Six forces are acting on a particle. Angle between two adjacent force is `60^(@)`. Five of the forces have magnitude `F_(1)` and the sixth has magnitude `F_(2)`. The resultant of all the forces will have magnitude of

Options

A. zero
B. `F_(1) + F_(2)`
C. `F_(1) - F_(2)`
D. `F_(2)`

Answer

To solve the problem, we need to analyze the forces acting on the particle step by step.

### Step 1: Understand the Forces
We have six forces acting on a particle:
- Five forces have a magnitude of \( F_1 \).
- One force has a magnitude of \( F_2 \).
- The angle between any two adjacent forces is \( 60^\circ \).

### Step 2: Visualize the Forces
We can visualize the forces as follows:
- The five forces \( F_1 \) are arranged in a circular pattern, each separated by an angle of \( 60^\circ \).
- The sixth force \( F_2 \) can be considered as acting in a specific direction (let's say along the x-axis for simplicity).

### Step 3: Analyze the Arrangement
Since the forces \( F_1 \) are evenly spaced at \( 60^\circ \), we can see that they will have symmetrical components. The resultant of these five forces can be calculated by considering their vector sum.

### Step 4: Calculate the Resultant of the Five Forces
The resultant \( R_1 \) of the five forces \( F_1 \) can be calculated using the formula for the resultant of forces at equal angles:
\[ R_1 = n \cdot F \cdot \cos\left(\frac{\theta}{2}\right) \]
where \( n \) is the number of forces, \( F \) is the magnitude of each force, and \( \theta \) is the angle between them.

In our case:
- \( n = 5 \)
- \( F = F_1 \)
- \( \theta = 60^\circ \)

Thus,
\[ R_1 = 5 \cdot F_1 \cdot \cos\left(30^\circ\right) = 5 \cdot F_1 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} F_1 \]

### Step 5: Consider the Sixth Force
Now, we have the resultant \( R_1 \) acting in one direction and the sixth force \( F_2 \) acting in the opposite direction.

### Step 6: Calculate the Final Resultant
The final resultant \( R \) of all the forces can be calculated as:
\[ R = R_1 - F_2 \]
Substituting the value of \( R_1 \):
\[ R = \frac{5\sqrt{3}}{2} F_1 - F_2 \]

### Conclusion
Thus, the magnitude of the resultant of all the forces acting on the particle is:
\[ R = \frac{5\sqrt{3}}{2} F_1 - F_2 \]