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 M.I. of Q about its axis is four times that of P about its axis (assuming the wire to be of diameter much smaller than either radius), the value of n is

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

Answer 1

Answer: (B)

$(4{)}^{1/3}$

Let the mass of ring P$=M$
$\therefore$ Mass of ring Q $=nM$ (As length of ring is n times that of ring P)
Moment of Inertia of ring P, $I_P=M{r}^2$
Moment of Inertia of ring Q, $I_Q=nM{\left(nr\right)}^2=n^{3}M{r}^2$
$I_Q=4I_P$
$n^{3}M{r}^2=4M{r}^{2}n=4^{\frac{1}{3}}$