Spherical Coordinate System Interpretation

In summary, the particle initially travels eastward and then turns to travel northward. After doing so, it crosses the equator at longitude ±90°.
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Homework Statement



(a) Starting from a point on the equator of a sphere of radius R, a particle travels through an angle α eastward and then through an angle β along a great circle toward the north pole. If the initial position is taken to correspond to x = R, y = 0, z = 0, show that its final coordinates are (Rcosαcosβ, Rsinαcosβ, Rsinβ).

(b) Find the coordinates of the final position of the same particle if it first travels through an angle α northward, then changes course by 90° and travels through an angle β along a great circle that starts out eastward.

2. The attempt at a solution
This is basically geometry, and I've shown part (a) correctly via a diagram. I don't seem to get the correct answer for part (b) and I suspect I am not interpreting the question correctly.

Attached is a somewhat crude (though the clearest I can produce!) diagram showing how I see the situation for (a) and (b). I haven't added my annotations for my answer for clarity's sake. I get the following for (b):

x = Rcosαcosβ
y = Rcosαsinβ
z = Rsinα

If someone could tell me whether or not I am interpreting the information correctly (and hence whether it actually is my geometric analysis) I would very much appreciate it. :)
 

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  • #2
Your diagram for b is wrong. After changing course it travels on a great circle. That is not a circle parallel to the equator.
 
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  • #3
Hi Zatman! :smile:
Zatman said:
(b) Find the coordinates of the final position of the same particle if it first travels through an angle α northward, then changes course by 90° and travels through an angle β along a great circle that starts out eastward.

z = Rsinα

No, that's still on the (small) circle of latitude α

the great circle dips down towards the equator, crossing it at longitude ±90°. :wink:
 
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  • #4
Got it. I didn't realize "great" actually meant something here. Thanks to you both. :)
 

1. What is a spherical coordinate system?

A spherical coordinate system is a three-dimensional coordinate system used to locate points on a sphere. It is defined by three coordinates: radius (r), inclination (θ), and azimuth (φ).

2. How is a point represented in spherical coordinates?

In spherical coordinates, a point is represented by the distance from the origin (r), the angle between the positive z-axis and the line connecting the point to the origin (θ), and the angle between the positive x-axis and the projection of the point onto the xy-plane (φ).

3. What are the advantages of using a spherical coordinate system?

A spherical coordinate system is particularly useful for representing points on a sphere or points in three-dimensional space that involve angles. It is also helpful in solving problems involving symmetry, such as in physics and engineering.

4. How is the spherical coordinate system related to the Cartesian coordinate system?

The spherical coordinate system is related to the Cartesian coordinate system through the following equations:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
where r is the distance from the origin, θ is the inclination angle, and φ is the azimuth angle.

5. Can a point have more than one representation in spherical coordinates?

Yes, a point can have more than one representation in spherical coordinates. For example, a point with the coordinates (r, θ, φ) will have the same representation as a point with coordinates (r, π - θ, φ + π) or (r, θ + 2π, φ). Therefore, it is important to specify the range of values for each coordinate to avoid confusion.

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