Two forces F₁ and F2 act on the screw eye. If their lines of action are at an angle 9 apart and the magnitude of each force is F₁ = F₂ = F, determine the magnitude of the resultant force FD and the angle between FD and F₁.
Two forces F₁ and F2 act on the screw eye. If their lines of action are at an angle 9 apart and the magnitude of each force is F₁ = F₂ = F, determine the magnitude of the resultant force FD and the angle between FD and F₁.
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
Related questions
Question
Transcribed Image Text:**Problem Statement:**
An engineer developed the following solution to the given problem. Verify his calculations. Is the given final answer "true" or "false"?
**Problem Details:**
Two forces, \( F_1 \) and \( F_2 \), act on the screw eye. If their lines of action are at an angle \( \theta \) apart and the magnitude of each force is \( F_1 = F_2 = F \), determine:
1. The magnitude of the resultant force \( F_R \).
2. The angle between \( F_R \) and \( F_1 \).
**Diagram Explanation:**
The diagram shows two forces acting on a screw eye. The screw eye is mounted on a flat wooden surface, and a hook shape is illustrated. Force vectors \( F_1 \) and \( F_2 \) would typically be depicted acting at an angle \( \theta \) relative to each other, converging at the point where the screw eye is attached.
![### Transcription and Explanation
#### Mathematical Derivation
\[
\frac{F}{\sin \theta} = \frac{F}{\sin(\theta - \phi)}
\]
\[
\sin(\theta - \phi) = \sin \phi
\]
\[
\theta - \phi = \phi
\]
Therefore,
\[
\phi = \frac{\theta}{2}
\]
#### Solution
\[
F_R = \sqrt{(F)^2 + (F)^2 - 2(F)(F) \cos(180^\circ - \theta)}
\]
Since \(\cos(180^\circ - \theta) = -\cos \theta\),
\[
F_R = F (\sqrt{2}) \sqrt{1 + \cos \theta}
\]
Since \(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}}\),
Then,
\[
F_R = 2F \cos\left(\frac{\theta}{2}\right)
\]
#### Diagrams Explanation
1. **Figure (a):**
- This is a vector diagram showing two forces \( F \) at an angle \(\theta\) to each other.
- The angle of separation is \(180^\circ - \theta\).
- \(\phi\) is shown as half of \(\theta\).
2. **Additional Diagram:**
- This diagram is a triangle representing the vector addition of forces.
- The triangle's sides are labeled with \( F \) and the angles as \( \phi\), \(180^\circ - \theta \), and \(\theta - \phi\).
- The resultant force \( F_R \) is perpendicular to the base formed by \( \phi \).
Both diagrams illustrate the geometric approach to resolving two forces at an angle into a single resultant vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe65993c9-41ee-4daf-9618-d161554f12c8%2F52b62c44-1619-43c8-8f54-9c1f818a6030%2Fw1swa6_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription and Explanation
#### Mathematical Derivation
\[
\frac{F}{\sin \theta} = \frac{F}{\sin(\theta - \phi)}
\]
\[
\sin(\theta - \phi) = \sin \phi
\]
\[
\theta - \phi = \phi
\]
Therefore,
\[
\phi = \frac{\theta}{2}
\]
#### Solution
\[
F_R = \sqrt{(F)^2 + (F)^2 - 2(F)(F) \cos(180^\circ - \theta)}
\]
Since \(\cos(180^\circ - \theta) = -\cos \theta\),
\[
F_R = F (\sqrt{2}) \sqrt{1 + \cos \theta}
\]
Since \(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}}\),
Then,
\[
F_R = 2F \cos\left(\frac{\theta}{2}\right)
\]
#### Diagrams Explanation
1. **Figure (a):**
- This is a vector diagram showing two forces \( F \) at an angle \(\theta\) to each other.
- The angle of separation is \(180^\circ - \theta\).
- \(\phi\) is shown as half of \(\theta\).
2. **Additional Diagram:**
- This diagram is a triangle representing the vector addition of forces.
- The triangle's sides are labeled with \( F \) and the angles as \( \phi\), \(180^\circ - \theta \), and \(\theta - \phi\).
- The resultant force \( F_R \) is perpendicular to the base formed by \( \phi \).
Both diagrams illustrate the geometric approach to resolving two forces at an angle into a single resultant vector.
Expert Solution
Step 1
Given-Information:-
To-Determine:-
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY