A circular plate of diameter d is kept in contact with a square plate of edge d as show in figure. The density of the material and the thickness are same everywhere. The centre of mass of the composite system will be

#### Options

inside the circular plate

inside the square plate

at the point of contact

outside the system.

#### Solution

inside the square plate

Let m_{1} be the mass of circular plate and m_{2} be the mass of square plate.

The thickness of both the plates is t.

\[\text{ mass = density × volume}\]

\[ m_1 = \rho\pi \left( \frac{d}{2} \right)^2 t \]

\[ m_2 = \rho d^2 t\]

Centre of mass of the circular plate lies at its centre.

Let the centre of circular plate be the origin.

\[\vec{r}_1 = 0\]

Centre of mass of the square plate lies at its centre.

\[\vec{r}_2 = 2d\]

\[\text{Now,} \]

\[R = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2}\]

\[ = \frac{m_1 \times 0 + \rho d^2 t \times 2d}{\rho\pi \left( \frac{d}{2} \right)^2 t + \rho d^2 t}\]

\[ = \frac{2d}{\frac{\pi}{4} + 1} = 1 . 12d\]

\[ \Rightarrow \text{R > d}\]

\[\therefore\] Centre of mass of the system lies in the square plate.