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 when two forces of 8 N and 12 N act at an angle of 120 degrees, we can follow these steps:

### Step 1: Identify the 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 \( R \) can be calculated using the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)} \]

### Step 3: Substitute the Values
Substituting the values into the formula:
\[ R = \sqrt{8^2 + 12^2 + 2 \cdot 8 \cdot 12 \cdot \cos(120^\circ)} \]
We know that \( \cos(120^\circ) = -\frac{1}{2} \).

### Step 4: Calculate Each Term
Calculating each term:
- \( 8^2 = 64 \)
- \( 12^2 = 144 \)
- \( 2 \cdot 8 \cdot 12 \cdot \cos(120^\circ) = 2 \cdot 8 \cdot 12 \cdot -\frac{1}{2} = -96 \)

### Step 5: Combine the Results
Now, combine the results:
\[ R = \sqrt{64 + 144 - 96} \]
\[ R = \sqrt{112} \]

### Step 6: Simplify the Result
The square root of 112 can be simplified:
\[ R = \sqrt{16 \cdot 7} = 4\sqrt{7} \, \text{N} \]

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

### Final Answer
The third force required to keep the body in equilibrium is \( 4\sqrt{7} \, \text{N} \).

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