Given r(t)= , find the tangential and normal components of acceleration, for t>0.
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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![**Problem Statement: Finding Tangential and Normal Components of Acceleration**
Given the position vector function \( \mathbf{r}(t) = \langle t^2, t^2, t^3 \rangle \), find the tangential and normal components of acceleration, for \( t > 0 \).
**Explanation:**
The position vector \(\mathbf{r}(t)\) specifies the location of a particle in 3-dimensional space as a function of time \( t \).
To find the tangential and normal components of acceleration, we need to follow these steps:
1. **Calculate the Velocity Vector \(\mathbf{v}(t)\):**
\[
\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}
\]
Compute the derivative of \(\mathbf{r}(t)\) with respect to \( t \).
2. **Calculate the Speed \(v(t)\):**
\[
v(t) = \|\mathbf{v}(t)\|
\]
Compute the magnitude of the velocity vector.
3. **Calculate the Acceleration Vector \(\mathbf{a}(t)\):**
\[
\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt}
\]
Compute the derivative of \(\mathbf{v}(t)\) with respect to \( t \).
4. **Calculate the Tangential Component of Acceleration \( a_T \):**
\[
a_T = \frac{d}{dt}v(t)
\]
5. **Calculate the Normal Component of Acceleration \( a_N \):**
\[
a_N = \sqrt{\|\mathbf{a}(t)\|^2 - a_T^2}
\]
Determine the component of acceleration perpendicular to the tangential component.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd91b0d75-3013-4add-be37-5e57ef1ff904%2F5561dd9b-d2f2-47d3-a9d7-2bf1a26b4bed%2F2v2n1q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Finding Tangential and Normal Components of Acceleration**
Given the position vector function \( \mathbf{r}(t) = \langle t^2, t^2, t^3 \rangle \), find the tangential and normal components of acceleration, for \( t > 0 \).
**Explanation:**
The position vector \(\mathbf{r}(t)\) specifies the location of a particle in 3-dimensional space as a function of time \( t \).
To find the tangential and normal components of acceleration, we need to follow these steps:
1. **Calculate the Velocity Vector \(\mathbf{v}(t)\):**
\[
\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}
\]
Compute the derivative of \(\mathbf{r}(t)\) with respect to \( t \).
2. **Calculate the Speed \(v(t)\):**
\[
v(t) = \|\mathbf{v}(t)\|
\]
Compute the magnitude of the velocity vector.
3. **Calculate the Acceleration Vector \(\mathbf{a}(t)\):**
\[
\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt}
\]
Compute the derivative of \(\mathbf{v}(t)\) with respect to \( t \).
4. **Calculate the Tangential Component of Acceleration \( a_T \):**
\[
a_T = \frac{d}{dt}v(t)
\]
5. **Calculate the Normal Component of Acceleration \( a_N \):**
\[
a_N = \sqrt{\|\mathbf{a}(t)\|^2 - a_T^2}
\]
Determine the component of acceleration perpendicular to the tangential component.
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