Five equal forces each of `20N` are acting at a point in the same plane. If the angles between them are same, the resultant of these forces is

Options

A. `F=10N`
B. `F=5N`
C. `90^(@) lt alphalt180^(@)`
D. `180^(@) lt alphalt270^(@)`

Answer

To find the resultant of five equal forces, each of 20 N, acting at a point in the same plane with equal angles between them, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces and Angles**:
We have five equal forces, each of magnitude 20 N. The forces are acting at a point in the same plane, and the angles between them are equal. Since there are 5 forces, the total angle around the point is 360 degrees. Therefore, the angle between each pair of forces can be calculated as:
\[ \theta = \frac{360^\circ}{5} = 72^\circ \]

2. **Use the Formula for Resultant of Multiple Forces**:
When multiple forces of equal magnitude are acting at equal angles, the resultant can be calculated using the formula:
\[ R = F \cdot \frac{\sin(n \cdot \theta / 2)}{\sin(\theta / 2)} \]
where \( F \) is the magnitude of each force, \( n \) is the number of forces, and \( \theta \) is the angle between the forces.

3. **Substituting the Values**:
Here, \( F = 20 \, \text{N} \), \( n = 5 \), and \( \theta = 72^\circ \). Plugging these values into the formula:
\[ R = 20 \cdot \frac{\sin(5 \cdot 72^\circ / 2)}{\sin(72^\circ / 2)} \]
Simplifying the angles:
\[ R = 20 \cdot \frac{\sin(180^\circ)}{\sin(36^\circ)} \]

4. **Calculating the Sine Values**:
We know that \( \sin(180^\circ) = 0 \). Therefore:
\[ R = 20 \cdot \frac{0}{\sin(36^\circ)} = 0 \]

5. **Conclusion**:
The resultant of the five equal forces acting at equal angles is:
\[ R = 0 \, \text{N} \]