Newtonian Mechanics
Force and Motion
by David Baird
I haven't gone into Calculus, yet, for this section. Currently all the maths that are necessary are a little Trig ( sin =y/r, cos=x/r, tan=y/x ), a little Algebra, and a little Geometry. Don't say Calculus is useless! This is a very good example of where Calculus is very useful.
Table of Contents
- Variables
- Constants
- Unit conversions
- SI Units
- Equations
- Motion
- Displacement in one dimension
- Velocity in one dimension
- Acceleration in one dimension
- A quick look at air resistance
- Freely falling bodies
- Motion in two dimensions
- Force
- Newton's first law - an object in motion remains in motion until acted upon by a force
- Newton's second law - all the forces acting on an object equals the mass times acceleration of the object
- Newton's third law - equal and opposite
- ...more will be available in the next update. Free body diagrams, examples, and friction.
|
 = acceleration
 = force
 = frequency
 = height
 = impulse
 = kinetic energy
 = spring constant
 = length
 = mass
 = normal force
 = power
 = momentum
 = distance
 = displacement
 = period
 = time
 = potential energy
 = velocity
 = work
 = displacement
 = friction
= angle
 = torque
|
Notes on some of the variables (for beginners)
VECTORS AND SCALARS: Scalars have only a magnitude - they only have a value, and no direction. Vectors have a magnitude and direction. When you go somewhere, you could describe it as a scalar value by saying you travelled 5 miles, or you describe it as a set of vectors: You turned left on New Mexico Dr. and traveled for 500 ft, then turned right on Ranchvale Highway and travelled 5 miles-500 feet. Scalars: mass, time, distance, speed. Vectors: acceleration, force, weight (which is a force), displacement, velocity.
MATH OPERATIONS AND UNITS: Whenever you multiply a vector by a scalar, (mass by velocity, for example), you get a vector with the direction of your original vector. A scalar times a scalar is still just a scalar. I have to check and see what to do with a vector times a vector. Whenever adding or subtracting, you must be using the same units!! For example, (kg m/s2) + (kg m/s2), but not (kg m/s2) + (kg m/s).
DISPLACEMENT: is relative displacement. Displacement can be thought of as distance, but displacement is a vector, and distance has only a magnitude. If you fly your fighter plane 300 miles to Chicago on a straight road, then your displacement has a magnitude of 300, as well as your distance. Your displacement also tells the direction of Chicago from where you started at. If you flew from home to Chicago and then right back home, you have travled a distance of 600 miles, but your displacement has a whopping magnitude of 0.
VELOCITY: is relative velocity. Velocity and speed are different just like displacement and distance. Velocity is displacement/time. If you shoot a missile to Chicago, it may have some maximum velocity of 50 m/s. If something goes wrong and it comes back and hits you, then it may still have a speed of 50 m/s on it's trip, but it's velocity at the point that it hits you is 0 m/s.
ACCELERATION: is acceration. For the 1st 3 equations, acceleration must remain constant. You need to use Calculus to solve things with changing accelerations. Acceleration is the rate of change of velocity. If the velocity changes at a constant rate, then there is an acceleration. Gravity is an accerlation going towards the center of gravity (center of a planet, for example) with a magnitude of 9.8 m/s for Earth.
|
Before going into force, let's go over motion. We can use the first 3 equations to determine motion under constant acceleration. We have to use calculus when acceleration is not constant.
This is a position always located on along a straight line. This can be a vector, but the angle of the vector always stays the same. You always have an initial position, 0. You also always have a fixed position (usually this is set to 0). This is where your other positions are measured from. And you have a final position. Displacement is (final position) - (initial position), or,  =f - 0. For example, if you start at a postion of 5m and end at 3m, your displacement was, 3m - 5m=-2m. Your final and initial positions are actual displacements from position 0.
This is when something moves along a straight line. Average velocity is /=(f - 0)/(f - 0). The SI unit for velocity is meters/second. If you travel 10m in 5sec, your average velocity was 2m/s, but you may not have traveled the same speed the whole time. You can travel at a higher speed part of the time and a lower speed another part of the time, and maybe the average speed for the rest of the time. Instantaneous velocity is velocity at a specific point in time. If something travels at the exact same speed over time, then it's instantaneous velocity is equal to it's average velocity.
