Square of the resultant of two forces of equal magnitude is equal to three times the product of their magnitude. The angle between them is

Options

A. `0^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Explanation

To solve the problem, we need to find the angle between two forces of equal magnitude when the square of their resultant is equal to three times the product of their magnitudes.

Let's denote the magnitude of each force as \( F \).

### Step 1: Write the formula for the resultant of two forces
The resultant \( R \) of two forces \( F \) and \( F \) at an angle \( \theta \) is given by the formula:
\[ R = \sqrt{F^2 + F^2 + 2F \cdot F \cdot \cos(\theta)} \]
This simplifies to:
\[ R = \sqrt{2F^2 + 2F^2 \cos(\theta)} \]
or
\[ R = \sqrt{2F^2(1 + \cos(\theta))} \]

### Step 2: Square the resultant
We need to square the resultant:
\[ R^2 = 2F^2(1 + \cos(\theta)) \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the square of the resultant is equal to three times the product of their magnitudes:
\[ R^2 = 3 \cdot F \cdot F = 3F^2 \]

### Step 4: Equate the two expressions for \( R^2 \)
Now we can equate the two expressions we have for \( R^2 \):
\[ 2F^2(1 + \cos(\theta)) = 3F^2 \]

### Step 5: Simplify the equation
We can divide both sides by \( F^2 \) (assuming \( F \neq 0 \)):
\[ 2(1 + \cos(\theta)) = 3 \]
Now, divide both sides by 2:
\[ 1 + \cos(\theta) = \frac{3}{2} \]

### Step 6: Solve for \( \cos(\theta) \)
Subtract 1 from both sides:
\[ \cos(\theta) = \frac{3}{2} - 1 = \frac{1}{2} \]

### Step 7: Find the angle \( \theta \)
The cosine of \( \theta \) is \( \frac{1}{2} \). The angle \( \theta \) that corresponds to this value is:
\[ \theta = 60^\circ \]

### Final Answer
Thus, the angle between the two forces is \( 60^\circ \).

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