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. 0
B. `40N`
C. `20N`
D. `20sqrt2N`

Explanation

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

### Step 1: Determine the angle between the forces
Since there are five forces acting at a point and they are evenly distributed around a circle, we can calculate the angle between each force. The total angle around a point is 360 degrees. Therefore, the angle \( \theta \) between each force can be calculated as:
\[ \theta = \frac{360^\circ}{5} = 72^\circ \]

### Step 2: Use the formula for the resultant of forces
For \( n \) equal forces \( F \) acting at an angle \( \theta \) to each other, the resultant \( R \) can be calculated using the formula:
\[ R = nF \cdot \cos\left(\frac{\theta}{2}\right) \]
where \( n \) is the number of forces, \( F \) is the magnitude of each force, and \( \theta \) is the angle between them.

### Step 3: Substitute the values into the formula
Here, \( n = 5 \), \( F = 20 \, \text{N} \), and \( \theta = 72^\circ \). First, we need to calculate \( \frac{\theta}{2} \):
\[ \frac{\theta}{2} = \frac{72^\circ}{2} = 36^\circ \]
Now, we can substitute these values into the formula:
\[ R = 5 \cdot 20 \cdot \cos(36^\circ) \]

### Step 4: Calculate \( \cos(36^\circ) \)
Using a calculator or trigonometric tables, we find:
\[ \cos(36^\circ) \approx 0.809 \]

### Step 5: Calculate the resultant
Now, we can substitute \( \cos(36^\circ) \) back into the equation:
\[ R = 5 \cdot 20 \cdot 0.809 = 100.9 \, \text{N} \]

### Final Result
The resultant of the five equal forces of 20 N each acting at a point in the same plane with equal angles between them is approximately:
\[ R \approx 100.9 \, \text{N} \]

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