Two forces each of magnitude 2N, act at an angle of `60^(@)`. The magnitude of the resultant force

Options

A. `sqrt(1.1)` Newton
B. `sqrt(4)` Newton
C. `sqrt(12)` Newton
D. `sqrt(14.9)` Newton

Answer

To find the magnitude of the resultant force when two forces of equal magnitude act at an angle, we can use the formula for the resultant of two vectors. Here's the step-by-step solution:

### Step 1: Identify the given values
- Magnitude of the first force, \( F_1 = 2 \, \text{N} \)
- Magnitude of the second force, \( F_2 = 2 \, \text{N} \)
- Angle between the two forces, \( \theta = 60^\circ \)

### Step 2: Write the formula for the resultant force
The magnitude of the resultant force \( F_{\text{net}} \) can be calculated using the formula:
\[ F_{\text{net}} = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)} \]

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ F_{\text{net}} = \sqrt{(2)^2 + (2)^2 + 2 \cdot 2 \cdot 2 \cdot \cos(60^\circ)} \]

### Step 4: Calculate each term
1. Calculate \( F_1^2 \) and \( F_2^2 \):
\[ F_1^2 = 2^2 = 4 \]
\[ F_2^2 = 2^2 = 4 \]

2. Calculate \( 2 F_1 F_2 \):
\[ 2 \cdot 2 \cdot 2 = 8 \]

3. Calculate \( \cos(60^\circ) \):
\[ \cos(60^\circ) = \frac{1}{2} \]

4. Substitute \( \cos(60^\circ) \) back into the equation:
\[ 8 \cdot \cos(60^\circ) = 8 \cdot \frac{1}{2} = 4 \]

### Step 5: Combine all terms
Now substitute back into the resultant force equation:
\[ F_{\text{net}} = \sqrt{4 + 4 + 4} = \sqrt{12} \]

### Step 6: Calculate the final result
\[ F_{\text{net}} = \sqrt{12} = 2\sqrt{3} \, \text{N} \]

### Final Answer
The magnitude of the resultant force is \( 2\sqrt{3} \, \text{N} \).

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