A 120N weight hangs off point C. The structure is in equilibrium. Determine the tension in the cable CD and in the ropes AC and BC.
A 120N weight hangs off point C. The structure is in equilibrium. Determine the tension in the cable CD and in the ropes AC and BC.
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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(3.3) A 120N weight hangs off point C. The structure is in equilibrium. Determine the tension in the
cable CD and in the ropes AC and BC.
![### Equilibrium Analysis in a 3D Force System
**Description:**
This diagram illustrates a 3D force equilibrium problem involving a sphere connected by cables to three points A, B, and D. The sphere experiences a downward force of 120 N. The goal is to resolve the tension forces in the cables.
**Coordinate System:**
- The diagram uses a 3-dimensional coordinate system with axes labeled x, y, and z.
- The positions of the points are defined as follows:
- Point A is at coordinates (0, 3, 3).
- Point B is at coordinates (2, 7, 9).
- Point C, where the cables meet, is located at (6, 7, 0).
- Point D is at coordinates (6, 0, 1).
**Measurements:**
- The horizontal distance from point A to point C along the x-axis is 6 meters.
- The distance from point C to point B along the x-axis is 2 meters.
- The vertical distance from point C to point D along the y-axis is 7 meters.
- Point B is 9 meters above the x-y plane.
- Point A and point D have vertical distances of 3 meters and 1 meter from the x-y plane, respectively.
**Forces:**
- The sphere at point C exerts three tension forces denoted as \( T_{CA} \), \( T_{CB} \), and \( T_{CD} \) along the cables connected to points A, B, and D respectively.
- \( T_{CA} \): Tension in the cable from C to A.
- \( T_{CB} \): Tension in the cable from C to B.
- \( T_{CD} \): Tension in the cable from C to D.
**Equilibrium Condition:**
- The force balance equation in the vertical direction is shown:
\[ +(-120j) = 0 \]
This equation indicates that the sum of the vertical forces (including the weight of the sphere) is zero for equilibrium.
### Explanation:
To solve the equilibrium problem, one must apply vector algebra and equilibrium equations. We resolve the position vectors and write the equilibrium equations for each axis. The sum of forces in both the x, y, and z directions must be zero for the system to be in equilibrium. This involves calculating the contributions of each tension force along](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F136a9916-8a5d-490b-a1c4-57e644d5608a%2Fe97b889e-b1d9-4a21-9589-b55896c5813d%2F6juuh18_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Equilibrium Analysis in a 3D Force System
**Description:**
This diagram illustrates a 3D force equilibrium problem involving a sphere connected by cables to three points A, B, and D. The sphere experiences a downward force of 120 N. The goal is to resolve the tension forces in the cables.
**Coordinate System:**
- The diagram uses a 3-dimensional coordinate system with axes labeled x, y, and z.
- The positions of the points are defined as follows:
- Point A is at coordinates (0, 3, 3).
- Point B is at coordinates (2, 7, 9).
- Point C, where the cables meet, is located at (6, 7, 0).
- Point D is at coordinates (6, 0, 1).
**Measurements:**
- The horizontal distance from point A to point C along the x-axis is 6 meters.
- The distance from point C to point B along the x-axis is 2 meters.
- The vertical distance from point C to point D along the y-axis is 7 meters.
- Point B is 9 meters above the x-y plane.
- Point A and point D have vertical distances of 3 meters and 1 meter from the x-y plane, respectively.
**Forces:**
- The sphere at point C exerts three tension forces denoted as \( T_{CA} \), \( T_{CB} \), and \( T_{CD} \) along the cables connected to points A, B, and D respectively.
- \( T_{CA} \): Tension in the cable from C to A.
- \( T_{CB} \): Tension in the cable from C to B.
- \( T_{CD} \): Tension in the cable from C to D.
**Equilibrium Condition:**
- The force balance equation in the vertical direction is shown:
\[ +(-120j) = 0 \]
This equation indicates that the sum of the vertical forces (including the weight of the sphere) is zero for equilibrium.
### Explanation:
To solve the equilibrium problem, one must apply vector algebra and equilibrium equations. We resolve the position vectors and write the equilibrium equations for each axis. The sum of forces in both the x, y, and z directions must be zero for the system to be in equilibrium. This involves calculating the contributions of each tension force along
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