Two forces of magnitudes 3N and 4N respectively are acting on a body. Calculate the resultant force if the angle between them is : (i) `0^(@)` (ii) `180^(@)` (iii) `90^(@)`

Options

A. `1`
B. `5`
C. `3`
D. `4`

Explanation

To solve the problem of finding the resultant force when two forces of magnitudes 3N and 4N are acting on a body at different angles, we will use the formula for the resultant of two forces:

\[ F_{net} = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)} \]

where:
- \( F_1 \) and \( F_2 \) are the magnitudes of the two forces,
- \( \theta \) is the angle between the two forces.

Let's calculate the resultant force for each angle.

### (i) When the angle \( \theta = 0^\circ \)

1. **Substitute the values into the formula:**
\[ F_{net} = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cdot \cos(0)} \]

2. **Calculate \( \cos(0) \):**
\[ \cos(0) = 1 \]

3. **Now substitute \( \cos(0) \) into the equation:**
\[ F_{net} = \sqrt{9 + 16 + 2 \cdot 3 \cdot 4 \cdot 1} \]

4. **Calculate the values:**
\[ F_{net} = \sqrt{9 + 16 + 24} = \sqrt{49} \]

5. **Final calculation:**
\[ F_{net} = 7 \, \text{N} \]

### (ii) When the angle \( \theta = 180^\circ \)

1. **Substitute the values into the formula:**
\[ F_{net} = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cdot \cos(180)} \]

2. **Calculate \( \cos(180) \):**
\[ \cos(180) = -1 \]

3. **Now substitute \( \cos(180) \) into the equation:**
\[ F_{net} = \sqrt{9 + 16 + 2 \cdot 3 \cdot 4 \cdot (-1)} \]

4. **Calculate the values:**
\[ F_{net} = \sqrt{9 + 16 - 24} = \sqrt{1} \]

5. **Final calculation:**
\[ F_{net} = 1 \, \text{N} \]

### (iii) When the angle \( \theta = 90^\circ \)

1. **Substitute the values into the formula:**
\[ F_{net} = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cdot \cos(90)} \]

2. **Calculate \( \cos(90) \):**
\[ \cos(90) = 0 \]

3. **Now substitute \( \cos(90) \) into the equation:**
\[ F_{net} = \sqrt{9 + 16 + 2 \cdot 3 \cdot 4 \cdot 0} \]

4. **Calculate the values:**
\[ F_{net} = \sqrt{9 + 16} = \sqrt{25} \]

5. **Final calculation:**
\[ F_{net} = 5 \, \text{N} \]

### Summary of Results:
- For \( \theta = 0^\circ \), \( F_{net} = 7 \, \text{N} \)
- For \( \theta = 180^\circ \), \( F_{net} = 1 \, \text{N} \)
- For \( \theta = 90^\circ \), \( F_{net} = 5 \, \text{N} \)