Question

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.

Answer 1

inside the square plate

Let m₁ be the mass of circular plate and m₂ 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.