Monthly Archives: February 2010

Photographs

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Circular Motion

We reviewed a another Force problem, force applied to an object from an angle. This force can be though of to have two components. We used Newton’s second Law, and broke it down into three equations. One for each axis (x and y) and one for The vector sum. We thought about how both x and y must sum to 0, for y there is only one force to be concerned about (Fexternal sin θ). But for the x direction we need to consider friction as well. Since it is pushing against our origin we think of Friction as a negative force. So Friction and the x force must sum to zero as well.

We needed one more thing to solve the problem; the magnitude of the friction force. To find it we multiplied the Normal force by the Coefficient friction of kinetic force. (mu μ). We rearranged our equations, and then divided by each other to combine them into one. And now we know how to calculate the Normal Force and the external Force. Since we know those two we can calculate other things we may need too.

Circular Motion
We just started to touch on circular Motion. If you swing an object in a circle, the magnitude of velocity is constant, but the direction changes all the time. At one time it is in one spot, then a short time later it is a little further along.

The magnitude of the radius at time 1 is equal to the radius at time 2. But the directional change is caused by a little vector called “Delta r” which pushes the direction of the vector along.

The change in time is expressed by “Delta t” And the velocity is the change in r (Delta r) over the change in time (Delta t)

Velocity is the tangent formed by Change in r over change in t. This makes Velocity almost a right triangle, when the distance in the change in time is very small.

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Booleans

I am going to group my class notes, since I didn’t have much from the last two classes.

break;
Can be used to exit Loops.

There is a Boolean type. To use it you need to initiate

bool x = true;

The ? is a conditional operator. It says

var = (logic Check) ? TrueStatement: FalseStatement;

Don’t use it.

Dangling Else Watch out for these! Don’t write if if else.

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Newton’s Laws.

We continued reviewing Newton’s three laws.
1) Objects in motion stay in motion. Objects at rest say at rest.
2) F = m/a
3) I think this is the law about equal but opposite forces.

We analyzed two masses connected by a frictionless massless pulley and rope. Using just Force equals mass times acceleration we were able to determine the acceleration of the masses.

Then once we knew the acceleration we were able to calculate the Tension Force on the rope.

The professor checked his answers by thinking logically about his results. First he checked on the dimensions, then he thought about what would happen near the zero values and then near the equilibrium states.

Friction
We started on Friction today, because friction is really quite an important thing. If you look very closely at the borders between objects, you will see their small rough surfaces, trying to catch on to each other.

The friction force always points in opposition to external forces. As long as a body does not move, the External Force and Friction force are equal. Once the body moves the external Force has overpowered the Friction force.

If you place an object on a plane, then tilt the plane until the object slides off, you can find the value of the friction force.

Friction is expressed using a greek letter “μ” read this “mu”. There are two types of friction we are considering. Static Friction, which is the friction required to get an object moving. This is represented by the subscript s, and look like μs. It is a dimensionless quantity and represents a fraction of the normal force required for horizontal movement. The second type of friction is Kinetic Friction. It is represented by the subscript k, and looks like μk. Static Friction is neat. It is the tangent of the angle created by the tilting of the plane.

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Sums of Forces

Force is a vector. Its magnitude is the Vector sum of it’s x and y component parts. If you hang a bowling ball from a string, there are two forces acting on it. The tension force from the string, and the force of gravity, trying to drag the bowling ball to earth. If you look at the picture, it is the “ghost” bowling ball.

Things get more complicated if you drag your bowling ball to the side, by tying a string to it and pulling it. Now there is another force.

This force is a vector sum, and it can be broken out into

So now we fill in our x and y values based on what we know (which is F String and Mass)

We need to solve for Theta right away. We do this using a trick, since these are vectors they are related to each other and we can combine our x and y values. So we decide to divide our x and y.

This gives us a very neat solution. We can cancel out the |FT| which leaves us with sine of theta over cosine of theta, which is equal to the tangent of theta. So since the other side is already equal to the magnitude of the Force of the string (length of String) divided by mass times gravity… Does this check dimensionally? It doesn’t seem so… but I will continue! Maybe Tangent auto-ignores units.
Ok, now we can get started solving for FT, which we do by performing a neat trick and squaring both sides. I never knew you could play with your math so much.

Ok, here the professor did something to turn the Trig into an identity… and I didn’t catch it. And I’m not to sure how to swing it into the answer, which I do know! Maybe he added x and y instead of dividing? I need to remember I’m working with vectors.

Now, the next example took into consideration acceleration! This was just the example for static force.

