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Conical Pendulums

Today was a quick class since the Midterm is Friday. We went over conical pendulums.

~Radius equals length of the cord times the sin theta, where theta is the angle formed by the cord and the center of the circle.

~Velocity equals 2 pi times the radius over the Time it takes to complete one rotation.

~Time equals 2 pi times the sqaure root of length times cos theta divided by gravity.

Add comment March 3rd, 2010

The Beginnings of Derivatives

The meaning of f’(a) [read “F Prime”] is the slope (tangent) at point a.

There are three main ways of thinking of the derivative.
It is the slope of a line
It is the Instantaneous velocity
It is the rate of change

Some Basic Derivative problems

(Practice and be able to explain these, using the Derivative formula (Limit h approaching 0…)

Find the Tangent line at point x = 2

y minus initial y equals the slope times x minus initial x.

Domain in f(x) and f’(x)
The Domains of a function and it’s derivative are not always equal to each other, the Domain of f(x) is greater than or equal to the domain of f’(x). We say that f is differentiable in f if f’(x) exists. To prove f’(x) exists, the following three must exist.

We say that f is differentiable if f’(a) exists for all a in the domain.

The smooth graph is differentiable. A graph with corners is not. Eq 4, above, is an example of a graph which is not differentiable at 0. To be differentiable implies continuity.

PROVING CONTINUITY
A function is continuous if the following two limits exist, and when multiplied by each other equal something.

Meaning of Derivative
The Derivative is the line of the slope. When it crosses an axis you have reached a minimum or maximum value. The line of negative slope is going down. The line of positive slope is increasing. Look at Tangent lines, they approximate the graph and tell the direction up or down.

In Conclusion
f’(x) is negative, x is decreasing near x.
f’(x) is positive, x is increasing near x.

Add comment March 2nd, 2010

Arrays

To Declare an array
type name[#];
int fish[3];

To Access an Array
name[#]
fish[2]
int i=2;
fish[i];
fish[8-3*i]

The name of an array refers to the spot in memory to begin storing. The index tells how much to offset.

WATCH for Array index out of bounds. C doesn’t warn you, it just writes over something else.

float sample[5]= {[2]=50.2,[1]=110.0}
Will make all other indices 0.

int name[5] = {1,2,4,5,6};

int name[20] = {5,6,7};
the first three values are 5, 6 and 7 everything else is 0

Sparse Arrays, are very few non zero values.

You can create arrays of characters

char myString[]={“Hello!”}; // This is a 7 character string The memory uses the 7th spot to store \0 which is a numeric character for null.

Anything inside of “ “ is a string

char letters[5] {‘A’,’B’,’C’,’D’,’E’};

Add comment March 1st, 2010

Centrifugal Force.

What happens when an object is rotated in a circle?

A) Vector radius
B) Radius
C) Vector Velocity
D) radius again
E) instantaneous velocity
F) The change in the radius vector is equal to the second radius vector minus the first radius vector
G) The change in the velocity vector is equal to the second velocity vector minus the first.
H) Instantaneous velocity equals the change in radius over the change in time.
I) The change in radius is equal to the magnitude of the radius times the change in theta.
J) The change in Theta is equal to the change in the radius vector over the Radius magnitude.
K) The change in Theta is equal to the change in velocity over velocity.
L) Acceleration is equal to the change in velocity (direction) over the change in time, it is also equal to the velocity magnitude times the change in radius magnitude, over the Radius Magnitude times the change in time. Centrifugal acceleration is equal to velocity squared over Radius magnitude.

Now we put these components together to start talking about Force.

Centrifugal force equals mass times centrifugal acceleration, which is the same as mass times velocity squared over radius.

This force must be provided by something, to keep the mass moving in a circle. Otherwise it will just fly off in a straight line. Examples of the force provider are Tension in a String, Gravity and Electrostatic force.

We did some examples, of what happens when you swing a can over your head (on a string), What happens to Tarzan when he grabs a rope and jumps off a cliff, and how the rider of a Ferris wheel feels as it rounds the top. We thought about how water in a can stays in the can when it is swung overhead with enough force. And we finished up with looking at a conical pendulum.

Add comment March 1st, 2010

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 (F_external 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.

Add comment February 24th, 2010

C

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.

Add comment February 22nd, 2010

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 μ_s. 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 μ_k. Static Friction is neat. It is the tangent of the angle created by the tilting of the plane.

Add comment February 22nd, 2010

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 |F_T| 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 F_T, 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?

Add comment February 19th, 2010

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.

Add comment February 17th, 2010

Force – Newtons Laws

1. Law of inertia “Objects in motion tend to stay in motion, objects at rest tend to stay at rest”
2. F_net= 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.

Add comment February 17th, 2010