Question

From the uniform disc of radius $R$, an equilateral triangle of side $\sqrt{3}R$ is cut as shown in the figure. The new position of the centre of mass is :

• A. $(0,0)$
• B. $(0,R)$
• C. $(0,\frac{\sqrt{3}R}{2})$
• D. $Noneofthese$

Answer 1

The correct option is A $(0,0)$

Let $\sigma$ is mass per unit area.

Mass of disc $M_1=\sigma \pi R^2$ and mass of equilateral triangle $M_2=\sigma \frac{3\sqrt{3}}{4}{R}^2$,

Centre of mass (COM) of disc is at centre of the disc. COM of equilateral triangle is centroid of triangle.

COM in direction of $x$ axis , $x=\frac{M_1{x}_1+M_2{x}_2}{M_1+M_2}$

COM in direction of $y$ axis , $y=\frac{M_1{y}_1+M_2{y}_2}{M_1+M_2}$

A new position of COM from a center of the disc.

In direction $x$ axis, $x=\frac{M_1\times 0-M_2\times 0}{M_1+M_2}=0$

In direction $y$ axis, $y=\frac{M_1\times 0-M_2\times 0}{M_1+M_2}=0$\Hence, new COM $=(0,0)$