Dynamic Force with acceleration along one axis:

Ignoring my deadhead passengers, once the elevator takes off and starts accelerating, there is another force to deal with. We need to add “a” into our equations for Force. Lucky it is only in the y direction, which makes things a bit easier.
If acceleration is not 0

So that is it for Dynamic forces along the y axis.

What is you place an object on a frictionless surface and tilt it?

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More on Loops

printf(“%d”, root*root);
You can use an expression as a variable in a printf statement

Do… while, is a post test loop.
First it does it. Then it checks the condition.

initialization
do {
body
Update
} while (condition test);

Nested Loops
– IF outer loops executes n times, and inner loop m times, inner loop executes n * m times.

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Force – Newtons Laws

1. Law of inertia “Objects in motion tend to stay in motion, objects at rest tend to stay at rest”
2. Fnet= ma Net force equals mass times acceleration
3. For every action there is an equal and opposite reaction

~~~~So Far~~~~

The units of force is Kilograms meters per second squared, summarized as a Newton. That is to say, 1 Newton of force will accelerate 1 kilogram of mass by 1 meter per second squared.

Weight is a force. It is the acceleration of gravity on the mass of an object.

We use Free Body Diagrams to depict force on an object we want to think about.

For right now we are focusing on Static objects. These are objects whose forces sum to zero. They are either not moving or moving at a constant velocity.

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Instant Velocity, Average velocity and Acceleration

When an object is traveling at a constant velocity, it is special. It’s Instantaneous Velocity and Average Velocityare the same, and the position of the object can be stated by:

Read this: “The position of the object is at point x which is equal to the origin position plus Velocity times Time.”

The slope between two positions of the object tells us the average velocity. Slope can be in meters per second or seconds per meter, but it should have a unit attached to it, in the form of Unit per Unit. On a graph of velocity, time would occupy the x axis, and position would occupy the y axis.
GRAPHIC 2

Instantaneous Velocity can be thought of as constant velocity between two very close positions. So this starts to get into Calculus, which I need to review, but expressing Instantaneous Velocity is really just the same as average velocity as the x axis approaches 0. (Since x is Time, as the time is shorter the closer you are to an instant)

Acceleration tells you about how fast the velocity of an object is changing. There is also an Average and Instantaneous Acceleration. When speaking about acceleration, velocity is along the x axis and Time and along the y axis. The basic formula is the same as velocity, with the only difference the axis of the graphs.

The second half of the class the Professor spoke about some special equations for objects with a constant acceleration. You can use these to get the Average Velocity or The Position or Time, depending on what you have to work with!
x = position
v = velocity
t = time
a = Constant Acceleration

A note on Speed:
Speed is the same as velocity, but it does not tell about direction! A car going -30mph is traveling at a speed of 30 miles per hour somewhere, but a car at a velocity of -30mph is going in reverse at a speed of 30 miles per hour.

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Loops.

Iterative Programming Repeat same task over and over w/ different data

Loop Elements
1. Initialization
2. Test loop condition
3. Loop Body (Task to repeat)
4. Loop Update.

    3 Types of Loop

  • For: Loop update in header, fixed number of iterations, pretest.
  • While: Loop update in body, execute 0 or more. Pretest
  • Do…. while: execute at least once, post test.

for ( initialization; loop condition test; loop update)
{
loop body;
}

Fancy Loop header
for (i=0,j=1; i < 5; i++, j++) Loop headers can include calls to functions and omit headers. while (loop condition) { loop body loop update }

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Projectile Motion, Range, Range Maximum, and start of Forces.

Once again, there where many pictures in class. I’m going to summarize the basic formula’s and idea’s from today’s class. Just look at the sketches in the paper notes.

Projectile motion forms a parabola. The object’s velocity forms a tangent of the x velocity and the y velocity. Y velocity is generally subject to gravity. The two components are independent of each other. x does not care about y and y doesn’t care about x either.

Note: My equation editor seems unable to draw an arrow on top of a letter.

1. Tangent of initial angle theta is equal to the initial Velocity of the vector, which is equal to the initial velocity of the y component over the initial velocity of the x component.
2. The magnitude (length) of the initial Velocity Vector is equal to square root of A squared plus B squared. (Pythagoras)
3. The velocity in x direction at time t does not change.
4. The x coordinate is equal to the initial x velocity times time.
5. The Velocity in the y direction is equal to the initial velocity minus acceleration caused by gravity times time.
6. The y coordinate is equal to the initial position plus initial velocity times time minus gravity times time squared.
7. The y position as a function of x is equal to x times tangent of theta minus, gravity times positive squared, divided by two times initial velocity of the x factor squared.
8. Range is equal to two times the sin of theta times initial velocity squared, divided by g.
9. Height is equal to the range divided by two.
10. The maximum range is equal to initial velocity squared times gravity.

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