Real Life Uses Of Math Gears

The Uses of Math

How does trigonometry apply in real life? You might be wondering that yourself. Thinking when hell am I ever going to use trigonometry. One of the hundreds of ways you can use it is to determine the most efficient gear ratios either for cars or for bikes. When I say efficient I mean either fuel efficient, or the most acceleration, or the highest top speed, or combination of all of those things.

You see gears in just about everything that has spinning parts. Car engines and transmissions contain many gears. So how do they work? Gears are generally used for one of four different reasons: To reverse the direction of rotation, to increase or decrease the speed of rotation, to move rotational motion to a different axis, and to keep the rotation of two axes synchronized. The fact that one gear is spinning twice as fast as another is because of the ratio between the gears; the gear ratio. If the diameter of a gear is twice that of another gear, the gear ratio is 2:1. Every time the larger gear goes around once, the smaller gear goes around twice. If both gears had the same diameter, they would rotate at the same speed. Understanding the concept of the gear ratio is easy if you understand the concept of the circumference of a circle. Keep in mind that the circumference of a circle is equal to the diameter of the circle multiplied by Pi. Therefore, if you have a circle or a gear with a diameter of 1 inch, the circumference of that circle is 3.14159 inches. The circumference of a circle with a diameter of 1.27 inches is equal to a linear distance of 4 inches.

Most gears that you see in real life have teeth. The teeth have three advantages: They prevent slippage between the gears. Therefore, axles connected by gears are always synchronized exactly with one another; they make it possible to determine exact gear ratios. You just count the number of teeth in the two gears and divide. So if one gear has 60 teeth and another has 20, the gear ratio when these two gears are connected together is 3:1, they make it so that slight imperfections in the actual diameter and circumference of two gears don't matter. The gear ratio is controlled by the number of teeth even if the diameters are a bit off. To create large gear ratios, gears are often connected together in gear trains, as shown here. The right-hand (purple) gear in the train is actually made in two parts. A small gear and a larger gear are connected together, one on top of the other. Gear trains often consist of multiple gears in the train. In the train on the left, the smaller gears are one-fifth the size of the larger gears. That means that if you connect the purple gear to a motor spinning at 100 revolutions per minute (rpm), the green gear will turn at a rate of 500 rpm and the red gear will turn at a rate of 2,500 rpm. In the same way, you could attach a 2,500-rpm motor to the red gear to get 100 rpm on the purple gear. Now that there is a basic understanding of gears and their workings it will be easier to understand the way it relates to trigonometry.

To make the wheels of a car turn the crankshaft and drive shaft must turn first. All engines have a maximum number of rpm's some as low as 1000 others are as high as 15000. It is extremity inefficient to redline your car or do the maximum rpm's. All cars have an efficient range. For a particular six speed car the crankshaft to drive shaft rpm ration is: first gear 3.5 to 1, second gear 2 to 1, third gear 1.4 to 1, fourth gear 1 to 1, fifth gear .8 to 1 and sixth gear .6 to 1 and the efficient rpm range is 1800 to 3200 rpm's. There is a device in every car called the tachometer that monitors the rpm's of the crankshaft. When driving slowly the drive shaft turns slowly

...