Two forces F, F have an angle `alpha` , between them , their resultant has magnitude of

Answer

To find the magnitude of the resultant of two forces \( F \) and \( F \) that have an angle \( \alpha \) between them, we can use the law of cosines. Here’s a step-by-step solution:

### Step 1: Identify the Forces and Angle
We have two forces:
- Force 1: \( F \)
- Force 2: \( F \)
- Angle between them: \( \alpha \)

### Step 2: Apply the Law of Cosines
The magnitude of the resultant \( R \) of two vectors can be calculated using the formula:
\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(\alpha)} \]

### Step 3: Substitute the Values
Since both forces are equal in magnitude, we can substitute \( F_1 = F \) and \( F_2 = F \):
\[ R = \sqrt{F^2 + F^2 + 2F \cdot F \cos(\alpha)} \]

### Step 4: Simplify the Expression
Combine the terms:
\[ R = \sqrt{F^2 + F^2 + 2F^2 \cos(\alpha)} = \sqrt{2F^2 + 2F^2 \cos(\alpha)} \]

### Step 5: Factor Out Common Terms
Factor out \( 2F^2 \):
\[ R = \sqrt{2F^2(1 + \cos(\alpha))} \]

### Step 6: Further Simplify
Taking the square root of the factored expression gives:
\[ R = \sqrt{2}F \sqrt{1 + \cos(\alpha)} \]

### Final Result
Thus, the magnitude of the resultant force is:
\[ R = \sqrt{2}F \sqrt{1 + \cos(\alpha)} \]

---