Which of the following group of concurrent forces may be in equilibrium (R=0)?

Options

A. `{:(A,B,C),(10,20,40):}`
B. `{:(A,B,C),(3,5,1):}`
C. `{:(A,B,C),(20,20,20):}`
D. `{:(A,B,C),(40,30,5):}`

Answer

To determine which group of concurrent forces may be in equilibrium (where the resultant force R = 0), we can analyze each option based on the principle of equilibrium for concurrent forces. The conditions for equilibrium state that the sum of the forces acting in any direction must be zero. This can be mathematically represented as:

1. For three forces A, B, and C to be in equilibrium, the following condition must hold true:
- A + B = C (or any permutation thereof)
- The triangle inequality must also be satisfied: A + B ≥ C, A + C ≥ B, and B + C ≥ A.

Now, let’s analyze each option step by step.

### Step 1: Analyze Option A
- **Given:** A + B > C
- **Condition Check:** This does not satisfy the equilibrium condition since it does not fulfill A + B = C or A + B ≥ C.
- **Conclusion:** This option is not in equilibrium.

### Step 2: Analyze Option B
- **Given:** A + C < B
- **Condition Check:** This also does not satisfy the equilibrium condition since it does not fulfill A + C = B or A + C ≥ B.
- **Conclusion:** This option is not in equilibrium.

### Step 3: Analyze Option C
- **Given:** A + B ≥ C
- **Condition Check:** This condition satisfies the equilibrium requirement. If A + B = C, then the forces can be in equilibrium.
- **Conclusion:** This option may be in equilibrium.

### Step 4: Analyze Option D
- **Given:** B + C < A
- **Condition Check:** This does not satisfy the equilibrium condition since it does not fulfill B + C = A or B + C ≥ A.
- **Conclusion:** This option is not in equilibrium.

### Final Conclusion:
The only group of concurrent forces that may be in equilibrium is **Option C**.

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