Forces of 4 N and 5 N are applied at origin along X-axis and Y-axis respectively. The resultant force will be

Options

A. `sqrt41N, tan^(-1) (5/4)`
B. `sqrt41N, tan^(-1) (4/5)`
C. `-sqrt41N, tan^(-1) (5/4)`
D. `-sqrt41N , tan^(-1) (4/5)`

Explanation

To solve the problem of finding the resultant force when forces of 4 N and 5 N are applied at the origin along the X-axis and Y-axis respectively, we can follow these steps:

### Step 1: Understand the Forces
We have two forces:
- \( F_1 = 4 \, \text{N} \) acting along the X-axis.
- \( F_2 = 5 \, \text{N} \) acting along the Y-axis.

### Step 2: Represent the Forces Graphically
Draw a coordinate system with the X-axis and Y-axis.
- Draw a vector of 4 N along the X-axis.
- Draw a vector of 5 N along the Y-axis.

### Step 3: Use the Pythagorean Theorem to Find the Magnitude of the Resultant Force
Since the forces are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of the resultant force \( R \):
\[ R = \sqrt{F_1^2 + F_2^2} \]
Substituting the values:
\[ R = \sqrt{(4 \, \text{N})^2 + (5 \, \text{N})^2} = \sqrt{16 + 25} = \sqrt{41} \, \text{N} \]

### Step 4: Find the Direction of the Resultant Force
To find the angle \( \theta \) that the resultant makes with the X-axis, we can use the tangent function:
\[ \tan \theta = \frac{F_2}{F_1} = \frac{5 \, \text{N}}{4 \, \text{N}} \]
Thus,
\[ \theta = \tan^{-1}\left(\frac{5}{4}\right) \]

### Step 5: Conclusion
The magnitude of the resultant force is \( \sqrt{41} \, \text{N} \) and the angle \( \theta \) is \( \tan^{-1}\left(\frac{5}{4}\right) \).

### Final Answer
The resultant force is \( \sqrt{41} \, \text{N} \) at an angle of \( \tan^{-1}\left(\frac{5}{4}\right) \) with respect to the X-axis.

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