Question

A square plate of edge d and a circular disc of diameter d are placed touching each other at the midpoint of an edge of the plate as shown in figure. Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.

figure

Answer 1

Let m be the mass per unit area of the square plate and the circular disc.



$\Rightarrow$Mass of the square plate, M₁ = d²m
Mass of the circular disc, M₂ = $\frac{\mathrm{\pi }{d}^{\mathit{2}}}{4}m$

Let the centre of the circular disc be the origin of the system.
$\Rightarrow$ x₁ = d, y₁ = 0
x₂ = 0, y₂ = 0

$\Rightarrow$ Position of the centre of mass of circular disc and square plate:
$$\begin{gathered} =\left(\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2},\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}\right) \\ =\left(\frac{d^{2}md+\mathrm{\pi }\left(d^2/4\right)m\times 0}{d^{2}m+\mathrm{\pi }\left(d^{\mathit{2}}\mathit{/}\mathit{4}\right)m},\frac{0+0}{m_1+m_2}\right) \\ =\left(\frac{d^{3}m}{d^{2}m(1+\mathrm{\pi }/4)},0\right) \\ =\left(\frac{4d}{\mathrm{\pi }+4},0\right) \end{gathered}$$
Hence, the new centre of mass of the system (circular disc plus square plate) lies at distance $\frac{4d}{(\mathrm{\pi }+4)}$ from the centre of circular disc, towards right.