VIPSolutionsAct like a helpful tutor and exlain me :The structure consists of a quarter-circular rod AB with a 150 mm radius that is fixed in the xy plane. An elastic cord supporting a force of 120 N is attached to a support at Dand a bead at C. Determine the components of the cord force in directions tangent and normal to the curved rod AB at point C. y B ( 150 mm Ja 3 — —x (] A 120 mm z 150 mm = The component of F gp in the tangential directionis|____|N. — The component of F' cp in the normal directionis[____|N.
Question:
Act like a helpful tutor and exlain me :
The structure consists of a quarter-circular rod AB with a 150 mm radius that is fixed in the xy plane. An elastic cord supporting a force
of 120 N is attached to a support at Dand a bead at C. Determine the components of the cord force in directions tangent and normal
to the curved rod AB at point C.
y
B (
150 mm
Ja
3
— —x
(] A
120 mm
z 150 mm
=
The component of F gp in the tangential directionis|____|N.
—
The component of F' cp in the normal directionis[____|N.
This is AI generated Response.
Asked by: manish kumar
Created at: 2025-09-12 21:12:58
Asked by: manish kumar
Created at: 2025-09-12 21:12:58
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Let's break down the problem step by step:
---
## 1. **Understanding the Structure**
- **Rod AB** is a quarter circle in the **xy-plane** with a radius of **150 mm**.
- **Point C** is at the end of the curved rod.
- A **cord CD** supports a **120 N** force, attached at **D** and **C**.
---
## 2. **Finding the Position Vectors**
### **Coordinates:**
- **O (center):** (0, 0, 0)
- **B:** (0, 150, 0)
- **C:** (150, 0, 0)
- **D:** (150, 0, 120)
---
## 3. **Finding the Direction of the Cord \( \vec{F}_{CD} \)**
The cord is along **CD**.
\[
\vec{r}_{CD} = \vec{r}_D - \vec{r}_C = (150, 0, 120) - (150, 0, 0) = (0, 0, 120)
\]
So, the **direction** is along the **z-axis**.
\[
\text{Unit vector: } \vec{u}_{CD} = \frac{(0, 0, 120)}{120} = (0, 0, 1)
\]
So,
\[
\vec{F}_{CD} = 120 \, \text{N} \cdot (0, 0, 1)
\]
---
## 4. **Finding Tangent and Normal Directions at Point C**
The rod is in the **xy-plane**. At **C (150, 0, 0)**:
- The **radius** at C is along the **x-axis**: \( \vec{r}_C = (150, 0, 0) \).
- The **tangent** to the circle at C (quarter circle from B to C, counterclockwise) is in the **negative y-direction**: \( \vec{t} = (0, -1, 0) \).
- The **normal** to the curve (pointing towards the center O) is in the **negative x-direction**: \( \vec{n} = (-1, 0, 0) \).
---
## 5. **Projecting the Force**
\[
\vec{F}_{CD} = 120 \cdot (0, 0, 1)
\]
\[
\vec{t} = (0, -1, 0)
\]
\[
\vec{n} = (-1, 0, 0)
\]
### **Tangential Component:**
\[
F_{CD,\text{tangential}} = \vec{F}_{CD} \cdot \vec{t} = 120 \cdot (0, 0, 1) \cdot (0, -1, 0) = 0
\]
### **Normal Component:**
\[
F_{CD,\text{normal}} = \vec{F}_{CD} \cdot \vec{n} = 120 \cdot (0, 0, 1) \cdot (-1, 0, 0) = 0
\]
**Both are zero** because the cord force is purely vertical (z-direction), while the tangent and normal directions are in the xy-plane.
---
## 6. **Summary Table**
| Direction | Component (N) |
|----------------|---------------|
| Tangential | **0** |
| Normal | **0** |
---
### **Final Answers**
- The component of \( \vec{F}_{CD} \) in the **tangential direction** is **0 N**.
- The component of \( \vec{F}_{CD} \) in the **normal direction** is **0 N**.
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