Vectors and Forces

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Earth's Moving Fluids

Wind and ocean currents carry heat and essential matter around the planet, making most of it inhabitable.  But how fast and where do they move?  We need to use a quantity that contains both pieces of information, and they are called vectors.  


Note: You don't need to be a math wizard to use vectors.  Working through the concepts graphically allows you to make useful estimates of the behavior of winds and ocean currents around the planet. But if you want to explore the math to be more precise with your analyses, it is presented here as well..

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Vectors

 A vector is a quantity that has both magnitude and direction. When drawn, it is typically represented by an arrow whose direction is based on the accepted axes and whose length is proportional to the quantity's magnitude. Examples of vectors in Earth Systems: temperature gradient, displacement, velocity, acceleration, force, wind, and ocean currents. Quantities with only a magnitude are scalars, such as distance, temperature, energy, and mass.

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Convention of Vector Directions

For navigation, moving at a bearing of 0º means you are traveling northward, and the angle increases clockwise. A bearing of 90º means traveling toward the east.  In meteorology, wind vectors use a different convention.  A 0º wind blows from the north, not going toward the north. Wind direction increases clockwise, as  navigation does.  Trigonometry has 0º pointing along the positive x axis and increases counterclockwise. So it is very important to know the conventions of direction for the vectors you are working with.


The beauty of vectors is that we can change the orientation of the axes to make it easier to work with the math. This is very useful for studying many phenomena in our physical world. We often use ‘x’ and ‘y’ when working with two dimensions. If working in three dimensions, the third axis is usually ‘z’. Key for 2 or 3D: the axes must be perpendicular to each other. This allows for information to be separate from each axis. If you took one step exactly north, there would not be a change in your east/west position.

Right Angle Trigonometry

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Right Angle Triangles and the Pythagorean Theorem

A right angle triangle has one angle equal to 90º. The side opposite this ‘right’ angle is called the hypotenuse, and it will always be the longest side of a right triangle.


Right triangles mean we can apply the Pythagorean Theorem: the square of the length of the hypotenuse equals the sum of the square of the lengths of the two shorter sides or: 

 

                       opposite^2 + adjacent^2 = hypotenuse^2


This allows the ratio of the lengths of the triangle’s sides to be calculated, upon which we base the trigonometric functions: sine, cosine, and tangent.

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The Essential Trig Functions

Sine (sin) is the ratio of the length of the side opposite an angle of the triangle (⍬) divided by the length of the hypotenuse.


                            sin(⍬) = opposite / hypotenuse


Cosine (cos) is the ratio of the length of the side adjacent to an angle of the triangle divided by the length of the hypotenuse.

 

                            cos(⍬) = adjacent / hypotenuse


Tangent (tan) is ratio of the length of the side opposite to an angle divided by the length of the side adjacent to the angle.

 

                            tan(⍬) = opposite / adjacent


For additional information on trig functions and their inverses, illustrated examples, and practice problems with solutions, see the Primer on Trigonometry in Earth Science below.

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Axial Components and Magnitude & Direction

Since the axes are perpendicular to each other, we use trig functions to find the axial components of vectors, so they are essential when mathematically calculating vector addition. Key is to identify the angle to an axis of interest and then use the trig function that works. In all cases, the hypotenuse will be the magnitude of the vector.

Adding Vectors

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All quantities of the same type (so they have the same units) may be added together, but adding vectors is a bit more involved than adding scalar quantities (those with only a magnitude). When vectors are added together, the sum is called the ‘resultant’.

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'Tail to Tip' Graphical Addition

Graphical vector addition is often called “tail to tip”. The tail of a second vector is aligned to the tip of the first vector. If there is a third vector, the tail of the third vector is added to the tip of the sum of the first two vectors.


When using this technique, create a scale to draw the vectors so the proportions of the magnitudes are consistent.  The more accurately the vectors' magnitudes and directions are drawn, the more accurate the results.

