Three concurrent co-planer force `1N` , `2N` and `3N` acting along different directions on a body

Options

A. can keep the body in equilibrium if 2 N and 3 N act at right angle.
B. can keep the body in equilibrium if 1 N and 2 N act at right angle.
C. cannot keep the body in equilibrium.
D. can keep the body in equilibrium if 1 N and 3 N act at an acute angle

Explanation

To determine whether the three concurrent coplanar forces of 1N, 2N, and 3N can keep a body in equilibrium, we need to analyze the conditions under which forces can balance each other.

### Step-by-Step Solution:

1. **Understanding Equilibrium Condition**:
- For a body to be in equilibrium, the vector sum of all forces acting on it must be zero. This is mathematically represented as:
\[ \Sigma \vec{F} = 0 \]

2. **Identifying the Forces**:
- We have three forces acting on the body:
- \( F_1 = 1N \)
- \( F_2 = 2N \)
- \( F_3 = 3N \)
- These forces are acting along different directions.

3. **Analyzing the Forces**:
- If two forces are acting at right angles, we can use the Pythagorean theorem to find the resultant force.
- For example, if \( F_2 \) and \( F_3 \) are at right angles, the resultant \( R \) can be calculated as:
\[ R = \sqrt{F_2^2 + F_3^2} = \sqrt{(2N)^2 + (3N)^2} = \sqrt{4 + 9} = \sqrt{13}N \approx 3.6N \]
- This resultant force \( R \) must be balanced by \( F_1 \) for equilibrium.

4. **Comparing Forces**:
- Since \( R \approx 3.6N \) and \( F_1 = 1N \), the forces cannot balance each other because \( 1N \) is less than \( 3.6N \). Thus, the body cannot be in equilibrium when \( F_2 \) and \( F_3 \) are at right angles.

5. **Considering Other Angles**:
- If \( F_1 \) and \( F_2 \) act at right angles, we can calculate the resultant similarly:
\[ R = \sqrt{F_1^2 + F_2^2} = \sqrt{(1N)^2 + (2N)^2} = \sqrt{1 + 4} = \sqrt{5}N \approx 2.24N \]
- Again, \( R \approx 2.24N \) is greater than \( F_3 = 3N \), so equilibrium cannot be achieved.

6. **Final Conclusion**:
- Since no combination of the given forces acting at different angles can sum to zero, we conclude that:
\[ \text{The three forces cannot keep the body in equilibrium.} \]

### Final Answer:
The three concurrent coplanar forces of 1N, 2N, and 3N cannot keep the body in equilibrium.

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