Two forces, each of magnitude F have a resultant of the same magnitude F. The angle between the two forces is

Options

A. `45^@`
B. `120^@`
C. `150^@`
D. `60^@`

Explanation

To solve the problem, we need to find the angle between two forces, each of magnitude \( F \), that have a resultant of the same magnitude \( F \).

### Step-by-Step Solution:

1. **Identify the Forces and Resultant**:
Let the two forces be represented as \( \vec{A} \) and \( \vec{B} \), each with a magnitude \( F \). The resultant \( \vec{R} \) of these two forces is also given to have a magnitude \( F \).

2. **Use the Formula for Resultant of Two Vectors**:
The formula for the resultant \( R \) of two vectors \( A \) and \( B \) at an angle \( \theta \) is given by:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
Since both forces have the same magnitude \( F \):
\[ R = \sqrt{F^2 + F^2 + 2F \cdot F \cos \theta} \]

3. **Substitute the Known Values**:
Since \( R \) is also equal to \( F \):
\[ F = \sqrt{F^2 + F^2 + 2F^2 \cos \theta} \]
Simplifying this gives:
\[ F = \sqrt{2F^2 + 2F^2 \cos \theta} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ F^2 = 2F^2 + 2F^2 \cos \theta \]

5. **Rearrange the Equation**:
Rearranging the equation gives:
\[ F^2 - 2F^2 = 2F^2 \cos \theta \]
\[ -F^2 = 2F^2 \cos \theta \]

6. **Solve for \( \cos \theta \)**:
Dividing both sides by \( 2F^2 \):
\[ \cos \theta = -\frac{1}{2} \]

7. **Find the Angle \( \theta \)**:
The angle \( \theta \) for which \( \cos \theta = -\frac{1}{2} \) corresponds to:
\[ \theta = 120^\circ \quad \text{(in the range of } 0^\circ \text{ to } 180^\circ\text{)} \]

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

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