Arc Length In Polar Coordinates: Calculus 3 project by Meredith Lapidas
Overview
This Project has been completed as part of a standard 10 weeks Calculus 3 asynchronous online course with optional WebEx sessions during Summer 2021 Semester at MassBay Community College, Wellesley Hills, MA.
Arc Length In Polar Coordinates
Introduction
Imagine a clock set to 3;30, so that the hour hand is on 3 and the minute hand is on 6. If you has to describe the location of the hour hand with respect to the minute hand, what would you say? Rather than measuring the distance between both numbers with exactmeasurements, it would be much easier to say that the 3 and 6 are 90 degrees apart.
Just like this clock example, polar coordinates can be used to simplify locating points on a graph.
A Polar Coordinate is set of values that specify the location of a point based on both the distance between the point and a fixed origin, as well as the angle between the point and a direction.
Polar curves are points that are a set distance from the origin/pole, depending on the angle measured in counterclockwise direction from a polar axis
Examples of polar curves are ellipses, cardioids and lemniscates:
Any equation written in Cartesian coordinates can be converted to one in polar coordinates using the trigonometric relationships:
| \(\cos \theta =\dfrac{x}{r}\) | \(x=r\cos \theta\) |
| \(\sin \theta =\dfrac{y}{r}\) | \(y=r\sin \theta\) |
| \(\tan \theta =\dfrac{x}{y}\) | \(x^2+y^2=r^2\) |
Definition of Arc Length
Arc length is the distance between two points along a section of a curve
- Deriving the arc length formula
In rectangular coordinates, the arc length of a parameterized curve (𝑥(𝑡),𝑦(𝑡)) for 𝑎≤𝑡≤𝑏 is given by:
\(\displaystyle{L}={\int_{{a}}^{{b}}}\sqrt{{{\left(\frac{{\left.{d}{x}\right.}}{{\left.{d}{t}\right.}}\right)}^{2}+{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{t}\right.}}\right)}^{2}}}{\left.{d}{t}\right.}\)
- To find the arc length in terms of polar coordinates, first write the curve in terms of parametric equations:
\(r=f(\theta) \\ x=rcos(\theta)=f(\theta)\cos(\theta) \\ x=rsin (\theta)=f(\theta)\sin(\theta)\)
- We can derive the parametric formula for finding the arc length by replacing the parameter t by θ:
Example
Find the arc length of the cardioid for r(θ) = 1 + cos(θ) for 0 ≤ θ ≤ 2π
\(x(\theta)=(1+\sin(\theta))\cos(\theta) \\ y(\theta)=(1+\sin(\theta))\sin(\theta) \\ r~'(\theta)=-\sin(\theta)\)
... therefore
\(L= \displaystyle {\int_0^{2\pi} \sqrt { (r ( \theta))^2 + (r'(\theta))^2 } ~d \theta = \\ \int_0^{2\pi} \sqrt { (1+\cos \theta)^2 + (-\sin \theta )^2 } ~d \theta = \\ \int_0^{2\pi} \sqrt { 2+ 2\cos \theta } ~d \theta = \\ \int_0^{2\pi} \sqrt { 4{\cos}^{2}{\left(\frac{\theta}{{2}}\right)}}~d \theta =~8 }\)
Applications
Polar coordinates are appropriate in situations where the object being considered is linked to direction and length from a center point in a plane.
In celestial mechanics, polar coordinates can be used to plot planetary orbits. Since there are bodies moving around a central point, it is easiest to use polar coordinates for such planets.
Given the fact that any planetary orbit is an ellipse, the Cartesian form of an elliptical curve is: \(1=\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}\)
Where h is the horizontal displacement of the curve, k is the vertical displacement of the curve a is the radius of the curve in the x direction, and b is the radius of the curve in the y direction.
In a planet’s orbital curve, there is no displacement in the y direction. Therefore, the value k is 0.
One can convert the Cartesian equation into the parametric form of an elliptical curve in a few simple steps:
Finally, one can use values from planetary data (values a,b, and c) to get the polar form of a curve for any planet:
Now, we can plug in each of the values for every planet to obtain...
Nasa explains solar system dynamics well through orbits and Kepler's Laws:
User: n/a - Added: 11/9/10
Links
http://study.com/academy/lesson/polar-coordinates-definition-equation-examples.html