Q.1) Use the Parallelogram law and the laws of sine and cosine to find the resultant of the two forces shown below. Also determine its orientation from x- axis. 170 N 5 12 210 N
Q.1) Use the Parallelogram law and the laws of sine and cosine to find the resultant of the two forces shown below. Also determine its orientation from x- axis. 170 N 5 12 210 N
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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![### Problem Statement
**Q.1) Use the Parallelogram law and the laws of sine and cosine to find the resultant of the two forces shown below. Also, determine its orientation from the x-axis.**
### Diagram Explanation
The given diagram is an x-y coordinate system showing two forces acting at an angle from a common origin point. The forces are:
1. A force of 170 N that points upwards and to the left, making an angle with the y-axis.
2. A force of 210 N that points downwards and to the right, making an angle with the x-axis.
The two vectors form a parallelogram with the x-axis and y-axis. The dimensions below the x-axis indicate the horizontal (12 units along the x-axis) and vertical (5 units along the y-axis) components necessary to form the parallelogram, creating a right triangle with a base of 12 units and a height of 5 units.
### Solution Steps
1. **Determine the angle between the forces:**
- Use the given horizontal and vertical components to find the internal angle, θ, between the 170 N and 210 N forces.
- Using trigonometric ratios, \( \tan^{-1} \left(\frac{5}{12}\right) \) can be used to find part of the angle.
2. **Find the magnitude of the resultant force (R):**
- Use the Parallelogram law. The magnitude of the resultant force \( R \) can be found using the law of cosines.
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos(\theta)} \]
- Here, \( F_1 = 170 \text{ N} \) and \( F_2 = 210 \text{ N} \).
3. **Determine the direction of the resultant force:**
- Use vector components or trigonometric identities (law of sines) to find the angle of the resultant vector relative to the x-axis.
### Detailed Calculation Explanation
**Step 1: Find the internal angle, θ:**
\[ \theta = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.62^\circ \]
Since the diagram implies an angle greater than the calculated internal angle, the included angle between the two vectors is:
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Transcribed Image Text:### Problem Statement
**Q.1) Use the Parallelogram law and the laws of sine and cosine to find the resultant of the two forces shown below. Also, determine its orientation from the x-axis.**
### Diagram Explanation
The given diagram is an x-y coordinate system showing two forces acting at an angle from a common origin point. The forces are:
1. A force of 170 N that points upwards and to the left, making an angle with the y-axis.
2. A force of 210 N that points downwards and to the right, making an angle with the x-axis.
The two vectors form a parallelogram with the x-axis and y-axis. The dimensions below the x-axis indicate the horizontal (12 units along the x-axis) and vertical (5 units along the y-axis) components necessary to form the parallelogram, creating a right triangle with a base of 12 units and a height of 5 units.
### Solution Steps
1. **Determine the angle between the forces:**
- Use the given horizontal and vertical components to find the internal angle, θ, between the 170 N and 210 N forces.
- Using trigonometric ratios, \( \tan^{-1} \left(\frac{5}{12}\right) \) can be used to find part of the angle.
2. **Find the magnitude of the resultant force (R):**
- Use the Parallelogram law. The magnitude of the resultant force \( R \) can be found using the law of cosines.
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos(\theta)} \]
- Here, \( F_1 = 170 \text{ N} \) and \( F_2 = 210 \text{ N} \).
3. **Determine the direction of the resultant force:**
- Use vector components or trigonometric identities (law of sines) to find the angle of the resultant vector relative to the x-axis.
### Detailed Calculation Explanation
**Step 1: Find the internal angle, θ:**
\[ \theta = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.62^\circ \]
Since the diagram implies an angle greater than the calculated internal angle, the included angle between the two vectors is:
\[
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