The densities of two solid spheres A and B of the same radius R vary with radial distance as and respectively, where is a constant. The moments of inertia of the individual spheres about axes passing through their centres are and , respectively. If

Question

The densities of two solid spheres A and B of the same radius R vary with radial distance $r$ as $ρ_{A} (r) = k (\frac{r}{R})$ and $ρ_{B} (r) = k {(\frac{r}{R})}^{5},$ respectively, where $k$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $I_{A}$ and $I_{B}$ , respectively. If $\frac{I_{B}}{I_{A}} = \frac{n}{10},$ the value of $n$ is _______. [2015]

Smart Achievers · Accepted Answer

(6)

For solid sphere A, density

$ρ_{A} (r) = k (\frac{r}{R})$

$And ρ_{B} (r) = k {(\frac{r}{R})}^{s}$

Consider a spherical shell of radius $x$ and thickness $d x$ .

$\frac{I_{B}}{I_{A}} = \frac{n}{10},$ $= (k \frac{x}{R}) (4 π x^{2} d x)$

So, moment of inertia of shell about its diameter,

$d I = \frac{2}{3} (d m) x^{2} = \frac{2}{3} (k \frac{x}{R}) (4 π x^{2} d x) x^{2} = (\frac{8 π k}{3 R}) x^{5} d x$

$∴$ $Moment of inertia of the sphere A, I_{A} = \int_{0}^{R} d I$

$= \frac{8 π k}{3 R} \int_{0}^{R} x^{5} d x = \frac{8 π k}{3 R} {[\frac{x^{6}}{6}]}_{0}^{R}$

$∴$ $I_{A} = (\frac{8 π k}{18}) R^{5} (i)$

$Similarly, for sphere B$

$I_{B} = \frac{8 π k}{3 R^{5}} \int_{0}^{R} x^{9} d x = (\frac{8 π k}{3 R^{5}}) {[\frac{x^{10}}{10}]}_{0}^{R}$

$∴$ $I_{B} = \frac{8 π k}{30} R^{5}$

$From eqns. (i) and (ii)$

$\frac{I_{B}}{I_{A}} = \frac{18}{30} = \frac{6}{10} = \frac{n}{10}$ $∴$ $n = 6$

Solution

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