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Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x 2 + y 2 + z 2 = c 2 has the simple equation ρ = c in spherical coordinates.

In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in [link] . Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 4000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.

This figure is an image of the Earth. It has the prime meridian labeled, which is a circle on the surface circumnavigating the Earth vertically through the poles. The equator is also labeled which is a horizontal circle circumnavigating the Earth. Three vectors extend out from the center of Earth. Two of them extend to the equator and indicate a measurement of longitude. Two of them extend to a vertical polar circle and indicate a measurement of latitude.
In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian.

Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive z -axis. The prime meridian represents the trace of the surface as it intersects the xz -plane. The equator is the trace of the sphere intersecting the xy -plane.

Converting latitude and longitude to spherical coordinates

The latitude of Columbus, Ohio, is 40 ° N and the longitude is 83 ° W, which means that Columbus is 40 ° north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is 40 ° . In the same way, measuring from the prime meridian, Columbus lies 83 ° to the west. Express the location of Columbus in spherical coordinates.

The radius of Earth is 4000 mi, so ρ = 4000 . The intersection of the prime meridian and the equator lies on the positive x -axis. Movement to the west is then described with negative angle measures, which shows that θ = −83 ° , Because Columbus lies 40 ° north of the equator, it lies 50 ° south of the North Pole, so φ = 50 ° . In spherical coordinates, Columbus lies at point ( 4000 , −83 ° , 50 ° ) .

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Sydney, Australia is at 34 ° S and 151 ° E . Express Sydney’s location in spherical coordinates.

( 4000 , 151 ° , 124 ° )

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Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one.

Choosing the best coordinate system

In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem. Note : There is not enough information to set up or solve these problems; we simply select the coordinate system ( [link] ).

  1. Find the center of gravity of a bowling ball.
  2. Determine the velocity of a submarine subjected to an ocean current.
  3. Calculate the pressure in a conical water tank.
  4. Find the volume of oil flowing through a pipeline.
  5. Determine the amount of leather required to make a football.
    This figure has 5 images. The first image shows bowling balls. The second image is a submarine traveling on an ocean surface. The third image is a traffic cone. The fourth image is a pipline across some barren land. The fifth image is a football.
    (credit: (a) modification of work by scl hua, Wikimedia, (b) modification of work by DVIDSHUB, Flickr, (c) modification of work by Michael Malak, Wikimedia, (d) modification of work by Sean Mack, Wikimedia, (e) modification of work by Elvert Barnes, Flickr)
  1. Clearly, a bowling ball is a sphere, so spherical coordinates would probably work best here. The origin should be located at the physical center of the ball. There is no obvious choice for how the x -, y - and z -axes should be oriented. Bowling balls normally have a weight block in the center. One possible choice is to align the z -axis with the axis of symmetry of the weight block.
  2. A submarine generally moves in a straight line. There is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice. The z -axis should probably point upward. The x - and y -axes could be aligned to point east and north, respectively. The origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port.
  3. A cone has several kinds of symmetry. In cylindrical coordinates, a cone can be represented by equation z = k r , where k is a constant. In spherical coordinates, we have seen that surfaces of the form φ = c are half-cones. Last, in rectangular coordinates, elliptic cones are quadric surfaces and can be represented by equations of the form z 2 = x 2 a 2 + y 2 b 2 . In this case, we could choose any of the three. However, the equation for the surface is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice. In addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a component that represents height and depth directly. Based on this reasoning, cylindrical coordinates might be the best choice. Choose the z -axis to align with the axis of the cone. The orientation of the other two axes is arbitrary. The origin should be the bottom point of the cone.
  4. A pipeline is a cylinder, so cylindrical coordinates would be best the best choice. In this case, however, we would likely choose to orient our z -axis with the center axis of the pipeline. The x -axis could be chosen to point straight downward or to some other logical direction. The origin should be chosen based on the problem statement. Note that this puts the z -axis in a horizontal orientation, which is a little different from what we usually do. It may make sense to choose an unusual orientation for the axes if it makes sense for the problem.
  5. A football has rotational symmetry about a central axis, so cylindrical coordinates would work best. The z -axis should align with the axis of the ball. The origin could be the center of the ball or perhaps one of the ends. The position of the x -axis is arbitrary.
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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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