elocity of the plate C is VC=6.0 m/s to the right. The velocity of the plate D is VD=2.0 m/s to the left. Using the instantaneous center of zero velocity (IC) to determine (3) The magnitude of the angular velocity of the cylinder ω=____________rad/s
elocity of the plate C is VC=6.0 m/s to the right. The velocity of the plate D is VD=2.0 m/s to the left. Using the instantaneous center of zero velocity (IC) to determine (3) The magnitude of the angular velocity of the cylinder ω=____________rad/s
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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A cylinder rolls without slipping between two moving plates C and D. The radius of the cylinder is r=4.0m. The velocity of the plate C is VC=6.0 m/s to the right. The velocity of the plate D is VD=2.0 m/s to the left. Using the instantaneous center of zero velocity (IC) to determine
(3) The magnitude of the angular velocity of the cylinder ω=____________rad/s
![### Angular Velocity of a Cylinder in Motion
#### Scenario Description:
A cylinder rolls without slipping between two moving plates, labeled C and D. The key parameters of the system are:
- Radius of the cylinder: \( r = 4.0 \) m.
- Velocity of plate C: \( V_C = 6.0 \) m/s to the right.
- Velocity of plate D: \( V_D = 2.0 \) m/s to the left.
Using the instantaneous center of zero velocity (IC), we aim to determine the magnitude of the angular velocity of the cylinder, denoted as \( \omega \).
#### Problem Statement:
Determine the magnitude of the angular velocity of the cylinder, \( \omega \) in rad/s.
#### Diagram Explanation:
The accompanying diagram shows:
- A cylindrical object in contact with two horizontal plates (C and D).
- Plate C is moving to the right with velocity \( V_C \) and Plate D is moving to the left with velocity \( V_D \).
- The forces of friction, \( F_{\text{ric}} \), act at the points of contact between the cylinder and the plates.
- The cylinder rolls without slipping, indicating a non-sliding contact at both interfacing points.
Below the cylinder, the 'Instantaneous Center of Rotation' (IC) is labeled. This represents the point where the velocity of the cylinder is zero relative to the ground, around which the cylinder can be considered to be rotating.
#### Solution Steps:
To determine \( \omega \), the required steps involve kinematic analysis leveraging the concept of the instantaneous center of zero velocity (IC).
By analyzing the velocities and positions with respect to the IC, we can express angular velocity \( \omega \) in terms of the given velocities and the radius of the cylinder.
#### Calculation:
Let's derive the angular velocity \( \omega \):
1. As the cylinder rolls without slipping, the tangential velocity at the top of the cylinder equals the velocity of plate C, and the tangential velocity at the bottom equals the velocity of plate D but in the opposite direction.
2. The distance from IC to the cylinder's surface (top and bottom) is the radius \( r \):
\[
\omega = \frac{V_C + V_D}{2r}
\]
Substituting the values:
\[
\omega = \frac{6.0 \, \text{m/s} + 2.0 \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0daff6b4-7b3b-4798-aa81-781ef422795b%2Fa6d3a4de-c884-4cca-ac56-054f6a027d60%2Fwohgaw_processed.png&w=3840&q=75)
Transcribed Image Text:### Angular Velocity of a Cylinder in Motion
#### Scenario Description:
A cylinder rolls without slipping between two moving plates, labeled C and D. The key parameters of the system are:
- Radius of the cylinder: \( r = 4.0 \) m.
- Velocity of plate C: \( V_C = 6.0 \) m/s to the right.
- Velocity of plate D: \( V_D = 2.0 \) m/s to the left.
Using the instantaneous center of zero velocity (IC), we aim to determine the magnitude of the angular velocity of the cylinder, denoted as \( \omega \).
#### Problem Statement:
Determine the magnitude of the angular velocity of the cylinder, \( \omega \) in rad/s.
#### Diagram Explanation:
The accompanying diagram shows:
- A cylindrical object in contact with two horizontal plates (C and D).
- Plate C is moving to the right with velocity \( V_C \) and Plate D is moving to the left with velocity \( V_D \).
- The forces of friction, \( F_{\text{ric}} \), act at the points of contact between the cylinder and the plates.
- The cylinder rolls without slipping, indicating a non-sliding contact at both interfacing points.
Below the cylinder, the 'Instantaneous Center of Rotation' (IC) is labeled. This represents the point where the velocity of the cylinder is zero relative to the ground, around which the cylinder can be considered to be rotating.
#### Solution Steps:
To determine \( \omega \), the required steps involve kinematic analysis leveraging the concept of the instantaneous center of zero velocity (IC).
By analyzing the velocities and positions with respect to the IC, we can express angular velocity \( \omega \) in terms of the given velocities and the radius of the cylinder.
#### Calculation:
Let's derive the angular velocity \( \omega \):
1. As the cylinder rolls without slipping, the tangential velocity at the top of the cylinder equals the velocity of plate C, and the tangential velocity at the bottom equals the velocity of plate D but in the opposite direction.
2. The distance from IC to the cylinder's surface (top and bottom) is the radius \( r \):
\[
\omega = \frac{V_C + V_D}{2r}
\]
Substituting the values:
\[
\omega = \frac{6.0 \, \text{m/s} + 2.0 \,
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