The resultant of two forces (A + B) and (A - B) is a force `sqrt(3A^(2) + B^(2))` The angle between two given forces is

Options

A. `pi/4`
B. `pi/3`
C. `pi/2`
D. `pi`

Explanation

To find the angle between the two forces \( A + B \) and \( A - B \), we can use the formula for the resultant of two vectors. Here are the steps to solve the problem:

### Step 1: Understand the Resultant Force
The resultant of two forces \( \vec{F_1} = A + B \) and \( \vec{F_2} = A - B \) is given as:
\[ R = \sqrt{3A^2 + B^2} \]

### Step 2: Use the Formula for the Magnitude of the Resultant
The magnitude of the resultant \( R \) of two vectors can be expressed as:
\[ R^2 = F_1^2 + F_2^2 + 2F_1F_2 \cos \theta \]
Where \( F_1 = A + B \) and \( F_2 = A - B \).

### Step 3: Calculate \( F_1^2 \) and \( F_2^2 \)
Calculate the squares of the magnitudes of the forces:
\[ F_1^2 = (A + B)^2 = A^2 + 2AB + B^2 \]
\[ F_2^2 = (A - B)^2 = A^2 - 2AB + B^2 \]

### Step 4: Substitute into the Resultant Formula
Now substitute \( F_1^2 \) and \( F_2^2 \) into the resultant formula:
\[ R^2 = (A^2 + 2AB + B^2) + (A^2 - 2AB + B^2) + 2(A + B)(A - B) \cos \theta \]
This simplifies to:
\[ R^2 = 2A^2 + 2B^2 + 2(A^2 - B^2) \cos \theta \]

### Step 5: Set the Equation Equal to Given Resultant
Now we know that \( R^2 = 3A^2 + B^2 \), so we set the two expressions for \( R^2 \) equal to each other:
\[ 3A^2 + B^2 = 2A^2 + 2B^2 + 2(A^2 - B^2) \cos \theta \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 3A^2 + B^2 - 2A^2 - 2B^2 = 2(A^2 - B^2) \cos \theta \]
This simplifies to:
\[ A^2 - B^2 = 2(A^2 - B^2) \cos \theta \]

### Step 7: Solve for \( \cos \theta \)
Dividing both sides by \( A^2 - B^2 \) (assuming \( A^2 \neq B^2 \)):
\[ 1 = 2 \cos \theta \]
Thus,
\[ \cos \theta = \frac{1}{2} \]

### Step 8: Find the Angle \( \theta \)
The angle \( \theta \) corresponding to \( \cos \theta = \frac{1}{2} \) 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.

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