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)`e
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 one force is double the other and the resultant is equal to the greater force. Let's break it down step by step.

### Step 1: Define the Forces
Let:
- \( F_1 \) be the smaller force.
- \( F_2 = 2F_1 \) be the greater force (since one force is double the other).

### Step 2: Write the Resultant Force Equation
According to the problem, the resultant force \( R \) is equal to the greater force \( F_2 \):
\[ R = F_2 = 2F_1 \]

### Step 3: Use the Formula for Resultant of Two Forces
The formula for the resultant \( R \) of two forces \( F_1 \) and \( F_2 \) acting at an angle \( \theta \) is given by:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]
Substituting \( F_2 = 2F_1 \) into the equation:
\[ R = \sqrt{F_1^2 + (2F_1)^2 + 2F_1(2F_1) \cos \theta} \]
This simplifies to:
\[ R = \sqrt{F_1^2 + 4F_1^2 + 4F_1^2 \cos \theta} \]
\[ R = \sqrt{5F_1^2 + 4F_1^2 \cos \theta} \]

### Step 4: Set the Resultant Equal to the Greater Force
Since we know \( R = 2F_1 \), we can set the two expressions equal to each other:
\[ 2F_1 = \sqrt{5F_1^2 + 4F_1^2 \cos \theta} \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ (2F_1)^2 = 5F_1^2 + 4F_1^2 \cos \theta \]
\[ 4F_1^2 = 5F_1^2 + 4F_1^2 \cos \theta \]

### Step 6: Rearrange the Equation
Rearranging gives us:
\[ 4F_1^2 - 5F_1^2 = 4F_1^2 \cos \theta \]
\[ -F_1^2 = 4F_1^2 \cos \theta \]

### Step 7: Solve for \( \cos \theta \)
Dividing both sides by \( F_1^2 \) (assuming \( F_1 \neq 0 \)):
\[ -1 = 4 \cos \theta \]
\[ \cos \theta = -\frac{1}{4} \]

### Step 8: Find the Angle \( \theta \)
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(-\frac{1}{4}\right) \]

### Conclusion
The angle between the two forces \( F_1 \) and \( F_2 \) is \( \theta = \cos^{-1}\left(-\frac{1}{4}\right) \).