A force of 3 Newtons and a force of 5 Newtons act concurrently to produce a resultant of 8 Newtons. The angle between the forces must be 180° 90° 60° O 0⁰
A force of 3 Newtons and a force of 5 Newtons act concurrently to produce a resultant of 8 Newtons. The angle between the forces must be 180° 90° 60° O 0⁰
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Physics Problem on Concurrent Forces
**Problem Statement:**
A force of 3 Newtons and a force of 5 Newtons act concurrently to produce a resultant of 8 Newtons. The angle between the forces must be:
- 180°
- 90°
- 60°
- 0°
**Explanation:**
This problem involves the concept of vector addition of forces. When two forces act concurrently, their resultant vector depends on the magnitude and the angle between them. Given that the resultant force is 8 Newtons, we need to determine the angle between the two forces using the law of cosines.
The law of cosines states:
\[ R^2 = F_1^2 + F_2^2 + 2 \cdot F_1 \cdot F_2 \cdot \cos(\theta) \]
where:
- \( R \) is the resultant force,
- \( F_1 \) and \( F_2 \) are the magnitudes of the two forces,
- \( \theta \) is the angle between the forces.
Given:
\[ R = 8 \, \text{N} \]
\[ F_1 = 3 \, \text{N} \]
\[ F_2 = 5 \, \text{N} \]
Substitute the values into the law of cosines:
\[ 8^2 = 3^2 + 5^2 + 2 \cdot 3 \cdot 5 \cdot \cos(\theta) \]
\[ 64 = 9 + 25 + 30 \cdot \cos(\theta) \]
\[ 64 = 34 + 30 \cdot \cos(\theta) \]
\[ 64 - 34 = 30 \cdot \cos(\theta) \]
\[ 30 = 30 \cdot \cos(\theta) \]
\[ \cos(\theta) = 1 \]
\[ \theta = 0° \]
**Conclusion:**
The angle between the forces must be \( 0° \).
This indicates that the forces are acting in the same direction. Thus, the correct answer is:
- **0°**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ed0702b-be25-4e8d-bf00-15df157283ad%2F524f0439-2863-4647-b582-a4b87ef206e7%2F9ztwq4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Physics Problem on Concurrent Forces
**Problem Statement:**
A force of 3 Newtons and a force of 5 Newtons act concurrently to produce a resultant of 8 Newtons. The angle between the forces must be:
- 180°
- 90°
- 60°
- 0°
**Explanation:**
This problem involves the concept of vector addition of forces. When two forces act concurrently, their resultant vector depends on the magnitude and the angle between them. Given that the resultant force is 8 Newtons, we need to determine the angle between the two forces using the law of cosines.
The law of cosines states:
\[ R^2 = F_1^2 + F_2^2 + 2 \cdot F_1 \cdot F_2 \cdot \cos(\theta) \]
where:
- \( R \) is the resultant force,
- \( F_1 \) and \( F_2 \) are the magnitudes of the two forces,
- \( \theta \) is the angle between the forces.
Given:
\[ R = 8 \, \text{N} \]
\[ F_1 = 3 \, \text{N} \]
\[ F_2 = 5 \, \text{N} \]
Substitute the values into the law of cosines:
\[ 8^2 = 3^2 + 5^2 + 2 \cdot 3 \cdot 5 \cdot \cos(\theta) \]
\[ 64 = 9 + 25 + 30 \cdot \cos(\theta) \]
\[ 64 = 34 + 30 \cdot \cos(\theta) \]
\[ 64 - 34 = 30 \cdot \cos(\theta) \]
\[ 30 = 30 \cdot \cos(\theta) \]
\[ \cos(\theta) = 1 \]
\[ \theta = 0° \]
**Conclusion:**
The angle between the forces must be \( 0° \).
This indicates that the forces are acting in the same direction. Thus, the correct answer is:
- **0°**
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