VIPSolutionsAct like a helpful tutor and exlain me : Give step-by-step solution with explanation and final answer:Qé. As shown in the image below, the force of 443 N directs from point A to point B. The coordinates are A: (18, 0, 0) cm, and B: (0, 18, 26) cm. Determine the z- component of the moment vector caused by the force about point Oin N-m. Negative sign must be included when applicable. Please pay attention: the numbers may change since they are randomized. Your answer must include 1 place after the decimal point. B F - y [@] A
Question:
Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:Qé. As shown in the image below, the force of 443 N directs from point A to point B.
The coordinates are A: (18, 0, 0) cm, and B: (0, 18, 26) cm. Determine the z-
component of the moment vector caused by the force about point Oin N-m.
Negative sign must be included when applicable. Please pay attention: the numbers
may change since they are randomized. Your answer must include 1 place after the
decimal point.
B
F
- y
[@]
A
This is AI generated Response.
Asked by: manish kumar
Created at: 2025-09-15 07:23:10
Asked by: manish kumar
Created at: 2025-09-15 07:23:10
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Let's break down the problem step by step:
## **Step 1: Write Given Data**
- **Force** \(\vec{F}\) of **443 N** from \(A\) to \(B\)
- **Coordinates:**
- \(A: (18, 0, 0)\) cm
- \(B: (0, 18, 26)\) cm
- **Origin:** \(O: (0, 0, 0)\)
Convert coordinates to meters:
- \(A: (0.18, 0, 0)\) m
- \(B: (0, 0.18, 0.26)\) m
---
## **Step 2: Find the Force Vector \(\vec{F}\) Direction**
1. **Find direction vector** from \(A\) to \(B\):
\[
\overrightarrow{AB} = (B - A) = (0 - 0.18, 0.18 - 0, 0.26 - 0) = (-0.18, 0.18, 0.26)
\]
2. **Find magnitude of \(\overrightarrow{AB}\):**
\[
|\overrightarrow{AB}| = \sqrt{(-0.18)^2 + (0.18)^2 + (0.26)^2} = \sqrt{0.0324 + 0.0324 + 0.0676} = \sqrt{0.1324} \approx 0.364
\]
3. **Unit vector in direction of \(\overrightarrow{AB}\):**
\[
\hat{u}_{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} = \left(\frac{-0.18}{0.364}, \frac{0.18}{0.364}, \frac{0.26}{0.364}\right) \approx (-0.494, 0.494, 0.714)
\]
4. **Force vector:**
\[
\vec{F} = 443 \times \hat{u}_{AB} \approx 443 \times (-0.494, 0.494, 0.714) \approx (-218.8, 218.8, 316.5) \text{ N}
\]
---
## **Step 3: Position Vector from \(O\) to \(A\) (\(\vec{r}\))**
\[
\vec{r}_A = (0.18, 0, 0) \text{ m}
\]
---
## **Step 4: Moment About \(O\) (\(\vec{M}_O = \vec{r} \times \vec{F}\))**
\[
\vec{M}_O = \vec{r}_A \times \vec{F}
\]
Write vectors:
- \(\vec{r}_A = (0.18, 0, 0)\)
- \(\vec{F} = (F_x, F_y, F_z) = (-218.8, 218.8, 316.5)\)
Use determinant:
\[
\vec{M}_O =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
0.18 & 0 & 0 \\
-218.8 & 218.8 & 316.5 \\
\end{vmatrix}
\]
The **z-component** (\(\hat{k}\)) is:
\[
M_{O,z} = (r_x F_y - r_y F_x) = (0.18 \times 218.8 - 0 \times -218.8) = 39.384
\]
---
## **Step 5: Final Answer (with 1 decimal place)**
\[
\boxed{M_{O,z} = 39.4 \ \text{N}\cdot\text{m}}
\]
---
### **Summary Table**
| Step | Result |
|------------------------|---------------------------------------|
| Direction vector (\(\overrightarrow{AB}\)) | \((-0.18, 0.18, 0.26)\) m |
| Unit vector (\(\hat{u}_{AB}\)) | \((-0.494, 0.494, 0.714)\) |
| Force vector (\(\vec{F}\)) | \((-218.8, 218.8, 316.5)\) N |
| Position vector (\(\vec{r}_A\)) | \((0.18, 0, 0)\) m |
| \(z\)-component of moment (\(M_{O,z}\)) | \(39.4 \text{ N}\cdot\text{m}\) |
If you need the full vector moment or any other component, let me know!
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