The resultant of two forces, one double the other in magnitude is perpendicular to the smaller of the two forces. The angle between the two forces is

Options

A. `120^@`
B. `60^@`
C. `90^@`
D. `150^@`

Explanation

To solve the problem, let's break it down step by step:

### Step 1: Understand the Forces
Let the smaller force be \( F \) and the larger force be \( 2F \) (since one force is double the other).

### Step 2: Set Up the Problem
We know that the resultant of these two forces is perpendicular to the smaller force \( F \). This means that if we represent the smaller force \( F \) along the x-axis, the larger force \( 2F \) will make an angle \( \theta \) with it.

### Step 3: Use the Resultant Force Condition
The resultant \( R \) of the two forces can be expressed using the formula:
\[ R^2 = F^2 + (2F)^2 + 2F \cdot 2F \cdot \cos(\theta) \]
Since \( R \) is perpendicular to \( F \), we have:
\[ R^2 = F^2 + (2F)^2 \]
This simplifies to:
\[ R^2 = F^2 + 4F^2 = 5F^2 \]

### Step 4: Relate the Resultant to the Angle
From the condition that \( R \) is perpendicular to \( F \), we also have:
\[ R = 2F \sin(\theta) \]
Setting the two expressions for \( R^2 \) equal gives:
\[ 5F^2 = (2F \sin(\theta))^2 \]
This simplifies to:
\[ 5F^2 = 4F^2 \sin^2(\theta) \]
Dividing both sides by \( F^2 \) (assuming \( F \neq 0 \)):
\[ 5 = 4 \sin^2(\theta) \]

### Step 5: Solve for \( \sin(\theta) \)
Rearranging gives:
\[ \sin^2(\theta) = \frac{5}{4} \]
This is incorrect since \( \sin^2(\theta) \) cannot exceed 1. Let's go back and use the correct approach.

### Step 6: Correct Approach
Since we know \( R \) is perpendicular to \( F \), we can use the sine rule:
\[ 2F \sin(\theta) = F \]
This implies:
\[ 2 \sin(\theta) = 1 \quad \Rightarrow \quad \sin(\theta) = \frac{1}{2} \]
Thus:
\[ \theta = 30^\circ \]

### Step 7: Find the Angle Between the Forces
The angle between the two forces is given by:
\[ \phi = 90^\circ + \theta = 90^\circ + 30^\circ = 120^\circ \]

### Conclusion
Thus, the angle between the two forces is \( 120^\circ \).