Two force vectors each having magnitude 6N are oriented as `30^(@)` from positive of x-axis and other at `90^(@)` with the same axis. Find out their magnitude along with position from positive of x-axis :-

Options

A. `6sqrt(3)N,30^(@)`
B. `6N, 60^(@)`
C. `6sqrt(2)N,60^(@)`
D. `6sqrt(3)N,60^(@)`

Answer

To solve the problem of finding the resultant of two force vectors, we will follow these steps:

### Step 1: Identify the forces and their angles
We have two forces:
- \( F_1 = 6 \, \text{N} \) at an angle of \( 30^\circ \) from the positive x-axis.
- \( F_2 = 6 \, \text{N} \) at an angle of \( 90^\circ \) from the positive x-axis.

### Step 2: Resolve the forces into their components
We can resolve each force into its x and y components.

For \( F_1 \):
- \( F_{1x} = F_1 \cdot \cos(30^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \, \text{N} \)
- \( F_{1y} = F_1 \cdot \sin(30^\circ) = 6 \cdot \frac{1}{2} = 3 \, \text{N} \)

For \( F_2 \):
- \( F_{2x} = F_2 \cdot \cos(90^\circ) = 6 \cdot 0 = 0 \, \text{N} \)
- \( F_{2y} = F_2 \cdot \sin(90^\circ) = 6 \cdot 1 = 6 \, \text{N} \)

### Step 3: Calculate the resultant components
Now, we can find the resultant components \( R_x \) and \( R_y \):
- \( R_x = F_{1x} + F_{2x} = 3\sqrt{3} + 0 = 3\sqrt{3} \, \text{N} \)
- \( R_y = F_{1y} + F_{2y} = 3 + 6 = 9 \, \text{N} \)

### Step 4: Calculate the magnitude of the resultant force
The magnitude of the resultant force \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} \]
Substituting the values:
\[ R = \sqrt{(3\sqrt{3})^2 + (9)^2} = \sqrt{27 + 81} = \sqrt{108} = 6\sqrt{3} \, \text{N} \]

### Step 5: Calculate the angle of the resultant force
The angle \( \theta \) that the resultant makes with the positive x-axis can be found using the tangent function:
\[ \tan(\theta) = \frac{R_y}{R_x} \]
Substituting the values:
\[ \tan(\theta) = \frac{9}{3\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]
Thus,
\[ \theta = 60^\circ \]

### Final Result
The magnitude of the resultant force is \( 6\sqrt{3} \, \text{N} \) and it makes an angle of \( 60^\circ \) with the positive x-axis.

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