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Magazine Chapter 2: Problem 406
For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle \(\varphi\) in radians rounded to four decimal places. \(\left(3,-\frac{\pi}{6}, 3\right)\)
Short Answer
Expert verified
Spherical coordinates are approximately \((4.2426, -0.5236, 0.7854)\).
Step by step solution
01
Identify Given Cylindrical Coordinates
Cylindrical coordinates are given as \( (r, \theta, z) \). In this exercise, the coordinates are \( r = 3 \), \( \theta = -\frac{\pi}{6} \), and \( z = 3 \).
02
Find Spherical Radius \( \rho \)
In spherical coordinates, \( \rho \) is the distance from the origin to the point. Use the formula \( \rho = \sqrt{r^2 + z^2} \). Substitute \( r = 3 \) and \( z = 3 \):\[\rho = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \approx 4.2426.\]
03
Find Polar Angle \( \phi \)
The polar angle \( \phi \) in spherical coordinates is the angle from the positive \( z \)-axis to the point. Use the formula \( \phi = \arccos\left(\frac{z}{\rho}\right) \). With \( z = 3 \) and \( \rho \approx 4.2426 \), calculate:\[\phi = \arccos\left(\frac{3}{3\sqrt{2}}\right) = \arccos\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \approx 0.7854 \text{ radians}.\]
04
Use Azimuthal Angle \( \theta \)
The azimuthal angle \( \theta \) is the same in both cylindrical and spherical coordinates. Thus, \( \theta = -\frac{\pi}{6} \) remains unchanged.
05
Write the Spherical Coordinates
Using the spherical coordinates \((\rho, \theta, \phi)\), we have:\[\left(3\sqrt{2}, -\frac{\pi}{6}, \frac{\pi}{4}\right) \approx (4.2426, -0.5236, 0.7854).\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cylindrical Coordinates
Cylindrical coordinates are a way to define the position of a point in a three-dimensional space. They are particularly useful when dealing with problems involving symmetry around an axis, such as those encountered in physics and engineering.
- Components: A point in cylindrical coordinates is described using three values:
- \( r \): The radial distance from the origin to the point's projection onto the XY-plane. Think of it as how far out the point is from the vertical \( z \) axis.
- \( \theta \): The azimuthal angle, which is the angle in radians from the positive x-axis to the line connecting the origin to the point's projection in the XY-plane.
- \( z \): The height above the XY-plane directly up to the point.
- Interpreting Cylindrical Coordinates: Consider the point given in the exercise, \( (3, -\frac{\pi}{6}, 3) \). Here, \( r = 3 \) which means the point is 3 units away from the \( z \) axis. The angle \( \theta = -\frac{\pi}{6} \) or \(-30^{\circ}\) indicates the direction along which the projection onto the XY-plane lies. The \( z \) coordinate, also 3, suggests that the point lies 3 units above the XY-plane.
Polar Angle
The polar angle, represented by \( \phi \), is an important component in spherical coordinates. It specifies the angle between the positive z-axis and the line connecting the origin to the point in space. Understanding how to find this angle is crucial for converting between coordinate systems.
- Formula for Polar Angle: When converting from cylindrical to spherical coordinates, the polar angle is determined using: \[ \phi = \arccos\left(\frac{z}{\rho}\right) \] This formula calculates the angle based on the \( z \)-height and the spherical radius \( \rho \).
- Example Calculation: In our example, \( z = 3 \) and \( \rho = 3\sqrt{2} \approx 4.2426 \). Substituting these values into the formula, the polar angle is \[ \phi = \arccos\left(\frac{3}{3\sqrt{2}}\right) = \frac{\pi}{4} \approx 0.7854 \text{ radians} \]. This angle tells us how far the point deviates from the zenith (the direct overhead direction).
Spherical Radius
The spherical radius, often denoted as \( \rho \), is a vital aspect of spherical coordinates. It represents the direct distance from the origin to a point in three-dimensional space.
- Determining the Spherical Radius: To find the spherical radius from cylindrical coordinates, we use the formula: \[ \rho = \sqrt{r^2 + z^2} \] This involves squaring the radial distance \( r \) and height \( z \), summing them up, and taking the square root of the result.
- Example Calculation: Given the coordinates \( (3, -\frac{\pi}{6}, 3) \), we calculate the spherical radius as: \[ \rho = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \approx 4.2426 \]. This value is crucial for transitioning between coordinate systems as it provides a universal measurement of radial distance in any direction from the origin.
Coordinate Conversion
Converting between different coordinate systems is a fundamental skill in mathematics and physics, as it allows for flexible problem-solving. In this case, we're converting from cylindrical to spherical coordinates. This process involves recalculating each component to fit into the new system.
- Understanding Conversion Steps: The conversion requires using specific relationships:
- The spherical radius \( \rho \) is determined by combining the components of the cylindrical coordinates with \( \rho = \sqrt{r^2 + z^2} \).
- The azimuthal angle \( \theta \) remains unchanged, as it is the same in both coordinate systems.
- The polar angle \( \phi \) is computed through the relationship \( \phi = \arccos\left(\frac{z}{\rho}\right) \).
- Application on Given Data: For the coordinates \( (3, -\frac{\pi}{6}, 3) \), conversion yields spherical coordinates \( \left(3\sqrt{2}, -\frac{\pi}{6}, \frac{\pi}{4}\right) \approx (4.2426, -0.5236, 0.7854) \). These translated values maintain the integrity of the point's position in space but describe it using different metrics.
