GDB Topic:
Discuss the applications of Polar coordinate System in our daily life with at least 3 examples.
Opening Date: Wednesday, 7th August, 2014 at 12:01 AM
Closing Date: Thursday, 8th August, 2014 at 11:59 PM
Instructions
Tags:
1)
GPS is a form of polar coordinate system, with 2 angles, North and East, and the radius isn't really necessary. This is obviously used by sat-navs, planes, maps on your phone etc.
2)
Electrical engineers use polar coordinates when designing AC components for your house. You learn at school V= I R but actually, it's a vector equation. The result impedance, influences how much voltage you get depending on the voltage. So electrical engineers use polar notation of vectors to make sure all your household products produce the power they're supposed to.
3)
Clocks. Analogue Clocks work on the principle of defining time by 2 angles and 2 lengths: the position of the point of each hand.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.
cylindrical coordinate system would be preferable to deal with a problem on a straight wire with an electrical charge on the surface. A cylindrical coordinate system is a coordinate system that is three-dimensional location of points determined by the distance from a chosen reference axis, the axis direction relative to a chosen reference direction and distance from the plane perpendicular to the chosen reference axis. The last distance is given as a positive or negative depending on which side of the reference plane is facing the issue. A spherical coordinate system would be preferable to deal with many problems on the climate of the earth as thermal radiation. In mathematics, the polar coordinate system is a system of two-dimensional coordinates that determines at each point a plane at a distance from a fixed point and an angle of a fixed address. The fixed point (similar to the origin of a Cartesian system) is called the pole, and the radius of the pole in the fixed direction is the polar axis. The distance from the pole, the radial coordinate or radius and angle is the angular coordinate, polar angle and azimuth.
GPS is a form of polar coordinate system, with 2 angles, North and East, and the radius isn't really necessary. This is obviously used by sat-navs, planes, maps on your phone etc.
Electrical engineers use polar coordinates when designing AC components for your house. You learn at school V= I R but actually, it's a vector equation. The result impedance, influences how much voltage you get depending on the voltage. So electrical engineers use polar notation of vectors to make sure all your household products produce the power they're supposed to.
Clocks. Analogue Clocks work on the principle of defining time by 2 angles and 2 lengths: the position of the point of each hand.
Hope this helps!
7. Polar Coordinates
For certain functions, rectangular coordinates (those using x-axis and y-axis) are very inconvenient. In rectangular coordinates, we describe points as being a certain distance along the x-axis and a certain distance along the y-axis.
polar coordinates
But certain functions are very complicated if we use the rectangular coordinate system. Such functions may be much simpler in the polar coordinate system, which allows us to describe and graph certain functions in a very convenient way.
Polar coordinates work in much the same way that we have seen in trignometry(radians and arc length, where we used r and θ) and in the polar form of complex numbers (where we also saw r and θ).
For instance, there is a phenomenon known as the "triple point," which is difficult to comprehend unless one sees it on a graph. For a chemical compound such as water or carbon dioxide, there is a point at which it is simultaneously a liquid, a solid, and a vapor. This, of course, seems to go against common sense, yet a graph makes it clear how this is possible.
Using the x-axis to measure temperature and the y-axis pressure, a number of surprises become apparent. For instance, most people associate water as a vapor (that is, steam) with very high temperatures. Yet water can also be a vapor—for example, the mist on a winter morning—at relatively low temperatures and pressures, as the graph shows.
The graph also shows that the higher the temperature of water vapor, the higher the pressure will be. This is represented by a line that curves upward to the right. Note that it is not a straight line along a 45° angle: up to about 68°F (20°C), temperature increases at a somewhat greater rate than pressure does, but as temperature gets higher, pressure increases dramatically.
As everyone knows, at relatively low temperatures water is a solid—ice. Pressure, however, is relatively high: thus on a graph, the values of temperatures and pressure for ice lie above the vaporization curve, but do not extend to the right of 32°F (0°C) along the x-axis. To the right of 32°F, but above the vaporization curve, are the coordinates representing the temperature and pressure for water in its liquid state.
Water has a number of unusual properties, one of which is its response to high pressures and low temperatures. If enough pressure is applied, it is possible to melt ice—thus transforming it from a solid to a liquid—at temperatures below the normal freezing point of 32°F. Thus, the line that divides solid on the left from liquid on the right is not exactly parallel to the y-axis: it slopes gradually toward the y-axis, meaning that at ultra-high pressures, water remains liquid even though it is well below the freezing point.
Nonetheless, the line between solid and liquid has to intersect the vaporization curve somewhere, and it does—at a coordinate slightly above freezing, but well below normal atmospheric pressure. This is the triple point, and though "common sense" might dictate that a thing cannot possibly be solid, liquid, and vapor all at once, a graph illustrating the triple point makes it clear how this can happen.
For convenience, we begin with a given xy -coordinate system and then take the origin as the pole and the nonnegative x -axis as the polar axis. Given the pole O and the polar axis, the point P with polar coordinates r and θ , written as the ordered pair (r,θ) , is located as follows. First find the terminal side of the angle θ , given in radians, where θ is measured counterclockwise (if θ>0 ) from the x -axis (the polar axis) as its initial side. If r≥0 , then P is on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the ray opposite the terminal side at the distance ∣r∣=-r>0 from the pole. The radial coordinate r can be described as the directed distance from P the pole along the terminal side of the angle θ . Thus, if r is positive, the point P lies in the same quadrant as θ , whereas if r is negative, then P lies in the opposite quadrant. If r=0 , the angle θ does not matter; the polar coordinates (0,θ) represent the origin whatever the angular coordinate θ might be. The origin, or pole, is the only point for which r=0 .
MTH301 GDB 1
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