The Gear Principle

Gears. They run everything. they're in your car, they're in bikes, they're in everything that 

moves (tricycles not included.) Why use gears? think about gears. Put two together interlocking (see first 

diagram) and you will get the same speed in as out; no 

matter the size of  the gears.

 the rotational speed differs, but not the speed it would be on the ground if it was a wheel. now think of this. set two gears next to each other (see second diagram.) the rotational speed of the two gears are the same-- though the speed it would be on the ground if it was a wheel would differ.

You're probably wondering how much the change in speed would be. The formula for the interlocking gears' speed is rotations per second in, times the diameter of the first gear( it can also be the radius or circumference, as long as both gear measurements use the same thing) divided by the diameter of the second gear equals the rotations per second out. The speed if it is a wheel on the ground for the interlocking gears would be the same in and out. The equation for the speed if it is a wheel on the ground for the gears set on each other is the speed in, times the second gear's diameter divided by the diameter of the first gear, equals the speed out. The rotational speed for each gear is the same.

What about the power? In any gear system the power in times the speed in (not rotational speed) equals the power out times the speed out. Going on some terrains and amounts of weight requires a certain amount of power. So if your source of power and speed (for now let's say an engine) has just enough power for that terrain and weight, you increase the speed with the gear system, what your carrying (for now let's say a car) couldn't make it. The power would go down below the power needed to make the car go. To understand why this happens , I'll rearrange the formula. The power in times the speed in divided by the speed out equals the power out. By increasing the gear ratio to make the speed out greater, the power would be smaller because the speed out would be greater than the speed in. If the speed in is smaller than the speed out when you divide the speed in by the speed out the number will be smaller than one. Anything times any number smaller than one will be smaller than the original number (using positive integers). So if the engine has the minimum power, the power out will be smaller than the power in, the least power needed to make it go. So it won't move.

You can solve this problem by making the power in greater.

What if you want to make your speed out a hundred fold, or a thousand fold. Your second gear would have to be very large in a stacked gear system to make that work. There is a more convenient way. Think about interlocking gears. Their rotational speed changes, but not the speed if it were on a track or the ground. Imagine taking two stacked gear systems, both with a gear ratio of ten.  Take the small gear of one and the large gear of the other and interlock them. Take the small gear on the two gear pair where the small gear is not connected to the other two gear pair. Have your power and speed come in that way. What is the out speed? It is one hundred times the in speed. The reason? If the out speed of the one is ten times the in speed, that ten times is the in speed of the other, which multiplies the speed by ten. Therefore, the output speed is 100 times the input. Put three together that way, one thousand fold. So the gear ratios of all the gear sets multiplied together is the total gear ratio.

What if you have more than one out? (see diagram.) the power and speed would be the same down each. They would all have less than if there were less outs. The formula is the power in, times the speed in, equals the number of outs times the power out times the speed out. To more easily understand that, I will rearrange the formula. Power in, times the speed in, divided by the number of outs equals the speed out times the power out, the same for each out. To make things simple for the moment, lets say the gear ratio to all the outs is one and there are five outs. The power out on all of them one fifth the power in. The speed on all of the outs are not only the same but are the same to the in. The speed does not change with the number of outs.

The rubber gear. you probably don't understand why it's called the rubber gear. This is a sliding gear. it has a conical shape instead of a circle. Think of a cone. It's made up an infinite number of successively larger gears now on the large end, place a gear. have that be the in gear. now pick a place on the cone. make that the out gear. now slide the place picked on the cone up. the gear ratio becomes successfully larger. not digitally, like a set of stairs, but in an analog fashion, like a ramp. so you could pick any speed out without changing the speed in, but you would have to change the power in to keep the power out the same.

Geometry to the Next Level

The Pythagorean theorem for Extra Dimensions is a new look at geometry.

Copyright © 2008 by Benjamin Bradshaw

 

You may know the Pythagorean theorem, A² + B² = C² (also known as the square root of A² + B² = C). This theorem works very well, but this only applies to right triangles and two dimensions. But there is a way to use the Pythagorean theorem in more than two dimensions.

            To understand this better, I will describe a different way to think about the Pythagorean theorem. The Pythagorean theorem was not designed for right triangles. It works for right triangles, but that’s because it has a right angle next to the two vectors. The Pythagorean theorem was designed to combine vectors. Vectors are the component quantities at right angles to each other that add up to the original line. The reason for doing this is to combine lines that are not vectors of each other. To add them together, you break each one up into its component vectors, add the vectors together, and then recombine the resultant vectors to get the resultant line (see picture below).

Thinking about vectors now, how would you combine the vectors if another vector were added sticking out in the third dimension? They are all vectors to each other because they are at right angles to each other. If you think about it, the resultant line of the original vectors is also at a right angle to the third vector, so the first resultant line is a vector compared to the first vector. Therefore, you should be able to combine the two to make another resultant line, the resultant line from all three vectors. In terms of A (vector 1), B (vector2), C (vector 3), and D (resultant line of all three vectors) the formula for the resultant line in three dimensions is (√ (A² + B²)) ² +C² = D². The square root of something squared is itself, so the formula could be rewritten as A² + B² + C² = D².

Scientists do not know if there are more than three dimensions, but for the argument let’s pretend that there are. As you may have guessed, the Pythagorean theorem for four dimensions is A² + B² + C² + D² = E². The general rule of thumb for the Pythagorean theorem is the sum of all of the variables squared for all of the dimensions of the vectors you are using equals the resultant line squared.

Introduction

Welcome to the Anything Blog

This blog is different than most blogs. A blog may tell you about recipes, skiing, computers, any thing. There is nothing wrong with those topics. But maybe your not interested in them. Maybe you’re interested in climbing, maybe math, maybe car repair. But those topics may not be easy to find. The difference with this blog is, if you comment about something you like, some subject you’re interested in, I will put in a section about that topic. That way, whatever you are specifically interested in, you can find easily. If you have questions about a topic, I will answer them at the bottom of the section about that topic, making answers to your questions even easier to find. At the end of each section, I will also put down sites of reference relating to that section.