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 (9-Q2).Locate the centre of mass of the combination,assuming same mass per unit area for the two plates.

Answer 1

Let m be the mass per unit area.

$\therefore$ Mass of the square plate=$M_1=d^{2}m$

Mass of the circular disc=$M_2=\frac{\pi d^2}{4}m$

Let the centre of the circular disc be the origin of the system,

$\therefore x_1=d,y_1=0$

$x_2=d,y_2=0$

$\therefore$ Position of centre of mass=$(\frac{d^{2}md+\pi \left(\frac{d^2}{4}\right)m\times 0}{d^{2}m+\pi \left(\frac{d^2}{4}\right)m},\frac{0+0}{m_1+m_2})$

=$(\frac{d^{3}m}{d^{2}m(1+\frac{\pi }{4})},0)$

=$(\frac{4d}{\pi +4},0)$

The new centre of mass is $\frac{4d}{\pi +4}$ right to the centre of circular disc.