Find N (t), T(t), aŢ, and añ: r(t) = t³i+ 8 cos tj + 6 sin tk; t = π
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help find the parts of the function shown in the photo provided
![### Vector Function Analysis
**Objective:**
Find \( N(t), T(t), a_T \), and \( a_N \) for the vector function defined as:
\[ \mathbf{r}(t) = t^3 \mathbf{i} + 8 \cos t \mathbf{j} + 6 \sin t \mathbf{k} \]
**Given:**
\[ t = \pi \]
### Steps to Determine the Components:
1. **Find the unit tangent vector \( \mathbf{T}(t) \)**:
- The unit tangent vector \( \mathbf{T}(t) \) is given by:
\[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} \]
2. **Find the unit normal vector \( \mathbf{N}(t) \)**:
- The unit normal vector \( \mathbf{N}(t) \) is given by:
\[ \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|} \]
3. **Find the tangential component of the acceleration \( a_T \)**:
- The tangential component \( a_T \) is given by:
\[ a_T = \mathbf{a}(t) \cdot \mathbf{T}(t) \]
4. **Find the normal component of the acceleration \( a_N \)**:
- The normal component \( a_N \) is given by:
\[ a_N = |\mathbf{a}(t) \cdot \mathbf{N}(t)| \]
**Note:** \( \mathbf{r}'(t) \) represents the first derivative and \( \mathbf{r}''(t) \) represents the second derivative of \( \mathbf{r}(t) \).
### Example Calculation:
Given the function:
\[ \mathbf{r}(t) = t^3 \mathbf{i} + 8 \cos t \mathbf{j} + 6 \sin t \mathbf{k} \]
At \( t = \pi \):
1. Compute \( \mathbf{r}'(t) \)
2. Compute \( \mathbf{r}''(t) \)
3. Use the derivatives to find \( \mathbf{T}(t) \) and \( \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fc33cc1-e822-44c8-a303-0b5fd2ff9f49%2F6421fe82-c68d-4c06-99f4-0f76c5b99d4f%2F7u09d3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Vector Function Analysis
**Objective:**
Find \( N(t), T(t), a_T \), and \( a_N \) for the vector function defined as:
\[ \mathbf{r}(t) = t^3 \mathbf{i} + 8 \cos t \mathbf{j} + 6 \sin t \mathbf{k} \]
**Given:**
\[ t = \pi \]
### Steps to Determine the Components:
1. **Find the unit tangent vector \( \mathbf{T}(t) \)**:
- The unit tangent vector \( \mathbf{T}(t) \) is given by:
\[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} \]
2. **Find the unit normal vector \( \mathbf{N}(t) \)**:
- The unit normal vector \( \mathbf{N}(t) \) is given by:
\[ \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|} \]
3. **Find the tangential component of the acceleration \( a_T \)**:
- The tangential component \( a_T \) is given by:
\[ a_T = \mathbf{a}(t) \cdot \mathbf{T}(t) \]
4. **Find the normal component of the acceleration \( a_N \)**:
- The normal component \( a_N \) is given by:
\[ a_N = |\mathbf{a}(t) \cdot \mathbf{N}(t)| \]
**Note:** \( \mathbf{r}'(t) \) represents the first derivative and \( \mathbf{r}''(t) \) represents the second derivative of \( \mathbf{r}(t) \).
### Example Calculation:
Given the function:
\[ \mathbf{r}(t) = t^3 \mathbf{i} + 8 \cos t \mathbf{j} + 6 \sin t \mathbf{k} \]
At \( t = \pi \):
1. Compute \( \mathbf{r}'(t) \)
2. Compute \( \mathbf{r}''(t) \)
3. Use the derivatives to find \( \mathbf{T}(t) \) and \( \
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