The boom of a crane extends from its base at point O to its tip at point Q, as shown in the figure below. A three-dimensional coordinate system is given in this figure with coordinates for both points of O = (0,0,0) and Q = (-2, 10, 10) ft. A force with a magnitude of 10.5 kip is applied at the tip of the boom by a cable that extends from point Q to point P in the x-y plane, which has coordinates P = (9,6, 0) ft. Determine the following quantities:
The boom of a crane extends from its base at point O to its tip at point Q, as shown in the figure below. A three-dimensional coordinate system is given in this figure with coordinates for both points of O = (0,0,0) and Q = (-2, 10, 10) ft. A force with a magnitude of 10.5 kip is applied at the tip of the boom by a cable that extends from point Q to point P in the x-y plane, which has coordinates P = (9,6, 0) ft. Determine the following quantities:
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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![**Crane Force and Moment Analysis**
The boom of a crane extends from its base at point \( O \) to its tip at point \( Q \), as illustrated in the figure. The figure displays a three-dimensional coordinate system with coordinates for \( O = (0, 0, 0) \) and \( Q = (-2, 10, 10) \) feet. A force \( \mathbf{F} \) with a magnitude of 10.5 kip is applied at the tip of the boom by a cable that extends from point \( Q \) to point \( P \) in the x–y plane, with coordinates \( P = (9, 6, 0) \) feet.
### Diagram Explanation:
- **Axes**: The diagram shows a three-dimensional coordinate system with \( x \), \( y \), and \( z \) axes intersecting at point \( O \).
- **Boom**: The crane's boom is represented as a line extending from \( O \) to \( Q \).
- **Force \(\mathbf{F}\)**: Illustrated as a vector applied at point \( Q \), directed along the cable towards point \( P \).
### Problem Requirements:
Determine the following quantities:
**(a) The force \(\mathbf{F}\) as a Cartesian vector:**
\[
\mathbf{F} = \text{number} \, \mathbf{i} + \text{number} \, \mathbf{j} + \text{number} \, \mathbf{k} \, \text{kip}
\]
**(b) The moment \(\mathbf{M}_O\) due to force \(\mathbf{F}\) about point \( O \):**
\[
\mathbf{M}_O = \text{number} \, \mathbf{i} + \text{number} \, \mathbf{j} + \text{number} \, \mathbf{k} \, \text{kip} \cdot \text{ft}
\]
**(c) The magnitude of moment \(\mathbf{M}_O\):**
\[
M_O = \text{number} \, \text{kip} \cdot \text{ft}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee563bfe-badf-4a9d-9a7e-d47066dc4e0e%2F542e324d-db5f-4495-b67a-56ef07d1097c%2Fhxkcqvf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Crane Force and Moment Analysis**
The boom of a crane extends from its base at point \( O \) to its tip at point \( Q \), as illustrated in the figure. The figure displays a three-dimensional coordinate system with coordinates for \( O = (0, 0, 0) \) and \( Q = (-2, 10, 10) \) feet. A force \( \mathbf{F} \) with a magnitude of 10.5 kip is applied at the tip of the boom by a cable that extends from point \( Q \) to point \( P \) in the x–y plane, with coordinates \( P = (9, 6, 0) \) feet.
### Diagram Explanation:
- **Axes**: The diagram shows a three-dimensional coordinate system with \( x \), \( y \), and \( z \) axes intersecting at point \( O \).
- **Boom**: The crane's boom is represented as a line extending from \( O \) to \( Q \).
- **Force \(\mathbf{F}\)**: Illustrated as a vector applied at point \( Q \), directed along the cable towards point \( P \).
### Problem Requirements:
Determine the following quantities:
**(a) The force \(\mathbf{F}\) as a Cartesian vector:**
\[
\mathbf{F} = \text{number} \, \mathbf{i} + \text{number} \, \mathbf{j} + \text{number} \, \mathbf{k} \, \text{kip}
\]
**(b) The moment \(\mathbf{M}_O\) due to force \(\mathbf{F}\) about point \( O \):**
\[
\mathbf{M}_O = \text{number} \, \mathbf{i} + \text{number} \, \mathbf{j} + \text{number} \, \mathbf{k} \, \text{kip} \cdot \text{ft}
\]
**(c) The magnitude of moment \(\mathbf{M}_O\):**
\[
M_O = \text{number} \, \text{kip} \cdot \text{ft}
\]
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