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

Answer

To solve the problem, we need to find the angle between two forces of equal magnitude given that the square of the resultant of these forces is equal to three times the product of their magnitudes. Let's go through the solution step by step.

### Step 1: Define the Forces
Let the magnitudes of the two forces be \( F_1 \) and \( F_2 \). Since the forces are of equal magnitude, we can write:
\[ F_1 = F_2 = F \]

### Step 2: Write the Expression for the Resultant
The resultant \( R \) of two vectors \( F_1 \) and \( F_2 \) can be expressed using the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta} \]
Substituting \( F_1 \) and \( F_2 \) with \( F \):
\[ R = \sqrt{F^2 + F^2 + 2F \cdot F \cos \theta} \]
\[ R = \sqrt{2F^2 + 2F^2 \cos \theta} \]
\[ R = \sqrt{2F^2(1 + \cos \theta)} \]
\[ R = F \sqrt{2(1 + \cos \theta)} \]

### Step 3: Square the Resultant
Now, we need to square the resultant:
\[ R^2 = (F \sqrt{2(1 + \cos \theta)})^2 \]
\[ R^2 = F^2 \cdot 2(1 + \cos \theta) \]
\[ R^2 = 2F^2(1 + \cos \theta) \]

### Step 4: Set Up the Equation
According to the problem, the square of the resultant is equal to three times the product of their magnitudes:
\[ R^2 = 3 \cdot F_1 \cdot F_2 \]
Substituting \( F_1 \) and \( F_2 \):
\[ R^2 = 3F \cdot F \]
\[ R^2 = 3F^2 \]

### Step 5: Equate the Two Expressions for \( R^2 \)
Now we equate the two expressions we have for \( R^2 \):
\[ 2F^2(1 + \cos \theta) = 3F^2 \]

### Step 6: Simplify the Equation
Dividing both sides by \( F^2 \) (assuming \( F \neq 0 \)):
\[ 2(1 + \cos \theta) = 3 \]
\[ 1 + \cos \theta = \frac{3}{2} \]
\[ \cos \theta = \frac{3}{2} - 1 \]
\[ \cos \theta = \frac{1}{2} \]

### Step 7: Find the Angle
The angle \( \theta \) for which \( \cos \theta = \frac{1}{2} \) is:
\[ \theta = 60^\circ \text{ or } \theta = \frac{\pi}{3} \text{ radians} \]

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

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