Two homogeneous spheres `A and B` of masses `m and 2 m` having radii `2 a` and `a` respectively are placed in contact. The ratio of distance of c.m from first sphere to the distance of c.m from second sphere is :

Explanation

To solve the problem of finding the ratio of the distance of the center of mass (c.m.) from the first sphere (A) to the distance of the center of mass from the second sphere (B), we can follow these steps:

### Step 1: Identify the masses and positions of the spheres
- Let the mass of sphere A be \( m \) and its radius be \( 2a \).
- Let the mass of sphere B be \( 2m \) and its radius be \( a \).
- Place sphere A on the left and sphere B on the right, touching each other.

### Step 2: Define the positions of the centers of the spheres
- The center of sphere A can be taken as the origin, so its position \( x_1 = 0 \).
- The center of sphere B will be at a distance equal to the sum of their radii, which is \( 2a + a = 3a \). Thus, the position of sphere B is \( x_2 = 3a \).

### Step 3: Calculate the center of mass of the system
The formula for the center of mass \( x_{cm} \) of a system of particles is given by:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
Substituting the values:
- \( m_1 = m \), \( x_1 = 0 \)
- \( m_2 = 2m \), \( x_2 = 3a \)

Thus, we have:
\[ x_{cm} = \frac{m \cdot 0 + 2m \cdot 3a}{m + 2m} = \frac{6ma}{3m} = 2a \]

### Step 4: Calculate the distances from the center of mass to each sphere
- The distance from the center of mass \( x_{cm} = 2a \) to sphere A (at \( x_1 = 0 \)):
\[ d_A = x_{cm} - x_1 = 2a - 0 = 2a \]
- The distance from the center of mass \( x_{cm} = 2a \) to sphere B (at \( x_2 = 3a \)):
\[ d_B = x_2 - x_{cm} = 3a - 2a = a \]

### Step 5: Find the ratio of distances
Now we can find the ratio of the distance from sphere A to the distance from sphere B:
\[ \text{Ratio} = \frac{d_A}{d_B} = \frac{2a}{a} = 2 \]

### Final Answer
The ratio of the distance of the center of mass from the first sphere to the distance of the center of mass from the second sphere is \( 2 \).

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