The square of resultant of two equal forces is three times their product. Angle between the force is

Options

A. (a)`pi`
B. (b)`( pi)/( 2)`
C. (c)`( pi)/( 4)`
D. (d)`( pi)/( 3)`

Answer

To solve the problem, we need to find the angle between two equal forces given that the square of their resultant is three times their product. Let's denote the equal forces as \( P \) and \( Q \), and the angle between them as \( \theta \).

### Step-by-Step Solution:

1. **Understanding the Forces**:
Let \( P = Q \). Since both forces are equal, we can denote them as \( P \).

2. **Using the Resultant Formula**:
The formula for the resultant \( R \) of two forces \( P \) and \( Q \) at an angle \( \theta \) is given by:
\[ R^2 = P^2 + Q^2 + 2PQ \cos(\theta) \]
Since \( P = Q \), we can rewrite this as:
\[ R^2 = P^2 + P^2 + 2P \cdot P \cos(\theta) = 2P^2 + 2P^2 \cos(\theta) \]
Simplifying this gives:
\[ R^2 = 2P^2(1 + \cos(\theta)) \]

3. **Using the Given Condition**:
According to the problem, the square of the resultant is three times their product:
\[ R^2 = 3PQ = 3P^2 \]

4. **Setting the Equations Equal**:
Now we can set the two expressions for \( R^2 \) equal to each other:
\[ 2P^2(1 + \cos(\theta)) = 3P^2 \]

5. **Dividing by \( P^2 \)**:
Since \( P^2 \) is not zero, we can divide both sides by \( P^2 \):
\[ 2(1 + \cos(\theta)) = 3 \]

6. **Solving for \( \cos(\theta) \)**:
Rearranging gives:
\[ 1 + \cos(\theta) = \frac{3}{2} \]
\[ \cos(\theta) = \frac{3}{2} - 1 = \frac{1}{2} \]

7. **Finding the Angle \( \theta \)**:
The cosine of \( \theta \) is \( \frac{1}{2} \). The angle \( \theta \) that satisfies this condition is:
\[ \theta = 60^\circ \quad \text{or} \quad \theta = \frac{\pi}{3} \text{ radians} \]

### Final Answer:
The angle between the two forces is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.