The resultant of two forces 3P and 2P is R. If the first force is doubled then resultant is also doubled.The angle between the two forces is

Options

A. `60^@`
B. `120^@`
C. `30^@`
D. `135^@`

Explanation

To solve the problem, we need to find the angle between two forces, given that the resultant of the forces changes when one of the forces is doubled. Let's go through the solution step by step.

### Step 1: Write the expression for the resultant of the two forces
The two forces are \(3P\) and \(2P\). The resultant \(R\) of two forces can be expressed using the formula:
\[ R = \sqrt{(3P)^2 + (2P)^2 + 2 \cdot (3P) \cdot (2P) \cdot \cos \theta} \]
Calculating the squares:
\[ R = \sqrt{9P^2 + 4P^2 + 12P^2 \cos \theta} \]
Simplifying this gives:
\[ R = \sqrt{13P^2 + 12P^2 \cos \theta} \]

### Step 2: Write the expression for the new resultant when the first force is doubled
When the first force is doubled, it becomes \(6P\). The new resultant \(R'\) is given by:
\[ R' = \sqrt{(6P)^2 + (2P)^2 + 2 \cdot (6P) \cdot (2P) \cdot \cos \theta} \]
Calculating the squares:
\[ R' = \sqrt{36P^2 + 4P^2 + 24P^2 \cos \theta} \]
Simplifying this gives:
\[ R' = \sqrt{40P^2 + 24P^2 \cos \theta} \]

### Step 3: Set up the relationship between the two resultants
According to the problem, when the first force is doubled, the resultant also doubles:
\[ R' = 2R \]
Substituting the expressions for \(R\) and \(R'\):
\[ \sqrt{40P^2 + 24P^2 \cos \theta} = 2 \sqrt{13P^2 + 12P^2 \cos \theta} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 40P^2 + 24P^2 \cos \theta = 4(13P^2 + 12P^2 \cos \theta) \]
Expanding the right side:
\[ 40P^2 + 24P^2 \cos \theta = 52P^2 + 48P^2 \cos \theta \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ 40P^2 - 52P^2 = 48P^2 \cos \theta - 24P^2 \cos \theta \]
This simplifies to:
\[ -12P^2 = 24P^2 \cos \theta \]

### Step 6: Solve for \(\cos \theta\)
Dividing both sides by \(12P^2\):
\[ -1 = 2 \cos \theta \]
Thus:
\[ \cos \theta = -\frac{1}{2} \]

### Step 7: Determine the angle \(\theta\)
The angle \(\theta\) for which \(\cos \theta = -\frac{1}{2}\) is:
\[ \theta = 120^\circ \]

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