Two forces `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 find the angle between two forces \( F_1 \) and \( F_2 \) given that \( F_1 = 2F_2 \) and the resultant force is equal to the greater force \( F_1 \).

### Step-by-Step Solution:

1. **Define the Forces**:
Let \( F_2 = F \). Then, according to the problem, \( F_1 = 2F_2 = 2F \).

2. **Express the Resultant**:
The resultant \( R \) of two forces \( F_1 \) and \( F_2 \) 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 \( F_1 \):
\[ R = F_1 \]

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

4. **Simplifying the Equation**:
Simplifying the equation:
\[ 2F = \sqrt{4F^2 + F^2 + 4F^2 \cos \theta} \]
This simplifies to:
\[ 2F = \sqrt{5F^2 + 4F^2 \cos \theta} \]

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

6. **Rearranging the Equation**:
Rearranging the equation:
\[ 4F^2 - 5F^2 = 4F^2 \cos \theta \]
This simplifies to:
\[ -F^2 = 4F^2 \cos \theta \]

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

8. **Finding the Angle**:
To find \( \theta \):
\[ \theta = \cos^{-1}(-\frac{1}{4}) \]

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