The three ropes in the figure are tied to a small, very light ring. Two of the ropes are anchored to the walls at right angles, and the third rope pulls as shown. What are T₁ and T2, the magnitudes of the tension forces in the first two ropes? (Figure 1) Figure ▼ Part A 1 of 20 Rope 1 Express your answer in newtons. Rope 2 30° 100 N 1 of 1
The three ropes in the figure are tied to a small, very light ring. Two of the ropes are anchored to the walls at right angles, and the third rope pulls as shown. What are T₁ and T2, the magnitudes of the tension forces in the first two ropes? (Figure 1) Figure ▼ Part A 1 of 20 Rope 1 Express your answer in newtons. Rope 2 30° 100 N 1 of 1
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Katz, Debora M.
Chapter6: Applications Of Newton’s Laws Of Motion
Section: Chapter Questions
Problem 40PQ
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![### Understanding Tension Forces in a Three-Rope System
#### Problem Statement:
The three ropes in the figure are tied to a small, very light ring. Two of the ropes are anchored to the walls at right angles, and the third rope pulls as shown. What are \( T_1 \) and \( T_2 \), the magnitudes of the tension forces in the first two ropes?
#### Diagram Description:
The provided figure illustrates three ropes connected to a small ring.
- **Rope 1** extends horizontally to the left.
- **Rope 2** extends vertically upward.
- **Rope 3** is pulling downwards to the right, forming a 30-degree angle with the horizontal axis. The rope has a tension force of 100 N directed at this angle.
#### Figure:
1. The ring is the central connecting point of the three ropes.
2. **Rope 1** is horizontal.
3. **Rope 2** is vertical.
4. **Rope 3** is angled at 30 degrees below the horizontal, with an indicated force of 100 N.
#### Part A:
To solve for the tension forces \( T_1 \) and \( T_2 \), we need to use the equilibrium equations in both horizontal and vertical directions because the ring is stationary.
- **Horizontal Equilibrium**: \( T_1 = T_3 \cos(30^\circ) \)
- **Vertical Equilibrium**: \( T_2 = T_3 \sin(30^\circ) \)
Given:
\[ T_3 = 100 \, \text{N} \]
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
\[ \sin(30^\circ) = \frac{1}{2} \]
Now, apply these to find \( T_1 \) and \( T_2 \):
\[ T_1 = 100 \, \text{N} \times \frac{\sqrt{3}}{2} = 50\sqrt{3} \, \text{N} \approx 86.6 \, \text{N} \]
\[ T_2 = 100 \, \text{N} \times \frac{1}{2} = 50 \, \text{N} \]
#### Answer Submission:
Express your answer in newtons.
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Transcribed Image Text:### Understanding Tension Forces in a Three-Rope System
#### Problem Statement:
The three ropes in the figure are tied to a small, very light ring. Two of the ropes are anchored to the walls at right angles, and the third rope pulls as shown. What are \( T_1 \) and \( T_2 \), the magnitudes of the tension forces in the first two ropes?
#### Diagram Description:
The provided figure illustrates three ropes connected to a small ring.
- **Rope 1** extends horizontally to the left.
- **Rope 2** extends vertically upward.
- **Rope 3** is pulling downwards to the right, forming a 30-degree angle with the horizontal axis. The rope has a tension force of 100 N directed at this angle.
#### Figure:
1. The ring is the central connecting point of the three ropes.
2. **Rope 1** is horizontal.
3. **Rope 2** is vertical.
4. **Rope 3** is angled at 30 degrees below the horizontal, with an indicated force of 100 N.
#### Part A:
To solve for the tension forces \( T_1 \) and \( T_2 \), we need to use the equilibrium equations in both horizontal and vertical directions because the ring is stationary.
- **Horizontal Equilibrium**: \( T_1 = T_3 \cos(30^\circ) \)
- **Vertical Equilibrium**: \( T_2 = T_3 \sin(30^\circ) \)
Given:
\[ T_3 = 100 \, \text{N} \]
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
\[ \sin(30^\circ) = \frac{1}{2} \]
Now, apply these to find \( T_1 \) and \( T_2 \):
\[ T_1 = 100 \, \text{N} \times \frac{\sqrt{3}}{2} = 50\sqrt{3} \, \text{N} \approx 86.6 \, \text{N} \]
\[ T_2 = 100 \, \text{N} \times \frac{1}{2} = 50 \, \text{N} \]
#### Answer Submission:
Express your answer in newtons.
\[ T
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