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Mathematical Vector Addition

The mathematical way to add vectors is to to add the magnitudes of the axial components. Add all of the magnitudes of the x components and all of the magnitudes of the y components.  This requires proficiency in converting magnitude and direction to axial components.  If needed, then use the Pythagorean Theorem and trig functions (most likely the inverse tangent) to calculate the magnitude and direction of the resultant.

Forces and Newton's Laws of Motion

Energy has the ability to make things move, but forces are what actually get things to speed up, slow down, and change direction. A force is the amount of push or pull acting on an object, so a force is a vector. The amount is the magnitude, and the push or pull is the direction.


Isaac Newton identified three key relationships (or laws) about forces and motion:


1) The acceleration of an object is equal to the force acting on an object divided by the mass of the object. This may be rewritten to be:  F = ma


Acceleration is the rate that the velocity is changing; velocity is the rate that displacement is changing; and displacement is the change in position of an object. All three are vectors. This means that acceleration occurs if the magnitude or direction of motion change, so a force must be applied to do so. 


2) A body only accelerates when acted upon by an unbalanced force. If only one force is acting on a body, it must be unbalanced. But if two or more forces are acting on an object, if they add to zero, then the forces are balanced, and there won’t be a changing in motion. If the object is moving in straight line, it will continue to move in the same direction and speed, but if the object is at rest, it will stay at rest.


3) For every action there is an equal and opposite reaction. When you were pushing your hands together and they weren’t moving, each hand was pushing equally into the other.


We will focus on #2: a body only accelerates when acted upon by an unbalanced force, which means we will be adding force vectors to see if they add to zero (a balanced set of forces) or a vector with a magnitude greater than zero (an unbalanced force).

Balanced forces: the rock is not moving since the ground is pushing back equally against its weight.

Balanced forces: the rock is not moving since the ground is pushing back equally against its weight.

Software to Practice Adding Vectors and Working with Forces

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First: Magnitude and Direction and Axial Components

Explore how these fit together conceptually and mathematically.


Create unlimited vectors to practice converting between magnitude and direction to axial components and vice versa.  Math and science teachers, save any graphic to include in homework or tests.

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Navigate the Color World with Vectors

This activity comes from my classroom experience helping students explore vectors and how they may be added conceptually/graphically ('tail to tip'). We used color cubes as a relatively inexpensive 3D physical model to explore the world of color using vectors of the magnitude/intensity of red, green, and blue to study the world of color in light and vectors of yellow, magenta, and cyan to explore the world of dyes, pigments, and paints. Teachers, consider sharing the cost with the art department since these cubes are an excellent resource for art classes as well.


This also helped us explore reflection, transmission, and absorption of visible light, which was useful to reinforce concepts about electromagnetic radiation presented in the ‘Energy’ section.

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Add Vectors Graphically and Mathematically

Each technique supports the other.  Graphical vector addition helps catch any errors in the mathematical calculations, and the mathematical calculations help verify the proportions used in the graphical technique.


Create unlimited sets of 3 or 4 vectors to add.  Math and science teachers, save any graphic to include in homework or tests.

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Working with Balanced and Unbalanced Forces

Create unlimited sets of forces to decide if they are balanced or modify one of the forces to create balanced forces.  Physics teachers, this is a virtual force table.  Math and science teachers, save any graphic to include in homework or tests.

Download Earth, Wind, and Forces Software

Use the software to explore vectors, forces, and wind on Earth.

  

Version 1-1 was greatly improved by the feedback and creative suggestions from Elliot Blume-Pickle, Allison Culbert,              Mason Glidden, Hyowon (Raphi) Kang, and Corey Rost, 


Mac users: Download the zipped dmg file. Uncompress and double click on the dmg file.  Drag the folder that pops up to the application folder on your computer.

Windows users: Select the location you want to put the files before uncompressing everything in the zip file.

Suggested Time: 2 60-minute blocks of time to explore the many options.


Wind calculations are based on Roland Stull's on-line book "Practical Meteorology: An Algebra-based Survey of Atmospheric Science", in particular, Chapter 10 "Atmospheric Forces and Wind."  This book is an invaluable resource for those wanting to dive into the mathematical analysis of atmospheric phenomena.