The resultant of two forces 3P and 2P is R, if first force is doubled, the resultant is also doubled. Then the angle between the forces is

Options

A. `30^(@)`
B. `60^(@)`
C. `120^(@)`
D. `150^(@)`

Answer

To solve the problem, we need to find the angle between the two forces, 3P and 2P, given that when the first force is doubled, the resultant is also doubled. Let's break this down step by step.

### Step 1: Write the expression for the resultant of two forces
The resultant \( R \) of two forces \( F_1 \) and \( F_2 \) acting at an angle \( \theta \) is given by the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]
In our case, \( F_1 = 3P \) and \( F_2 = 2P \). Therefore, we can write:
\[ R = \sqrt{(3P)^2 + (2P)^2 + 2(3P)(2P) \cos \theta} \]

### Step 2: Substitute the values into the equation
Substituting the values into the expression, we get:
\[ R = \sqrt{9P^2 + 4P^2 + 12P^2 \cos \theta} \]
Simplifying this, we have:
\[ R = \sqrt{13P^2 + 12P^2 \cos \theta} \]

### Step 3: Consider the case when the first force is doubled
If the first force is doubled, it becomes \( 6P \). The new resultant \( R' \) is given by:
\[ R' = \sqrt{(6P)^2 + (2P)^2 + 2(6P)(2P) \cos \theta} \]
Substituting the values, we have:
\[ R' = \sqrt{36P^2 + 4P^2 + 24P^2 \cos \theta} \]
This simplifies to:
\[ R' = \sqrt{40P^2 + 24P^2 \cos \theta} \]

### Step 4: Set up the equation based on the condition that the resultant is doubled
According to the problem, when the first force is doubled, the resultant is also doubled:
\[ R' = 2R \]
Substituting the expressions for \( R' \) and \( R \):
\[ \sqrt{40P^2 + 24P^2 \cos \theta} = 2 \sqrt{13P^2 + 12P^2 \cos \theta} \]

### Step 5: Square both sides to eliminate the square roots
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 6: Rearranging the equation
Rearranging the equation, we get:
\[ 40P^2 - 52P^2 = 48P^2 \cos \theta - 24P^2 \cos \theta \]
This simplifies to:
\[ -12P^2 = 24P^2 \cos \theta \]
Dividing both sides by \( 12P^2 \):
\[ -1 = 2 \cos \theta \]

### Step 7: Solve for \( \theta \)
Thus, we find:
\[ \cos \theta = -\frac{1}{2} \]
This corresponds to:
\[ \theta = 120^\circ \quad \text{(or } 240^\circ \text{, but we consider the angle between forces)} \]

### Final Answer
The angle between the forces is \( \theta = 120^\circ \).

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