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 the diameter of wire is much smaller than either radii), find the value of $n$.

Answer 1

${\sigma }_1=r:{\sigma }_2=n\sigma$

Mass of wire per unit length $=d$

Mass of $P=m_1=2\pi r_{1}d=2\pi rd$

Mass of $Q=m_2=2\pi r_{2}d=2\pi nrd$

Now $I_1=\left(2\pi rd\right)r^2=2\pi d{r}^3$

$I_2=\left(2\pi nrd\right){\left(nr\right)}^2=2\pi d{n}^3{r}^3$

Given $I_2=4I$,

$n^3=4$

$n=\sqrt[3]{34}$ or $n={\left(4\right)}^{\frac{1}{3}}$