Question

A circular wire frame is made of four uniform quarter circular wires of same radius and of mass per unit length $\lambda ,2\lambda ,3\lambda$ and $4\lambda$ as shown in figure. Assuming the origin to be at the centre of the circle, which of the following options is (are) correct?

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

Answer 1

Answer: (A), (C)

$y_{cm}<0$

Given, four uniform quarter circular rings of same radius and mass per unit length $\lambda ,2\lambda ,3\lambda$ and $4\lambda$.
If mass of ring with density $\lambda =m$, then
Mass of ring with density $2\lambda =2m$
Mass of ring with density $3\lambda =3m$
Mass of ring with density $4\lambda =4m$
$COM$ of quarter ring of mass $m=(\frac{2R}{\pi },\frac{2R}{\pi })$
$COM$ of quarter ring of mass $2m=(\frac{-2R}{\pi },\frac{2R}{\pi })$
$COM$ of quarter ring of mass $3m=(\frac{-2R}{\pi },\frac{-2R}{\pi })$
$COM$ of quarter ring of mass $4m=(\frac{2R}{\pi },\frac{-2R}{\pi })$

To find COM of system
$x_{cm}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3+m_4{x}_4}{m_1+m_2+m_3+m_4}$
$=\frac{m\left(\frac{2R}{\pi }\right)+2m\left(\frac{-2R}{\pi }\right)+3m\left(\frac{-2R}{\pi }\right)+4m\left(\frac{2R}{\pi }\right)}{10m}$
$=\frac{\frac{2R}{\pi }(m-2m-3m+4m)}{10m}$
$=0$
$y_{cm}=\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3+m_4{y}_4}{m_1+m_2+m_3+m_4}$
$=\frac{m\left(\frac{2R}{\pi }\right)+2m\left(\frac{2R}{\pi }\right)+3m\left(\frac{-2R}{\pi }\right)+4m\left(\frac{-2R}{\pi }\right)}{10m}$
$=\frac{\frac{2R}{\pi }(m+2m-3m-4m)}{10m}$
$=\frac{-4R}{5\pi }<0$
Option (a) and (c) are correct.