Question

A circular wire frame is made of two uniform semicircular wires of same radius and of mass per unit lengths $\lambda$ and $2\lambda$ as shown in figure. Which of the following options is/are true?

• A. $x_{cm}=0$
• B. $x_{cm}<0$
• C. $y_{cm}<0$
• D. $y_{cm}=0$

Answer 1

Answer: (A), (C)

The correct options are
A $x_{cm}=0$
C $y_{cm}<0$

Given, two semi circular wires of same radius and mass per unit lenths $\lambda$ and $2\lambda$
$\therefore$ Masses of rings are $m$ and $2m$ respectively.
Position of COM of semicircular ring of density $\lambda$
$C_1=(x_1,y_1)=(0,\frac{2R}{\pi })$
Position of COM of semicircular ring of density $2\lambda$
$C_2=(x_2,y_2)=(0,\frac{-2R}{\pi })$


To find COM of system
$x_{cm}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}=\frac{m\left(0\right)+2m\left(0\right)}{3m}=0$
$y_{cm}=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}=\frac{m\left(\frac{2R}{\pi }\right)+2m\left(\frac{-2R}{\pi }\right)}{3m}=\frac{-2R}{3\pi }$
$\Rightarrow (x_{cm},y_{cm})=(0,\frac{-2R}{3\pi })\text{m}$
From this we can say that , $x_{cm}=0$ and $y_{cm}<0$
Thus, options (a) and (c) are correct answers.