### Rotational Kinematics Problem#### Problem StatementThe cylinder rotates about the fixed z-axis in the direction indicated. If the speed of point \( A \) is \( v_A = 3.1 \) ft/sec and the magnitude of its acceleration is \( a_A = 20.1 \) ft/sec², determine the magnitudes of the angular velocity and angular acceleration of the cylinder. Is knowledge of the angle \( \theta \) necessary?#### Diagram DescriptionThe diagram features a cylindrical object, with its axis aligned along the z-axis. This cylinder is shown to be rotating about this z-axis.- Point \( O \) is placed at the intersection of the z-axis with the circular base of the cylinder. - Point \( A \) is marked on the circumference of the cylinder.- An angular displacement \( \theta = 8° \) is indicated from the x-axis to the vector extending from the origin O to point A in the cylindrical x-y plane.The following notations are provided in the diagram to illustrate the angular parameters:- \( \omega \) represents the angular velocity vector.- The z-axis, x-axis, and y-axis are denoted to show the fixed reference coordinate system.**Note:** The diagram provides a visual representation to help understand the geometric and kinematic relationship between the points and the angular motion of the cylinder.#### Solution ApproachTo solve this problem, you need to use the rotational kinematics relations and consider the motion constraints provided:1. **Angular Velocity (\( \omega \)) Calculation:** - Relate the linear speed \( v_A \) of point \( A \) to the angular velocity using \( v_A = \omega r \), where \( r \) is the radius from the z-axis to point A.2. **Angular Acceleration (\( \alpha \)) Calculation:** - Use the given acceleration \( a_A \) and the relationship between linear acceleration and angular acceleration \( a_A = \alpha r \).Additionally, consider how or if the given angle \( \theta \) affects these calculations or if the problem can be solved without explicitly using \( \theta \).These fundamental principles and relations from rotational kinematics will enable you to determine the magnitudes of the required angular values. ### Part 2#### Find the magnitude of the normal component of the acceleration of point A.**Answer:** \( a_{An} = \) \_\_\_\_ ft/sec\(^2\)---### Additional Resources#### eTextbook and Media[Access eTextbook and Media]#### Hint[Click here for hints]---(Note: there are no graphs or diagrams in the image provided, only text and input fields)

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The cylinder rotates about the fixed z-axis in the direction indicated. If the speed of point A is VA = 3.1 ft/sec and the magnitude of its acceleration is aA = 20.1 ft/sec², determine the magnitudes of the angular velocity and angular acceleration of the cylinder. Is knowledge of the angle 8 necessary?

The cylinder rotates about the fixed z-axis in the direction indicated. If the speed of point A is VA = 3.1 ft/sec and the magnitude of its acceleration is aA = 20.1 ft/sec², determine the magnitudes of the angular velocity and angular acceleration of the cylinder. Is knowledge of the angle 8 necessary?

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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Question

### Rotational Kinematics Problem

#### Problem Statement
The cylinder rotates about the fixed z-axis in the direction indicated. If the speed of point \( A \) is \( v_A = 3.1 \) ft/sec and the magnitude of its acceleration is \( a_A = 20.1 \) ft/sec², determine the magnitudes of the angular velocity and angular acceleration of the cylinder. Is knowledge of the angle \( \theta \) necessary?

#### Diagram Description
The diagram features a cylindrical object, with its axis aligned along the z-axis. This cylinder is shown to be rotating about this z-axis.

- Point \( O \) is placed at the intersection of the z-axis with the circular base of the cylinder.
- Point \( A \) is marked on the circumference of the cylinder.
- An angular displacement \( \theta = 8° \) is indicated from the x-axis to the vector extending from the origin O to point A in the cylindrical x-y plane.

The following notations are provided in the diagram to illustrate the angular parameters:

- \( \omega \) represents the angular velocity vector.
- The z-axis, x-axis, and y-axis are denoted to show the fixed reference coordinate system.

**Note:** The diagram provides a visual representation to help understand the geometric and kinematic relationship between the points and the angular motion of the cylinder.

#### Solution Approach
To solve this problem, you need to use the rotational kinematics relations and consider the motion constraints provided:
1. **Angular Velocity (\( \omega \)) Calculation:**
- Relate the linear speed \( v_A \) of point \( A \) to the angular velocity using \( v_A = \omega r \), where \( r \) is the radius from the z-axis to point A.
2. **Angular Acceleration (\( \alpha \)) Calculation:**
- Use the given acceleration \( a_A \) and the relationship between linear acceleration and angular acceleration \( a_A = \alpha r \).

Additionally, consider how or if the given angle \( \theta \) affects these calculations or if the problem can be solved without explicitly using \( \theta \).

These fundamental principles and relations from rotational kinematics will enable you to determine the magnitudes of the required angular values.

### Part 2

#### Find the magnitude of the normal component of the acceleration of point A.

**Answer:** \( a_{An} = \) \_\_\_\_ ft/sec\(^2\)

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