Question

From a given sample of uniform wire, two circular loops $P$ and $Q$ are made, $P$ of radius $r$ and $Q$ of radius $nr$. If the moment of inertia of $Q$ about its axis is $4$ times that of $P$ about its axis (assuming wire diameter is much smaller than either raidii) the value of $n$ is:

• A. $(4{)}^{2/3}$
• B. $(4{)}^{1/3}$
• C. $(4{)}^{1/2}$
• D. $(4{)}^{1/4}$

Answer 1

The correct option is B $(4{)}^{1/3}$

Moment of inertia of a circular loop about an axis perpendicular to plane of loop and through centre$=\mu R^2$

$I_1=\left(2\pi rd\right)r^2-2\pi d{r}^3{I}_2=\left(2\pi nrd\right)\left({nr}^2\right)=2\pi d{n}^3{r}^3$

Given

$I_2=4I_1\therefore 2\pi d{n}^3{r}^3=4\times 2\pi d{r}^3\therefore n^3=4n=4^{\frac{1}{3}}$