Two forces 8 N and 12 act ay `120^(@)` The third force required to keep the body in equilbrium is

Options

A. 4 N
B. `4 sqrt(7)N`
C. 20 N
D. none of these

Explanation

To find the third force required to keep the body in equilibrium, we will follow these steps:

### Step 1: Identify the Given Forces
We have two forces:
- \( F_1 = 8 \, \text{N} \)
- \( F_2 = 12 \, \text{N} \)
- The angle between them, \( \theta = 120^\circ \)

### Step 2: Use the Law of Cosines to Find the Resultant Force
The resultant force \( R \) of the two forces can be calculated using the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(\theta)} \]
Substituting the values:
\[ R = \sqrt{8^2 + 12^2 + 2 \cdot 8 \cdot 12 \cdot \cos(120^\circ)} \]

### Step 3: Calculate \( \cos(120^\circ) \)
We know that:
\[ \cos(120^\circ) = -\frac{1}{2} \]

### Step 4: Substitute \( \cos(120^\circ) \) into the Resultant Force Equation
Now substituting \( \cos(120^\circ) \) into the equation:
\[ R = \sqrt{8^2 + 12^2 + 2 \cdot 8 \cdot 12 \cdot \left(-\frac{1}{2}\right)} \]

### Step 5: Simplify the Equation
Calculating each term:
\[ R = \sqrt{64 + 144 - 96} \]
\[ R = \sqrt{112} \]

### Step 6: Calculate the Resultant Magnitude
Now, we simplify \( \sqrt{112} \):
\[ R = \sqrt{16 \cdot 7} = 4\sqrt{7} \]

### Step 7: Find the Third Force
Since the body is in equilibrium, the third force \( F_3 \) must be equal in magnitude and opposite in direction to the resultant force \( R \):
\[ F_3 = R = 4\sqrt{7} \, \text{N} \]

### Conclusion
Thus, the third force required to keep the body in equilibrium is:
\[ F_3 = 4\sqrt{7} \, \text{N} \]

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