Two force `F_(1)" and "F_(2)` are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is :-

Options

A. `cos^(-1)(1//2)`
B. `cos^(-1)(-1//2)`
C. `cos^(-1)(-1//4)`
D. `cos^(-1)(1//4)`

Answer

To solve the problem, we need to analyze the forces acting on the body and apply the principles of vector addition.

### Step-by-Step Solution:

1. **Define the Forces:**
Let \( F_1 = F \) (the smaller force) and \( F_2 = 2F \) (the greater force).

2. **Use the Resultant Force Equation:**
The resultant \( R \) of two forces can be expressed using the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]
According to the problem, the resultant is equal to the greater force \( F_2 \):
\[ R = F_2 = 2F \]

3. **Substituting Values:**
Substitute \( F_1 \) and \( F_2 \) into the resultant equation:
\[ 2F = \sqrt{F^2 + (2F)^2 + 2 \cdot F \cdot 2F \cdot \cos \theta} \]

4. **Simplify the Equation:**
Calculate \( (2F)^2 \):
\[ (2F)^2 = 4F^2 \]
Now substitute this back into the equation:
\[ 2F = \sqrt{F^2 + 4F^2 + 4F^2 \cos \theta} \]
Combine like terms:
\[ 2F = \sqrt{5F^2 + 4F^2 \cos \theta} \]

5. **Square Both Sides:**
Square both sides to eliminate the square root:
\[ (2F)^2 = 5F^2 + 4F^2 \cos \theta \]
This simplifies to:
\[ 4F^2 = 5F^2 + 4F^2 \cos \theta \]

6. **Rearranging the Equation:**
Rearranging gives:
\[ 4F^2 - 5F^2 = 4F^2 \cos \theta \]
\[ -F^2 = 4F^2 \cos \theta \]

7. **Solving for Cosine:**
Divide both sides by \( F^2 \) (assuming \( F \neq 0 \)):
\[ -1 = 4 \cos \theta \]
Therefore:
\[ \cos \theta = -\frac{1}{4} \]

8. **Finding the Angle:**
Now, take the inverse cosine to find \( \theta \):
\[ \theta = \cos^{-1}\left(-\frac{1}{4}\right) \]

### Final Answer:
The angle between the two forces is:
\[ \theta = \cos^{-1}\left(-\frac{1}{4}\right) \]