Two semicircular rings of linear mass densities `lamda` and `2 lamda` and of radius 'R' each are joined to form a complete ring. The distance of the centre of the mass of complete ring from its geometrical centre is :

Options

A. `(3R)/(8pi)`
B. `(2R)/(3pi)`
C. `(3R)/(4pi)`
D. none of these

Explanation

To find the distance of the center of mass of the complete ring formed by two semicircular rings with different linear mass densities, we can follow these steps:

### Step 1: Define the Problem
We have two semicircular rings:
- The first semicircular ring has a linear mass density of \( \lambda \) and radius \( R \).
- The second semicircular ring has a linear mass density of \( 2\lambda \) and the same radius \( R \).

### Step 2: Calculate the Mass of Each Semicircular Ring
The mass \( m \) of a semicircular ring can be calculated using the formula:
\[ m = \text{linear mass density} \times \text{length} \]
The length of a semicircular ring is half the circumference of a full circle:
\[ \text{Length} = \frac{1}{2} \times 2\pi R = \pi R \]
Thus, the masses of the two semicircular rings are:
- For the first ring:
\[ m_1 = \lambda \times \pi R = \lambda \pi R \]
- For the second ring:
\[ m_2 = 2\lambda \times \pi R = 2\lambda \pi R \]

### Step 3: Determine the Center of Mass of Each Semicircular Ring
The center of mass for each semicircular ring can be found as follows:
- The center of mass of the first semicircular ring (upper part) is located at:
\[ (0, \frac{2R}{\pi}) \]
- The center of mass of the second semicircular ring (lower part) is located at:
\[ (0, -\frac{2R}{\pi}) \]

### Step 4: Calculate the Overall Center of Mass of the Complete Ring
The overall center of mass \( (X_{cm}, Y_{cm}) \) of the complete ring can be calculated using the formula:
\[ Y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \]
Substituting the values:
\[ Y_{cm} = \frac{(\lambda \pi R) \left(\frac{2R}{\pi}\right) + (2\lambda \pi R) \left(-\frac{2R}{\pi}\right)}{\lambda \pi R + 2\lambda \pi R} \]
Simplifying this:
\[ Y_{cm} = \frac{\lambda \pi R \cdot \frac{2R}{\pi} - 2\lambda \pi R \cdot \frac{2R}{\pi}}{3\lambda \pi R} \]
\[ Y_{cm} = \frac{2\lambda R^2 - 4\lambda R^2}{3\lambda \pi R} \]
\[ Y_{cm} = \frac{-2\lambda R^2}{3\lambda \pi R} \]
\[ Y_{cm} = -\frac{2R}{3\pi} \]

### Step 5: Conclusion
The distance of the center of mass of the complete ring from its geometrical center is:
\[ \frac{2R}{3\pi} \]
This indicates that the center of mass is located along the negative y-axis.