Question

An object comprises of a uniform ring of radius $R$ and its uniform chord $AB$ (not necessarily made of the same material) as shown. Which of the following cannot be the centre of mass of the object?

• A. $\left(\frac{R}{3} , \frac{R}{3}\right)$
• B. $\left(\frac{R}{\sqrt{2}} , \frac{R}{\sqrt{2}}\right)$
• C. $\left(\frac{R}{4} , \frac{R}{4}\right)$
• D. none of these

Answer 1

Answer: (B)

From figure $\left(\frac{R}{\sqrt{2}} , \, \frac{R}{\sqrt{2}}\right)$ is the coordinate of the rod only in absence of the ring,
When the ring is also introduced then $r_{c m}=\frac{m_{c h o r d} \, r_{c h o r d} + m_{r i n g} \, r_{r i n g}}{m_{c h o r d} + m_{r i n g}}$
Hence coordinate of the center of mass of system $r_{c m}=\frac{m_{c h o r d} \, r_{c h o r d}}{m_{c h o r d} + m_{r i n g}}$
Hence the value of coordinate will decrease and will not be $\left(\frac{R}{\sqrt{2}} , \, \frac{R}{\sqrt{2}}\right)$ , hence the answer is B