Question

From a uniform wire, two circular loops are made (i) $P$ of radius $r$ and (ii) $Q$ of radius $nr$. If the moment of inertia of $Q$ about an axis passing through its centre and perpendicular to its plane is $8$ times that of $P$ about a similar axis, the value of $n$ is (diameter of the wire is very much smaller than $r$ or $nr$)

• A. $8$
• B. $6$
• C. $4$
• D. $2$

Answer 1

The correct option is D $2$

Let mass per unit length of wire is $m$.

For wire P,

Length of wire $=2\pi r$

Mass of wire $=m_1=2\pi r\times m=2\pi rm$

Moment of Inertia $=I_1=\frac{1}{2}{m}_1{r}_1^2$

=> $I_1=\frac{1}{2}\times 2\pi rm\times r^2$

=> $I_1=\pi m{r}^3$

For wire Q,

Radius $=r_2=nr$

Mass of wire $=m_1=2\pi \times nr\times m=2\pi nrm$

Moment of Inertia $=I_2=\frac{1}{2}{m}_2{r}_2^2$

=> $I_2=\frac{1}{2}\times 2\pi nrm\times n^2{r}^2$

=> $I_2=\pi m{r}^3{n}^3$

Also $I_2=8I_1$

=> $n^3=8$

=> $n=2$