Which of the following sets of concurrent force may be in equilibrium?

Options

A. `F_(1)=3N, F_(2)=5N, F_(3)=1 N`
B. `F_(1)=3N, F_(2)=5N, F_(3)=6 N`
C. `F_(1)=3N, F_(2)=5N, F_(3)=9 N`
D. `F_(1)=3N, F_(2)=5N, F_(3)=16 N`

Explanation

To determine which set of concurrent forces may be in equilibrium, we need to apply the principle of equilibrium for concurrent forces. The condition for equilibrium states that the vector sum of all forces acting at a point must be zero. This can be mathematically expressed as:

\[ F_1 + F_2 + F_3 = 0 \]

For three forces to be in equilibrium, they must satisfy the triangle inequality. The triangle inequality states that for any three lengths \( a, b, c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Let's analyze the given options step by step:

### Step 1: Analyze Each Set of Forces

1. **First Set: \( F_1 = 5, F_2 = 3, F_3 = 1 \)**
- Check \( F_1 + F_2 > F_3 \): \( 5 + 3 > 1 \) (True)
- Check \( F_1 + F_3 > F_2 \): \( 5 + 1 > 3 \) (True)
- Check \( F_2 + F_3 > F_1 \): \( 3 + 1 > 5 \) (False)
- **Conclusion**: This set does not satisfy the triangle inequality.

2. **Second Set: \( F_1 = 5, F_2 = 3, F_3 = 6 \)**
- Check \( F_1 + F_2 > F_3 \): \( 5 + 3 > 6 \) (False)
- Check \( F_1 + F_3 > F_2 \): \( 5 + 6 > 3 \) (True)
- Check \( F_2 + F_3 > F_1 \): \( 3 + 6 > 5 \) (True)
- **Conclusion**: This set does not satisfy the triangle inequality.

3. **Third Set: \( F_1 = 5, F_2 = 1, F_3 = 3 \)**
- Check \( F_1 + F_2 > F_3 \): \( 5 + 1 > 3 \) (True)
- Check \( F_1 + F_3 > F_2 \): \( 5 + 3 > 1 \) (True)
- Check \( F_2 + F_3 > F_1 \): \( 1 + 3 > 5 \) (False)
- **Conclusion**: This set does not satisfy the triangle inequality.

4. **Fourth Set: \( F_1 = 5, F_2 = 2, F_3 = 8 \)**
- Check \( F_1 + F_2 > F_3 \): \( 5 + 2 > 8 \) (False)
- Check \( F_1 + F_3 > F_2 \): \( 5 + 8 > 2 \) (True)
- Check \( F_2 + F_3 > F_1 \): \( 2 + 8 > 5 \) (True)
- **Conclusion**: This set does not satisfy the triangle inequality.

### Step 2: Summary of Findings
- After analyzing all the sets of forces, none of the sets satisfy the triangle inequality, which means none of them can be in equilibrium.

### Final Conclusion
None of the given sets of concurrent forces may be in equilibrium.

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