The path r(t) = (t- sin t) i+ (1- cos t) j describes motion on the cycloid x=t- sin t, y=1- cos t. Find the particle's velocity and acceleration vectors at t=- and sketch them as vectors on the curve. .....
The path r(t) = (t- sin t) i+ (1- cos t) j describes motion on the cycloid x=t- sin t, y=1- cos t. Find the particle's velocity and acceleration vectors at t=- and sketch them as vectors on the curve. .....
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Cycloid Motion and Vector Analysis**
The path \( \mathbf{r}(t) = (t - \sin t) \, \mathbf{i} + (1 - \cos t) \, \mathbf{j} \) describes motion on the cycloid \( x = t - \sin t, \, y = 1 - \cos t \). Find the particle's velocity and acceleration vectors at \( t = \frac{\pi}{2} \), and sketch them as vectors on the curve.
**Solution Approach:**
1. **Position Function**: The particle’s position at time \( t \) is given by:
\[
\mathbf{r}(t) = (t - \sin t) \, \mathbf{i} + (1 - \cos t) \, \mathbf{j}
\]
2. **Velocity Vector**: Derive the velocity \( \mathbf{v}(t) \) by differentiating the position function \( \mathbf{r}(t) \) with respect to time \( t \):
\[
\mathbf{v}(t) = \frac{d}{dt}[(t - \sin t) \, \mathbf{i} + (1 - \cos t) \, \mathbf{j}]
\]
3. **Acceleration Vector**: Derive the acceleration \( \mathbf{a}(t) \) by differentiating the velocity function \( \mathbf{v}(t) \) with respect to time \( t \):
\[
\mathbf{a}(t) = \frac{d}{dt}[\mathbf{v}(t)]
\]
4. **Evaluate at \( t = \frac{\pi}{2} \)**: Substitute \( t = \frac{\pi}{2} \) into \(\mathbf{v}(t)\) and \(\mathbf{a}(t)\) to find the specific vectors at this point.
5. **Sketch the Vectors**: Illustrate both the velocity and acceleration vectors on the curve at \( t = \frac{\pi}{2} \).
This analysis helps in understanding the dynamics of motion along a cycloidal path commonly encountered in physics and engineering applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1bf229e-839d-42dd-b4c9-1bfa437f3607%2Fc86b37f3-6b1d-4dd3-b3cb-67e55b5a76ae%2F8e7kb2l_processed.png&w=3840&q=75)
Transcribed Image Text:**Cycloid Motion and Vector Analysis**
The path \( \mathbf{r}(t) = (t - \sin t) \, \mathbf{i} + (1 - \cos t) \, \mathbf{j} \) describes motion on the cycloid \( x = t - \sin t, \, y = 1 - \cos t \). Find the particle's velocity and acceleration vectors at \( t = \frac{\pi}{2} \), and sketch them as vectors on the curve.
**Solution Approach:**
1. **Position Function**: The particle’s position at time \( t \) is given by:
\[
\mathbf{r}(t) = (t - \sin t) \, \mathbf{i} + (1 - \cos t) \, \mathbf{j}
\]
2. **Velocity Vector**: Derive the velocity \( \mathbf{v}(t) \) by differentiating the position function \( \mathbf{r}(t) \) with respect to time \( t \):
\[
\mathbf{v}(t) = \frac{d}{dt}[(t - \sin t) \, \mathbf{i} + (1 - \cos t) \, \mathbf{j}]
\]
3. **Acceleration Vector**: Derive the acceleration \( \mathbf{a}(t) \) by differentiating the velocity function \( \mathbf{v}(t) \) with respect to time \( t \):
\[
\mathbf{a}(t) = \frac{d}{dt}[\mathbf{v}(t)]
\]
4. **Evaluate at \( t = \frac{\pi}{2} \)**: Substitute \( t = \frac{\pi}{2} \) into \(\mathbf{v}(t)\) and \(\mathbf{a}(t)\) to find the specific vectors at this point.
5. **Sketch the Vectors**: Illustrate both the velocity and acceleration vectors on the curve at \( t = \frac{\pi}{2} \).
This analysis helps in understanding the dynamics of motion along a cycloidal path commonly encountered in physics and engineering applications.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning