Posts Tagged ‘trigonometry’

Using Parallax to Measure Distance

Monday, January 19th, 2015

      Last time we introduced an optical effect known as the principle of parallax.   Today we’ll get a step closer to seeing how it’s instrumental in measuring distances.   For our example we’ll be viewing a point on a tree several blocks away.

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      Since we can’t measure the distance, r, to the tree directly by using a tape measure, we’ll have to use another approach that is made available to us through the principle of parallax.   The first step to doing this is to establish two different lines of sight to the red dot on the tree.   For our example these will originate at Points A and B, which are both viewing the same spot.

      Line of Sight A extends straight from Point A to the red spot, while Point B, which is situated only a few feet to the right of A, will provide us with a slanted line of sight to it.   We’ll refer to the path between Points A and B as Path AB, which is represented by a black dashed line and has a length, d.

      You may have noticed from the illustration that Line of Sight B, because it lies on a slant relative to Line of Sight A, provides us with an internal angle, θ, relative to Path AB.   This relationship is in fact required to exist if we are to use the principle of parallax, and you’ll see why later.

      Another requirement is that Path AB must be at a right angle to Line of Sight A, which it is.   The illustration bears this out by the fact that a right triangle is formed when Lines of Sight A, B, and Path AB intersect.

      Right triangles are generally regarded as special within mathematics, but they are particularly special when applying the principle of parallax.   The presence of a right triangle enables us to use a branch of mathematics known as trigonometry to correlate the triangle’s angles to the ratios of the lengths of its sides.   In other words, we can calculate r by using trigonometry if we are able to measure d and θ.

      We’ll see how that calculation is done 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.

gears and torque

      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.

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Torque Formula Symplified

Wednesday, April 2nd, 2014

      Last time we introduced the mathematical formula for torque, which is most simply defined as a measure of how much a force acting upon an object causes that object to rotate around a pivot point.   When manipulated, torque can produce a mechanical advantage in gear trains and tools, which we’ll see later.   The formula is:

Torque = Distance × Force × sin(ϴ)

      We learned that the factors Distance and Force are vectors, and sin(ϴ) is a trigonometric function of the angle ϴ which is formed between their two vectors.   Let’s return to our wrench example and see how the torque formula works.

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      Vectors have both a magnitude, that is, a size or extent, and a direction, and they are typically represented in physics and engineering problems by straight arrows.   In our illustration the vector for distance is represented by an orange arrow, while the vector for force is represented by a red arrow.   The orange distance vector has a magnitude of 6 inches, while the red force vector has a magnitude of 10 pounds, which is being supplied by the user’s arm muscle manipulating the nut.   That muscle force follows a path from the arm to the pivot point located at the center of the nut, a distance of 6 inches.

      Vector arrows point in a specific direction, a direction which is indicative of the way in which the vectors’ magnitudes — in our case inches of distance vs. pounds of force — are oriented with respect to one another.   In our illustration the orange distance vector points away from the pivot point.   This is according to engineering and physics convention, which dictates that, when a force vector is acting upon an object to produce a torque, the distance vector always points from the object’s pivot point to the line of force associated with the force vector.   The angle, ϴ, that is formed between the two vectors in our example is 90 degrees, as measured by any common, ordinary protractor.

      Next we must determine the trigonometric value for sin(ϴ).   This is easily accomplished by simply entering “90” into our calculator, then pressing the sin button.   An interesting fact is that when the angle ϴ ranges anywhere between 0 and 90 degrees, the values for sin(ϴ) always range between 0 and 1.   To see this in action enter any number between 0 and 90 into a scientific calculator, then press the sin button.

      For our angle of 90 degrees we find that,

sin(90) = 1

      Thus the formula for torque in our example, because the sin(ϴ) is equal to 1, simply becomes the product of the magnitudes of the Distance and Force vectors:

Torque = Distance × Force × sin(90)

Torque = Distance × Force × 1

Torque = Distance × Force

      Next time we’ll insert numerical values into the equation and see how easily torque can be manipulated.

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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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