Question

Two solid spheres of radii \(10\) cm and \(5\) cm are in contact with each other. If the density of the material of the smaller sphere is twice the density of the material of the larger sphere, then the distance of the centre of mass of the system from the centre of the smaller sphere is

• A. \(6\) cm
• B. \(9\) cm
• C. \(12\) cm
• D. \(8\) cm

Hint:

For objects of different densities, \[ \boxed{\text{Mass}=\rho\times\text{Volume}.} \] Then use \[ \boxed{ x_{CM} = \frac{\sum mx}{\sum m} } \] to locate the centre of mass.

Answer 1

The Correct Option is C

Step 1: Find the mass ratio. Mass is proportional to \[ \text{density}\times\text{volume}. \] Let the density of the larger sphere be \[ \rho. \] Then the density of the smaller sphere is \[ 2\rho. \] Hence, \[ m_{\text{small}} \propto 2\rho\left(\frac43\pi5^3\right) = 250\rho, \] and \[ m_{\text{large}} \propto \rho\left(\frac43\pi10^3\right) = 1000\rho. \] Therefore, \[ m_{\text{small}}:m_{\text{large}} = 1:4. \]

Step 2: Find the distance between the centres. Since the spheres are in contact, \[ d=5+10=15\text{ cm}. \] Take the centre of the smaller sphere as the origin. Then the larger sphere is at \[ x=15\text{ cm}. \]

Step 3: Find the centre of mass. Using \[ x_{CM} = \frac{m_1x_1+m_2x_2}{m_1+m_2}, \] we get \[ x_{CM} = \frac{1(0)+4(15)}{1+4} = \frac{60}{5} = 12\text{ cm}. \] Hence, \[ \boxed{12\text{ cm}}. \] Thus, \[ \boxed{(C)} \] is the correct answer.