See photo ### Calculating Curvature Using Parametric EquationsTo determine the curvature $ \kappa $ of a curve defined parametrically by $ x $ and $ y $ in terms of $ t $, use the following formula:\[\kappa = \frac{| \dot{x} \ddot{y} - \dot{y} \ddot{x} |}{(\dot{x}^2 + \dot{y}^2)^{3/2}}\]Here, the dots over $ x $ and $ y $ represent derivatives with respect to $ t $:- $ \dot{x} $ is the first derivative of $ x $ with respect to $ t $,- $ \ddot{x} $ is the second derivative of $ x $ with respect to $ t $,- $ \dot{y} $ is the first derivative of $ y $ with respect to $ t $,- $ \ddot{y} $ is the second derivative of $ y $ with respect to $ t $.### Example: Ellipse CurvatureFind the curvature of the ellipse defined by the parametric equations:- $ x = 3 \cos t $- $ y = 4 \sin t $Calculate the curvature at the points $(3, 0)$ and $(0, 4)$.The given formulas and process allow you to understand and compute how curves deviate from being straight.

Name: See photo ### Calculating Curvature Using Parametric EquationsTo deter
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: See photo ### Calculating Curvature Using Parametric EquationsTo determine the curvature $ \kappa $ of a curve defined parametrically by $ x $ and $ y $ in terms of $ t $, use the following formula:\[\kappa = \frac{| \dot{x} \ddot{y} - \dot{y

Answered: Use the formula to find the curvature.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Topic Video

Question

See photo

### Calculating Curvature Using Parametric Equations

To determine the curvature $ \kappa $ of a curve defined parametrically by $ x $ and $ y $ in terms of $ t $, use the following formula:

\[
\kappa = \frac{| \dot{x} \ddot{y} - \dot{y} \ddot{x} |}{(\dot{x}^2 + \dot{y}^2)^{3/2}}
\]

Here, the dots over $ x $ and $ y $ represent derivatives with respect to $ t $:
- $ \dot{x} $ is the first derivative of $ x $ with respect to $ t $,
- $ \ddot{x} $ is the second derivative of $ x $ with respect to $ t $,
- $ \dot{y} $ is the first derivative of $ y $ with respect to $ t $,
- $ \ddot{y} $ is the second derivative of $ y $ with respect to $ t $.

### Example: Ellipse Curvature

Find the curvature of the ellipse defined by the parametric equations:
- $ x = 3 \cos t $
- $ y = 4 \sin t $

Calculate the curvature at the points $(3, 0)$ and $(0, 4)$.

The given formulas and process allow you to understand and compute how curves deviate from being straight.

Expert Solution

Step 1

The equation of the given ellipse is $x = 3 \cos t, y = 4 \sin t$ .

Now,

$x = 3 \cos t \dot{x} = - 3 \sin t \ddot{x} = - 3 \cos t y = 4 \sin t \dot{y} = 4 \cos t \ddot{y} = - 4 \sin t$

Substitute the above equations in the equation of the curvature $κ = \frac{|\dot{x} \ddot{y} - \dot{y} \ddot{x}|}{{[{\dot{x}}^{2} + {\dot{y}}^{2}]}^{\frac{3}{2}}}$ as follows.

$\begin{array}{rcl} κ & = & \frac{|\dot{x} \ddot{y} - \dot{y} \ddot{x}|}{{[{\dot{x}}^{2} + {\dot{y}}^{2}]}^{\frac{3}{2}}} \\ = & \frac{|(- 3 \sin t) (- 4 \sin t) - (4 \cos t) (- 3 \cos t)|}{{[{(- 3 \sin t)}^{2} + {(4 \cos t)}^{2}]}^{\frac{3}{2}}} \\ = & \frac{|12 \sin^{2} t + 12 \cos^{2} t|}{{[9 \sin^{2} t + 16 \cos^{2} t]}^{\frac{3}{2}}} \\ = & \frac{|12 (\sin^{2} t + \cos^{2} t)|}{{[9 {(\sin t)}^{2} + 16 {(\cos t)}^{2}]}^{\frac{3}{2}}} \\ = & \frac{|12 (1)|}{{[9 {(\frac{y}{4})}^{2} + 16 {(\frac{x}{3})}^{2}]}^{\frac{3}{2}}} [∵ \sin^{2} t + \cos^{2} t = 1, x = 3 \cos t, y = 4 \sin t] \\ = & \frac{12}{{[\frac{9 y^{2}}{16} + \frac{16 x^{2}}{9}]}^{\frac{3}{2}}} \end{array}$

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Quadrilaterals$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions