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Chapter 2: Problem 417

Washington, DC, is located at \(39^{\circ} \mathrm{N}\) and \(77^{\circ} \mathrm{W}\) (see the following figure). Assume the radius of Earth is 4000 mi. Express the location of Washington, DC, in spherical coordinates.

Short Answer

Expert verified
The spherical coordinates are \((4000 \text{ mi}, 51^{\circ}, -77^{\circ})\)."

Step by step solution

01

Understand Spherical Coordinates

Spherical coordinates \((r, \theta, \phi)\) consist of three components: \(r\) is the radial distance from the origin, \(\theta\) is the polar angle (angle from the vertical z-axis), and \(\phi\) is the azimuthal angle (angle in the xy-plane from the x-axis). In our case, \(r\) is the radius of the Earth since the location is on the Earth's surface.
02

Determine Radius

For the location of Washington, DC, since it is on the Earth's surface, the radial distance \(r\) is the radius of the Earth. Thus, \(r = 4000 \text{ mi}\).
03

Compute Polar Angle \(\theta\)

The polar angle \(\theta\) is normally defined from the positive z-axis toward the point. \(\theta\) is derived from the colatitude, \(\theta = 90^{\circ} - \text{latitude}\). For Washington, DC, \(\text{latitude} = 39^{\circ}\), so \(\theta = 90^{\circ} - 39^{\circ} = 51^{\circ}\).
04

Compute Azimuthal Angle \(\phi\)

The azimuthal angle \(\phi\) is measured in the xy-plane from the positive x-axis usually looking east from the north, corresponding to the longitude. For Washington, DC, \(\text{longitude} = 77^{\circ}W\). Typically, west is negative, so \(\phi = -77^{\circ}\).
05

Express Spherical Coordinates

The spherical coordinates of Washington, DC based on the calculations are \((4000 \text{ mi}, 51^{\circ}, -77^{\circ})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Angle
The polar angle, denoted as \( \theta \), is an essential element in spherical coordinates. It is defined as the angle between the positive z-axis and the line vector pointing towards the position in question. In simpler terms, it tells us how far down from the North Pole a location is when visualizing it on a spherical model of the Earth.
  • In terms of latitude, the polar angle can be obtained using the formula \( \theta = 90^{\circ} - \text{latitude} \). This transformation is because the latitude starts at the equator (0掳) and ranges up to the poles (90掳).
  • For example, if you have a latitude of \( 39^{\circ} \), like Washington, DC, the polar angle would be \( \theta = 90^{\circ} - 39^{\circ} = 51^{\circ} \).
  • This conversion ensures that as you move from the equator towards the North Pole, \( \theta \) decreases, with 0掳 at the North Pole.
Azimuthal Angle
The azimuthal angle \( \phi \) describes positions in the xy-plane from a specific orientation. In the context of geography and Earth's spherical coordinates, it often represents the longitude.
  • Calculating the azimuthal angle involves a circular measurement around the equator, beginning from the positive x-axis and covering a 360掳 view around Earth.
  • Traditionally, eastward longitudes are considered positive and westward ones negative. So, for Washington, DC, which is located at a longitude of \( 77^{\circ}\text{W} \), \( \phi = -77^{\circ} \).
  • This means the angle is measured counter-clockwise when viewed from above, or in simpler terms, clockwise direction from the x-axis defined by the prime meridian.
Radius of Earth
The radius in spherical coordinates refers to the radial distance \( r \) from the center of the Earth to the location on its surface. For spherical coordinates, especially when dealing with geographical locations, \( r \) is set to the radius of Earth.
  • Typical value: For these calculations, a common approximation is \( r = 4000 \text{ mi} \). This value simplifies many computations when determining distances or plotting points on Earth鈥檚 surface.
  • Importance: Understanding the radius helps in computing not only distances but also shapes and sizes of objects or paths on the Earth, such as flight paths or satellite trajectories.
  • Application: For Washington, DC, the point is essentially positioned on the Earth's surface, so \( r \) is the Earth's radius. This ensures that spherical coordinates accurately reflect the true dimensions from the Earth's center to the surface.

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Most popular questions from this chapter

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