Find the curvature K of the curve. r(t) = 4 cos 2nti + 4 sin 2ntj K.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Curvature of a Parametric Curve**
**Problem Statement:**
Find the curvature \( K \) of the curve.
**Parametric Equation:**
\[
\mathbf{r}(t) = 4 \cos(2 \pi t) \, \mathbf{i} + 4 \sin(2 \pi t) \, \mathbf{j}
\]
**Objective:**
Calculate the curvature \( K \) of the given parametric curve.
**Curvature Formula:**
The formula for the curvature \( K \) of a plane curve given by \(\mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j}\) is:
\[
K = \frac{|x'(t)y''(t) - y'(t)x''(t)|}{\left( (x'(t))^2 + (y'(t))^2 \right)^{3/2}}
\]
**Solution:**
To find the curvature, calculate the first and second derivatives of \( x(t) \) and \( y(t) \) from the parametric equation, substitute them into the curvature formula, and simplify to find \( K \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F482445b4-5947-40a3-b7da-6005a8b8ce0f%2F88c068ae-d55a-4abc-ac33-408a0333925b%2F5n55txr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Curvature of a Parametric Curve**
**Problem Statement:**
Find the curvature \( K \) of the curve.
**Parametric Equation:**
\[
\mathbf{r}(t) = 4 \cos(2 \pi t) \, \mathbf{i} + 4 \sin(2 \pi t) \, \mathbf{j}
\]
**Objective:**
Calculate the curvature \( K \) of the given parametric curve.
**Curvature Formula:**
The formula for the curvature \( K \) of a plane curve given by \(\mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j}\) is:
\[
K = \frac{|x'(t)y''(t) - y'(t)x''(t)|}{\left( (x'(t))^2 + (y'(t))^2 \right)^{3/2}}
\]
**Solution:**
To find the curvature, calculate the first and second derivatives of \( x(t) \) and \( y(t) \) from the parametric equation, substitute them into the curvature formula, and simplify to find \( K \).
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