Two homogeneous spheres A and B of masses m and 2m having radii 2a and a respectively are placed in touch . The distance of the centre of mass from the first sphere is

Options

A. a
B. `2a`
C. `3a`
D. none of these

Answer

To find the distance of the center of mass from the first sphere (sphere A), 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 \( a \).
- The centers of the spheres are separated by the sum of their radii since they are in contact. Therefore, the distance between the centers of spheres A and B is \( 2a + a = 3a \).

### Step 2: Set up the coordinate system
- We can place the center of sphere A at the origin (0,0).
- The center of sphere B will then be at the position \( 3a \) along the x-axis.

### Step 3: Use the formula for the center of mass
The formula for the center of mass \( x_{cm} \) of two bodies is given by:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
Where:
- \( m_1 = m \) (mass of sphere A)
- \( x_1 = 0 \) (position of sphere A)
- \( m_2 = 2m \) (mass of sphere B)
- \( x_2 = 3a \) (position of sphere B)

### Step 4: Substitute the values into the formula
Substituting the values into the center of mass formula:
\[ x_{cm} = \frac{m \cdot 0 + 2m \cdot 3a}{m + 2m} \]
\[ x_{cm} = \frac{0 + 6ma}{3m} \]
\[ x_{cm} = \frac{6ma}{3m} = 2a \]

### Step 5: Conclusion
The distance of the center of mass from the first sphere (sphere A) is \( 2a \).

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