Three forces `20 sqrt(2)N,20 sqrt(2) N` and `40N` are acting along `X,Y` and `Z-`axes respectively on a `5sqrt2 kg` mass at rest at the origin. The magnitude of its displacement after `5s` is .

Options

A. `50m`
B. `25m`
C. `60m`
D. `100m`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the forces acting on the mass
The forces acting on the mass are:
- \( F_x = 20\sqrt{2} \, \text{N} \) along the X-axis
- \( F_y = 20\sqrt{2} \, \text{N} \) along the Y-axis
- \( F_z = 40 \, \text{N} \) along the Z-axis

### Step 2: Calculate the resultant force in the XY-plane
Since the forces along the X and Y axes are perpendicular, we can calculate the resultant force \( F_{xy} \) using the Pythagorean theorem:
\[ F_{xy} = \sqrt{F_x^2 + F_y^2} = \sqrt{(20\sqrt{2})^2 + (20\sqrt{2})^2} \]
Calculating this:
\[ F_{xy} = \sqrt{(400 \cdot 2) + (400 \cdot 2)} = \sqrt{800 + 800} = \sqrt{1600} = 40 \, \text{N} \]

### Step 3: Calculate the net resultant force in 3D
Now, we need to find the resultant force \( F_{net} \) considering the force along the Z-axis:
\[ F_{net} = \sqrt{F_{xy}^2 + F_z^2} = \sqrt{(40)^2 + (40)^2} \]
Calculating this:
\[ F_{net} = \sqrt{1600 + 1600} = \sqrt{3200} = 40\sqrt{2} \, \text{N} \]

### Step 4: Calculate the acceleration
Using Newton's second law, \( F = ma \), we can find the acceleration \( a \):
\[ a = \frac{F_{net}}{m} = \frac{40\sqrt{2}}{5\sqrt{2}} = \frac{40}{5} = 8 \, \text{m/s}^2 \]

### Step 5: Calculate the displacement after 5 seconds
Since the mass is initially at rest, we can use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( u = 0 \) (initial velocity)
- \( a = 8 \, \text{m/s}^2 \)
- \( t = 5 \, \text{s} \)

Substituting the values:
\[ s = 0 \cdot 5 + \frac{1}{2} \cdot 8 \cdot (5)^2 = 0 + \frac{1}{2} \cdot 8 \cdot 25 = 4 \cdot 25 = 100 \, \text{m} \]

### Final Answer
The magnitude of the displacement after 5 seconds is \( \boxed{100 \, \text{m}} \).

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