Acceleration is what brings about velocity. Things don't suddenly start going at a certain velocity when they were just prviously at rest. They gradually reach a particular velocity. When an object experiences a change in velocity, it has undergone acceleration. (This is where force comes into play). Average acceleration is /=(f - 0)/(f - 0). When acceleration is constant (which should be alright for many games), you can use the first 3 equations. If a police car is going 10 m/s, but has to chase another car, so it speeds up to 40 m/s in 3 seconds, what was the acceleration of the police care during this time interval? (40 m/s - 10 m/s)/(3 s) = (30 m/s)/(3 s) = 10 m/s2.
Everything falls at the same speed - or at least they would if there was no air. Gravity causes all objects to accelerate towards the center of gravity (on Earth, that would be the center of Earth). That same acceleration is applied to every object, pulling it towards the center of the Earth, but these objects must move through air which affects the acceleration - you will understand this better after studying force. Air does not apply a constant acceleration, therefore, it is very fun to ignore (but not very acurate all the time). The force of air is a function of velocity. Try riding a bike against the wind. At first it seems easy, but the faster you ride, the harder it seems the wind is pushing on you - and that's just what's happening. The wind is trying to cause you to go at the same speed it is. If you travel at the same speed as the wind, it will feel like the wind is not pushing on you at all, which is really the truth. When the air is not moving, it tries to keep other objects from moving. As you will learn in studying force, without air, the mass of an object does not change the rate at which it falls. The mass, plays an important role, however, when friction is involved. Friction has a lesser effect on greater the masses.
If you drop a rock, it goes down, doesn't it? If you through a rock up, it comes back down, right? These are freely falling bodies. A body is freely falling when the only acceleration (or force) of it is due to gravity. Note that this acceleration occurs only in the y axis. What is the velocity of a penny when it reaches the bottom of the Chrysler building (255 m high), if you drop it from the top?
0 = 255 m
= 0 m
0 = 0 m/s
= ?
= -9.8 m/s2 (remember: -9.8, beacuse it's going down)
Look at what information you're given. Which of the 1st 3 equations could be used to solve this problem? Equation 3: = 0 + 2( - 0).
- = 0 + 2( - 0)
- = 02 + 2(-9.8 m/s2)(0 m - 255 m)
- = (-19.6 m/s2)(-255 m)
- = 4998 m2/s2
- = 70.69653456853 m/s (we won't worry about significant figures, here)
This is exactly the same as doing things in one dimension, except now you have to do everything 2 times. Once for the "y dimension", and once for the "x dimension". You mu, orst convert all of your given information into these 2 dimensions, then use the 1st 3 equations 2 times, once for each dimension. You will get 2 answers - one for each dimension.
Okay, here's how to split your information up into x and y components and then put it back together - this is where trig comes in handy.
Here're the variables:
A = reference angle of vector
x = x-component of the vector
y = y-component of the vector
r = magnitude of the vector
Here're the equations:
x = r(cosA)
y = r(sinA)
r2 = x2 + y2
A = tan-1(y/x)
Given: A football player kicks 30º up from horizontal, giving it an initial velocity of 40 m/s. How far away from the football player's initial position will the ball hit the ground?
In the real world, motion is created by forces. Kicking a ball is a force. A rocket motor provides a force for a rocket. Gravity causes objects to have weight. Weight is a force. Do not confuse weight and mass. Mass always stays the same! Weight changes, for example, if you send something from Earth to the Moon. Force is equal to mass times acceleration. The SI unit for force is (kg m)/s2, or N - a Newton, after Sir Isaac Newton.
An object at rest will remain at rest and an object in motion will continue in motion with a constant velocity (in a straight line) unless it experiences a net external force.
The acceleration of an object is directly proportional to the resultant force (net force) acting on it and inversely proportional to its mass. The direction of acceleration is in the direction of the resultant force.
 =
If two bodies interact, the force exerted on body 1 by body 2 is equal in magnitude and opposite in direction to the force exerted on body 2 by body 1.
This explains how space ships travel through the vacuum of space. This is why an object on a table doesn't fall right through the table - the table is pushing up on the object just as hard as the object is pushing on the table. Likewise, the Earth is pushing on the table as much as the table is pushing on the Earth.
...There is still much more in store for this force section. Expect more in the next update!
|