Use the following figure to determine the spin rate so that the top will maintain 45 degrees. The find the nutation and precession at 3 times faster than the spin rate. Assume is constant 45 degrees.
Use the following figure to determine the spin rate so that the top will maintain 45 degrees. The find the nutation and precession at 3 times faster than the spin rate. Assume is constant 45 degrees.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Problem Statement:**
Use the following figure to determine the spin rate so that the top will maintain 45 degrees. Then find the nutation and precession at 3 times faster than the spin rate. Assume \(\theta\) is constant at 45 degrees.
**Diagram Explanation:**
The diagram depicts a three-dimensional coordinate system with axes labeled as \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). The system shows a spinning top with its center of mass (\(G\)) at a distance \(d\) from the origin \(O\). Various forces and motions affecting the top are indicated:
- The gravitational force (\(mg\)) acts downwards along the z-axis.
- The spin vector (\(\vec{\omega}_s\)) makes an angle with the vertical axis.
- The precession (\(\omega_p = \dot{\phi}\)) and nutation (\(\omega_n = \dot{\theta}\)) are illustrated, along with their respective directions.
- The angle \(\theta\) is marked at a constant 45 degrees.
- The circular path of the top's motion is indicated around the origin.
**Variables and Constants:**
- Mass \(m = 2.0\, \text{kg}\)
- Distance \(d = 0.3\, \text{m}\)
- Inertia matrix \([I]\) given as:
\[
\begin{bmatrix}
20 & 0 & 0 \\
0 & 20 & 0 \\
0 & 0 & 80
\end{bmatrix}
\]
**Analysis Objective:**
Calculate the appropriate spin rate to maintain a stable angle of 45 degrees, and evaluate the nutation and precession at three times the rate of spin.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12c62ea9-2423-4a35-a6cd-74646c6bbd41%2F205d7bfc-385f-46c2-9884-e2edf9a320aa%2F78liisw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the following figure to determine the spin rate so that the top will maintain 45 degrees. Then find the nutation and precession at 3 times faster than the spin rate. Assume \(\theta\) is constant at 45 degrees.
**Diagram Explanation:**
The diagram depicts a three-dimensional coordinate system with axes labeled as \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). The system shows a spinning top with its center of mass (\(G\)) at a distance \(d\) from the origin \(O\). Various forces and motions affecting the top are indicated:
- The gravitational force (\(mg\)) acts downwards along the z-axis.
- The spin vector (\(\vec{\omega}_s\)) makes an angle with the vertical axis.
- The precession (\(\omega_p = \dot{\phi}\)) and nutation (\(\omega_n = \dot{\theta}\)) are illustrated, along with their respective directions.
- The angle \(\theta\) is marked at a constant 45 degrees.
- The circular path of the top's motion is indicated around the origin.
**Variables and Constants:**
- Mass \(m = 2.0\, \text{kg}\)
- Distance \(d = 0.3\, \text{m}\)
- Inertia matrix \([I]\) given as:
\[
\begin{bmatrix}
20 & 0 & 0 \\
0 & 20 & 0 \\
0 & 0 & 80
\end{bmatrix}
\]
**Analysis Objective:**
Calculate the appropriate spin rate to maintain a stable angle of 45 degrees, and evaluate the nutation and precession at three times the rate of spin.
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