### Figure Explanation for Educational Website#### Description:This figure depicts a mechanical system in a three-dimensional coordinate space. The system consists of a structure connected at various joints, represented by nodes. The axes are labeled as x, y, and z, illustrating the 3D coordinate system.#### Components:- **Axes:** The three coordinate axes (x, y, z) are depicted in the diagram for orientation.- **Structure:** The structure is composed of blue rods or beams connected by joints at different points in the space.- **Joints:** The joints are shown as points where the beams meet, marked with spherical connections.- **Forces:** Two forces, \(F_1\) and \(F_2\), are acting on a joint labeled \(B\). - \(F_1\) acts in a direction along the plane formed by \(x_1\) and \(y_1\). - \(F_2\) is pointing along the y-axis.- **Reference Points:** - The origin \(O\) is the reference point of the coordinate system. - Dimensions \(x_1\), \(y_1\), and \(z_1\) indicate the positions of the structure components relative to the origin.#### Annotations:- **\(O\):** Represents the origin of the coordinate system where all axes intersect.- **\(B\):** Represents the point at which the forces \(F_1\) and \(F_2\) are applied. The dimensions and forces are crucial for understanding the equilibrium and reaction forces in the structural analysis. This figure is particularly relevant for topics in mechanics, statics, and structural engineering. **Learning Goal:**To use the vector cross product to calculate the moment produced by a force, or forces, about a specified point on a member.The moment of a force **F** about the moment axis passing through **O** and perpendicular to the plane containing **O** and **F** can be expressed using the vector cross product, **M₀ = r x F**. In a properly constructed Cartesian coordinate system, the vector cross product can be calculated using a matrix determinant:\[ \mathbf{M} = \mathbf{r} \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\r_x & r_y & r_z \\F_x & F_y & F_z \end{vmatrix}\]Notice that the order of the elements of the matrix determinant is important; switching rows 2 and 3 of the determinant would change the sign of the moment from positive to negative (or vice versa).### Part A - Moment due to a force specified by magnitude and endpointsAs shown, a member is fixed at the origin, point **O**, and has an applied force **F**, the tension in the rope, applied at the free end, point **B**. (Figure 1)The force has a magnitude **F = 180 N** and is directed as shown. The dimensions are \( x_1 = 0.350 \, \text{m}, \, x_2 = 1.90\, \text{m}, \, y_1 = 2.30\, \text{m}, \, \text{and} \, z_1 = 1.20\, \text{m} \).What is the moment about the origin due to the applied force **F**? Express the individual components of the Cartesian vector to three significant figures, separated by commas.#### View Available Hint(s)\[ M_0 = [ \ \ \ \ \ ] \quad \text{i,j,k} \ \text{N} \cdot \text{m} \]\[ ( \text{Diagram depicting the Cartesian coordinate system and vectors}) \]**Submit**\[ \text{Incorrect; Try Again; 3 attempts remaining} \]### Part B - Moment due to a force specified as a Cartesian vectorAs shown, a member is fixed at the

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Engineering

Mechanical Engineering

F2 B

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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Concept explainers

Design Against Fluctuating Loads

Machine elements are subjected to varieties of loads, some components are subjected to static loads, while some machine components are subjected to fluctuating loads, whose load magnitude tends to fluctuate. The components of a machine, when rotating at a high speed, are subjected to a high degree of load, which fluctuates from a high value to a low value. For the machine elements under the action of static loads, static failure theories are applied to know the safe and hazardous working conditions and regions. However, most of the machine elements are subjected to variable or fluctuating stresses, due to the nature of load that fluctuates from high magnitude to low magnitude. Also, the nature of the loads is repetitive. For instance, shafts, bearings, cams and followers, and so on.

Design Against Fluctuating Load

Stress is defined as force per unit area. When there is localization of huge stresses in mechanical components, due to irregularities present in components and sudden changes in cross-section is known as stress concentration. For example, groves, keyways, screw threads, oil holes, splines etc. are irregularities.

Question

### Figure Explanation for Educational Website

#### Description:
This figure depicts a mechanical system in a three-dimensional coordinate space. The system consists of a structure connected at various joints, represented by nodes. The axes are labeled as x, y, and z, illustrating the 3D coordinate system.

#### Components:
- **Axes:** The three coordinate axes (x, y, z) are depicted in the diagram for orientation.
- **Structure:** The structure is composed of blue rods or beams connected by joints at different points in the space.
- **Joints:** The joints are shown as points where the beams meet, marked with spherical connections.
- **Forces:** Two forces, \(F_1\) and \(F_2\), are acting on a joint labeled \(B\).
- \(F_1\) acts in a direction along the plane formed by \(x_1\) and \(y_1\).
- \(F_2\) is pointing along the y-axis.
- **Reference Points:**
- The origin \(O\) is the reference point of the coordinate system.
- Dimensions \(x_1\), \(y_1\), and \(z_1\) indicate the positions of the structure components relative to the origin.

#### Annotations:
- **\(O\):** Represents the origin of the coordinate system where all axes intersect.
- **\(B\):** Represents the point at which the forces \(F_1\) and \(F_2\) are applied.

The dimensions and forces are crucial for understanding the equilibrium and reaction forces in the structural analysis. This figure is particularly relevant for topics in mechanics, statics, and structural engineering.

**Learning Goal:**

To use the vector cross product to calculate the moment produced by a force, or forces, about a specified point on a member.

The moment of a force **F** about the moment axis passing through **O** and perpendicular to the plane containing **O** and **F** can be expressed using the vector cross product, **M₀ = r x F**. In a properly constructed Cartesian coordinate system, the vector cross product can be calculated using a matrix determinant:

\[ \mathbf{M} = \mathbf{r} \times \mathbf{F} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
r_x & r_y & r_z \\
F_x & F_y & F_z
\end{vmatrix}
\]

Notice that the order of the elements of the matrix determinant is important; switching rows 2 and 3 of the determinant would change the sign of the moment from positive to negative (or vice versa).

### Part A - Moment due to a force specified by magnitude and endpoints

As shown, a member is fixed at the origin, point **O**, and has an applied force **F**, the tension in the rope, applied at the free end, point **B**. (Figure 1)

The force has a magnitude **F = 180 N** and is directed as shown. The dimensions are \( x_1 = 0.350 \, \text{m}, \, x_2 = 1.90\, \text{m}, \, y_1 = 2.30\, \text{m}, \, \text{and} \, z_1 = 1.20\, \text{m} \).

What is the moment about the origin due to the applied force **F**?

Express the individual components of the Cartesian vector to three significant figures, separated by commas.

#### View Available Hint(s)
\[ M_0 = [ \ \ \ \ \ ] \quad \text{i,j,k} \ \text{N} \cdot \text{m} \]

\[ ( \text{Diagram depicting the Cartesian coordinate system and vectors}) \]

**Submit**
\[ \text{Incorrect; Try Again; 3 attempts remaining} \]

### Part B - Moment due to a force specified as a Cartesian vector

As shown, a member is fixed at the

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