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Last time we introduced the vertical asymptote and the fact that it’s associated with the tangent function when angles approach 90° in value. On a graph that looks like this:
Figure 1This asymptotic relationship exists when attempting to use an optical rangefinder to determine the distance to objects that are extremely far away– as in so far away they can barely be seen by the naked eye. When this is the case, is it even possible to use the optical rangefinder? Theoretically, yes. But not without complications. Our attempt to use the rangefinder to do this is illustrated in Figure 2.
Figure 2You’ll note that the lines of sight extending from mirrors A and B on our rangefinder are almost parallel to each other, creating a situation where a vertical asymptote will form with regard to θ’s tangent. In plain English this means that even minute changes in θ will result in huge changes to tan(θ). We’ll explore that subject next time.
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Posts Tagged ‘theta’
Vertical Asymptotes and Optical Distance Measurement
Friday, March 13th, 2015
Parallax and Trigonometry
Monday, January 26th, 2015|
We’ve been working on a way to calculate the distance to a tree situated blocks from our viewing point. Flying raptors, such as our beloved bald eagle, wouldn’t find this in the least bit challenging. They’re able to accurately judge distances due to their special shape-shifting eye lenses which are capable of actually changing curvature spontaneously. But human physiology isn’t equipped for this task, so we’ll have to employ other methods. Today we’ll see how a branch of mathematics known as trigonometry comes into play.
Referring to the illustration, we’d like to calculate the distance, r. You’ll note that a right triangle is formed by Line of Sight A, Path AB, and Line of Sight B. As mentioned in our last blog, right triangles are special because the relationship that exists between their sides and their internal angles is well defined within mathematics. In fact, we can calculate r by using this trigonometric formula, r = d × tan(θ) where d is the length of Path AB, θ is the angle between Line of Sight B and Path AB, and tan(θ) is the trigonometric function known as the tangent. Tangent, and other trigonometric functions like sine and cosine, relate the angles in a right triangle to the ratios of the lengths of the sides of the triangle. If we know two of the variables present in the equation presented above, we can determine the third, and the fact that there’s a right triangle present makes that task so much easier. As things stand now we have two unknowns, d and θ. As pointed out last week, the distance, d, is short, so we’ll use a tape measure to determine its length. Let’s say it measures out to be three feet. Now we need to solve for the angle θ that’s formed between Path AB and Line of Sight B. That’s a bit more challenging. There are a number of devices that can be used to measure θ, including a handheld magnetic compass. However, using a compass often yields inaccurate results, thereby increasing the likelihood of mistakes. A more accurate device to use would be an optical rangefinder, as shown below.
The optical rangefinder is a device that’s often used in the military to measure long distances by using the principle of parallax. It functions much like binoculars do, but with a twist, literally, as we’ll see next time. ____________________________________
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The Mathematical Link Between Gears in a Gear Train
Wednesday, May 14th, 2014|
Last time we analyzed the angular relationship between the Force and Distance vectors in this simple gear train. Today we’ll discover a commonality between the two gears in this train which will later enable us to develop individual torque calculations for them.
From the illustration it’s clear that the driving gear is mechanically linked to the driven gear by their teeth. Because they’re linked, force, and hence torque, is transmitted by way of the driving gear to the driven gear. Knowing this we can develop a mathematical equation to link the driving gear Force vector F1 to the driven gear Force vector F2, then use that linking equation to develop a separate torque formula for each of the gears in the train. We learned in the previous blog in this series that F1 and F2 travel in opposite directions to each other along the same line of action. As such, both of these Force vectors are situated in the same way so that they are each at an angle value ϴ with respect to their Distance vectors D1 and D2. This fact allows us to build an equation with like terms, and that in turn allows us to use trigonometry to link the two force vectors into a single equation: F = [F1 × sin(ϴ)] – [F2 × sin(ϴ)] where F is called a resultant Force vector, so named because it represents the force that results when the dead, or inert, weight that’s present in the resisting force F2 cancels out some of the positive force of F1. Next week we’ll simplify our gear train illustration and delve into more math in order to develop separate torque computations for each gear in the train. _______________________________________ |
Vectors, Sin(ϴ), and the Torque Formula
Wednesday, March 26th, 2014|
Last time we introduced a physics concept known as torque and how it, together with modified gear ratios, can produce a mechanical advantage in devices whose motors utilize gear trains. Now we’ll familiarize ourselves with torque’s mathematical formula, which involves some terminology, symbols, and concepts which you may not be familiar with, among them, vectors, and sin(ϴ). Torque = Distance × Force × sin(ϴ) In this formula, Distance and Force are both vectors. Generally speaking, a vector is a quantity that has both a magnitude — that is, any measured quantity associated with a vector, whether that be measured in pounds or inches or any other unit of measurement — and a direction. Vectors are typically represented graphically in engineering and physics illustrations by pointing arrows. The arrows are indicative of the directionality of the magnitudes involved. Sin(ϴ), pronounced sine thay-tah, is a function found within a field of mathematics known as trigonometry , which concerns itself with the lengths and angles of triangles. ϴ, or thay-tah, is a Greek symbol used to represent the angle present between the Force and Distance vectors as they interact to create torque. The value of sin(ϴ) depends upon the number of degrees in the angle ϴ. Sin(ϴ) can be found by measuring the angle ϴ, entering its value into a scientific calculator, and pressing the Sin button. We’ll dive into the math behind the vectors next time, when we return to our wrench and nut example and apply vector force quantities.
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