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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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Transcribed Image Text:The image shows a roller with weight \( W \) and diameter \( R \) being pushed to the left over a tile of thickness \( \delta \). The roller is in a state of momentary mechanical equilibrium as it rolls over the sharp corner of the tile.
### Diagram Explanation
- The roller is depicted with a horizontal surface underneath and a deviation at the point of the tile edge.
- A force \( P \) is applied at an angle \( \theta \) to the horizontal.
- The weight \( W \) acts vertically downward at the roller’s center.
- The radius from the center of the roller to the contact point with the tile edge is labeled \( r \).
- Two angles are indicated within the roller:
- \( \alpha \), the angle between the vertical line through \( W \) and the radius \( r \).
- \( \theta \), the angle of the force \( P \) from the horizontal.
### Tasks
a. **Free Body Diagram**:
- Sketch a free body diagram showing all the forces acting on the roller at this moment of equilibrium.
b. **Force Triangle Sketch**:
- Illustrate the closed force vector triangle that represents the equilibrium condition. Note that this sketch is not to scale.
c. **Interior Angles of Force Triangle**:
- Express each angle in the force triangle in terms of \( \alpha \) and \( \theta \), and label each force vector and angle.
d. **Relation Establishment**:
- Derive a relationship between \( \alpha \), \( \delta \), and \( r \).
e. **Expression for Force \( P \)**:
- Use static equilibrium and the law of sines to derive an expression for \( P \) in terms of \( W \), \( r \), \( \delta \), and \( \theta \).
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