Two equal forces acting at `60^(@)` to each other. Find the forces if resultant force is 300 N.

Options

A. 155.3N
B. 173.2 N
C. 162.4 N
D. None of these

Explanation

To solve the problem of finding the equal forces acting at an angle of 60 degrees to each other, given that the resultant force is 300 N, we can follow these steps:

### Step 1: Understand the Problem
We have two equal forces, \( F_1 \) and \( F_2 \), acting at an angle of \( 60^\circ \) to each other. The resultant force \( R \) is given as 300 N.

### Step 2: Use the Formula for Resultant Force
The formula for the resultant force when two forces are acting at an angle is given by:
\[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\alpha)} \]
Where:
- \( R \) is the resultant force,
- \( F_1 \) and \( F_2 \) are the magnitudes of the two forces,
- \( \alpha \) is the angle between the two forces.

### Step 3: Substitute Known Values
Since \( F_1 = F_2 = F \) (the forces are equal), and \( \alpha = 60^\circ \), we can rewrite the equation as:
\[ R = \sqrt{F^2 + F^2 + 2 F F \cos(60^\circ)} \]
This simplifies to:
\[ R = \sqrt{2F^2 + 2F^2 \cos(60^\circ)} \]

### Step 4: Calculate \( \cos(60^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
Substituting this value into the equation gives:
\[ R = \sqrt{2F^2 + 2F^2 \cdot \frac{1}{2}} \]
This simplifies further to:
\[ R = \sqrt{2F^2 + F^2} = \sqrt{3F^2} \]

### Step 5: Set Up the Equation
Now we can set the resultant force equal to 300 N:
\[ 300 = \sqrt{3F^2} \]

### Step 6: Square Both Sides
To eliminate the square root, we square both sides:
\[ 300^2 = 3F^2 \]
This gives:
\[ 90000 = 3F^2 \]

### Step 7: Solve for \( F^2 \)
Now, divide both sides by 3:
\[ F^2 = \frac{90000}{3} = 30000 \]

### Step 8: Find \( F \)
Taking the square root of both sides gives:
\[ F = \sqrt{30000} = \sqrt{3 \times 10000} = \sqrt{3} \times 100 \]
Calculating this gives:
\[ F \approx 173.2 \text{ N} \]

### Final Answer
The magnitude of each force \( F \) is approximately **173.2 N**.

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