In a system of particles `8 kg` mass is subjected to a force of `16 N` along + ve `x`-axis and another `8 kg` mass is subjected to a force of `8 N` along + ve ` y`-axis. The magnitude of acceleration of centre of mass and the angle made by it with `x`-axis are given, respectively, by

Options

A. `(sqrt(5))/2ms^(2),theta=45^(@)`
B. `3sqrt(5)ms^(2),theta=tan^(-1)(2/3)`
C. `(sqrt(5))/2ms^(2),theta=tan^(-1)(1/2)`
D. `1ms^(2),theta=tan^(-1)sqrt(3)`

Explanation

To solve the problem, we need to determine the acceleration of the center of mass of the system of particles and the angle it makes with the x-axis. Here’s the step-by-step solution:

### Step 1: Identify the Forces and Masses
We have two masses:
- Mass \( m_1 = 8 \, \text{kg} \) subjected to a force \( F_1 = 16 \, \text{N} \) along the positive x-axis.
- Mass \( m_2 = 8 \, \text{kg} \) subjected to a force \( F_2 = 8 \, \text{N} \) along the positive y-axis.

### Step 2: Calculate the Acceleration of Each Mass
Using Newton's second law \( F = ma \), we can find the acceleration for each mass.

For mass \( m_1 \):
\[ a_1 = \frac{F_1}{m_1} = \frac{16 \, \text{N}}{8 \, \text{kg}} = 2 \, \text{m/s}^2 \quad \text{(along x-axis)} \]

For mass \( m_2 \):
\[ a_2 = \frac{F_2}{m_2} = \frac{8 \, \text{N}}{8 \, \text{kg}} = 1 \, \text{m/s}^2 \quad \text{(along y-axis)} \]

### Step 3: Calculate the Total Mass of the System
The total mass \( M \) of the system is:
\[ M = m_1 + m_2 = 8 \, \text{kg} + 8 \, \text{kg} = 16 \, \text{kg} \]

### Step 4: Calculate the Acceleration of the Center of Mass
The acceleration of the center of mass \( a_{cm} \) can be calculated using the formula:
\[ a_{cm} = \frac{m_1 a_1 + m_2 a_2}{M} \]
Substituting the values:
\[ a_{cm} = \frac{8 \cdot 2 + 8 \cdot 1}{16} = \frac{16 + 8}{16} = \frac{24}{16} = \frac{3}{2} \, \text{m/s}^2 \]

### Step 5: Calculate the Components of the Acceleration
The x-component of the acceleration is:
\[ a_{cm,x} = \frac{m_1 a_1}{M} = \frac{8 \cdot 2}{16} = 1 \, \text{m/s}^2 \]

The y-component of the acceleration is:
\[ a_{cm,y} = \frac{m_2 a_2}{M} = \frac{8 \cdot 1}{16} = 0.5 \, \text{m/s}^2 \]

### Step 6: Calculate the Magnitude of the Acceleration of the Center of Mass
Using the Pythagorean theorem:
\[ |a_{cm}| = \sqrt{(a_{cm,x})^2 + (a_{cm,y})^2} = \sqrt{(1)^2 + (0.5)^2} = \sqrt{1 + 0.25} = \sqrt{1.25} = \frac{\sqrt{5}}{2} \, \text{m/s}^2 \]

### Step 7: Calculate the Angle Made with the x-axis
The angle \( \theta \) can be calculated using the tangent function:
\[ \tan \theta = \frac{a_{cm,y}}{a_{cm,x}} = \frac{0.5}{1} = 0.5 \]
Thus,
\[ \theta = \tan^{-1}(0.5) \]

### Final Answer
The magnitude of the acceleration of the center of mass is \( \frac{\sqrt{5}}{2} \, \text{m/s}^2 \) and the angle made with the x-axis is \( \tan^{-1}(0.5) \).