball snaps, or Dr Von Awesome shoots the satellite with his evil gravity-blocking rays), the object would just fly off in a straight line forever.

Centripetal force is also why, if you're inside a body that's moving in a circle, you'll feel centrifugal force, which is the thing that makes you feel like you're being pushed out of the circle. If you ride the Gravitron at the county fair you'll get pinned to the walls. If you swing a bucket of water fast enough the water'll stay in even when it's upside down. If you take a corner at high speed in your car, you'll feel like you're being pushed towards the outside of the corner.

Not that there's anything mysterious about centrifugal force, it's just the same sort of reactive force you always feel if you're inside an object that's accelerating (eg being pushed back into your seat when you hit the accelerator, feeling heavier for the first couple of moments when you go up in an elevator).
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quote:Originally posted by The Drake: There's little point to describing mechanics without the use of Calculus.

Until you realize that

v=at

and

d = v0 + 1/2 at^2

for a good reason, basic dynamic motion will elude you.

Yeah but explaining to folks who don't remember calculus how the basic equations of motion are derived from each other can be a bit tricky without the aid of graphs.

Plus as Stephen Hawking famously said in the introduction to "A Brief History Of Time", every equation you include cuts your readership by half
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Everard
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I've asked the mods to move the calculus discussion into another thread, if possible, where I would be glad to discuss it. Here, though, I'd like to focus on "teaching" mechanics to those who are interested, but don't have a math background.

posted
Awesome stuff Ev. I never took physics in HS or college and have always regretted it. This stuff sounds pretty interesting though. Maybe I'll take a course when I go back to school. When you say that "positive" and "negative" denote direction, do you mean that something with a negative velocity is moving *away* from its eventual end-point? How might that work (or do I have it wrong?)
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All number systems require a reference point.

You are somewhere between Boston and San Francisco. If you walk toward SF, you are increasing the distance from Boston. So the same velocity that would be reducing the distance to SF is increasing the distance relative to Boston.

So, you must define the direction of travel you are interested in. Do you wish to accelerate toward San Francisco, or toward Boston? This all depends on the question you are trying to answer:

How long will it take to get to San Francisco?

OR

How long will it take to be 900 miles away from Boston?

Of course, this gets more complicated when you talk about two spatial dimensions.

If you walk northwest instead of due west, only a portion of your travel brings you farther west (closer to SF), while a portion of your travel takes you farther north (away from SF).

At some point, you will have got as close to SF as you can on that heading, and you will actually start moving away from it (as you pass through Canada)

Kinematics is a description of motion. Mathematically, we use the kinematics equations for a variety of situations... perhaps most common are ballistics or projectile problems. But, this is essentially "mapping." Take some object, and find where it is at certain times, or how fast its going. The kinematics equations rely on constant accelerations, and constant acceleration is unusual in our lives.

But accelerating objects are more interesting then objects at constant velocity... constant velocity means motion in a straight line, at constant speed.

So we start to talk about why things change their motion.

Historically, the natural state of objects was quite a point of contention, until gallileo, and then newton, basically settled the question. gallileo performed a series of experiments, using ramps and balls, which demonstrated that "Objects tend to maintain their current state of motion." Essentially, he discovered that a ball will roll further the smoother the surface it is rolling on. He postulated that, given a sufficiently smooth surface, the ball would roll forever. What this told him is that some attribute of the surface was causing the ball to slow down, or accelerate... change its state of motion. Newton formulated that into what is now known as Newton's first law...

"An object will maintain indefinetely its current motion until acted on by an unbalanced force."

Forces are pushes, or pulls, that are capable of changing an objects state of motion.

Two important things about that statement. 1) Try pushing on nothing. Can you do it? No. A force is generated by the interaction between two objects. Because of this, a property of an object (such as its mass) cannot be a force. Forces only arrise because two objects interact.

2) Forces don't always change an objects state of motion. That couch you are sitting on? There are currently lots of forces acting on it, but they are all being opposed by other forces, and cancel each other out. We can tell from simple observation that, like velocity, or displacement, forces have associated directions. They are vector quantities.

Mathematically, how a force changes motion is the relation,

F=MA, where F is the value of the force, M is the mass of the object, or perhaps more accurately its inertia, and A is the acceleration.

This relationship is more accurately expressed as the force is equal to the time derivative of momentum, which we'll get to later, but this is newton's second law.

Finally, newton's third law comes from the fact that forces derive from interactions between two objects. Because you can't strike a nail with a hammer without the nail also striking the hammer, anytime one force acts on one object, a second force must act on the object that causes the first force.

Forces are INTERACTIONS. They always come in pairs. A acts on B, and simultaneously, B acts on A... but in the opposite direction.

For every action, there is a reaction, equal in magnitude, but opposite in direction.

Its winter time. Get out your skates, find a buddy, and go iceskating. come to rest with your hands against your buddy, and push. You'll both start moving. You both changed your state of motion, even though only one person pushed.

Because forces are vectors, its important to understand that a force along one axis will only change motion along that axis. We often talk about components of forces... when you push your lawnmower, you push down, and out. You are pushing with a certain force, but only part of it is moving the force forward. Through trigonometry, we can figure out how much of a force is along each of our axes.