Four forces are acting on a particle. One forces of magnitude 3N is directed upward, another is directed `37^(@)` East of North having magnitude 5N third is directed in South-West direction is of magnitude `4 sqrt2N` and fourth force is `sqrt5n N`. If the partice in equilibrium. Find the value of n.

Answer

To solve the problem of finding the value of \( n \) such that the particle is in equilibrium under the influence of four forces, we will follow these steps:

### Step 1: Identify the Forces
1. **Force 1**: \( F_1 = 3 \, \text{N} \) (upward)
2. **Force 2**: \( F_2 = 5 \, \text{N} \) (directed \( 37^\circ \) East of North)
3. **Force 3**: \( F_3 = 4\sqrt{2} \, \text{N} \) (South-West direction)
4. **Force 4**: \( F_4 = \sqrt{5}n \, \text{N} \) (unknown direction)

### Step 2: Resolve Forces into Components
- **Force 2**:
- \( F_{2x} = 5 \sin(37^\circ) \) (East component)
- \( F_{2y} = 5 \cos(37^\circ) \) (North component)

Using the trigonometric values:
- \( \sin(37^\circ) = \frac{3}{5} \)
- \( \cos(37^\circ) = \frac{4}{5} \)

Calculating the components:
- \( F_{2x} = 5 \cdot \frac{3}{5} = 3 \, \text{N} \)
- \( F_{2y} = 5 \cdot \frac{4}{5} = 4 \, \text{N} \)

- **Force 3** (South-West direction):
- The angle is \( 45^\circ \) from both South and West.
- \( F_{3x} = -4\sqrt{2} \cdot \frac{1}{\sqrt{2}} = -4 \, \text{N} \) (West component)
- \( F_{3y} = -4\sqrt{2} \cdot \frac{1}{\sqrt{2}} = -4 \, \text{N} \) (South component)

### Step 3: Sum the Forces in Each Direction
- **Net Force in the x-direction**:
\[ F_{net_x} = F_{2x} + F_{3x} = 3 - 4 = -1 \, \text{N} \]

- **Net Force in the y-direction**:
\[ F_{net_y} = F_{1} + F_{2y} + F_{3y} = 3 + 4 - 4 = 3 \, \text{N} \]

### Step 4: Calculate the Resultant Force
The resultant force \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{(F_{net_x})^2 + (F_{net_y})^2} = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \, \text{N} \]

### Step 5: Set the Fourth Force Equal to the Resultant Force
For the particle to be in equilibrium, the fourth force must balance the resultant force:
\[ F_4 = \sqrt{5}n = \sqrt{10} \]

### Step 6: Solve for \( n \)
\[ n = \frac{\sqrt{10}}{\sqrt{5}} = \sqrt{\frac{10}{5}} = \sqrt{2} \]

### Conclusion
The value of \( n \) is \( \sqrt{2} \).

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