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Q. 20.10

A star is modelled as a sphere of radius a whose density is given in terms of the distance
r from the centre by the function $r = r_{0}\left(1 – \frac{r^{2}}{a^{2}}\right).$
i) Given that $ρ_{0}$ is constant, find the total mass M of the sphere.
ii) Find its moment of inertia about a diameter.

Verified Solution

Imagine that the sphere is divided into elementary shells and a typical shell has radius r and thickness δr.
i)     The mass of the shell is

$4πr²ρδr = \frac{4πρ_{0}}{a²}(a²r² – r^{4}) δr$              ①

⇒         $M = \int_{0}^{a}{\frac{4πρ_{0}}{a²}(a²r² – r^{4}) dr}$

$= \frac{4πρ_{0}}{a²}\left[\frac{a²r³}{3} – \frac{r^{5}}{5}\right]^{a}_{0}$

$= \frac{8πρ_{0}}{15}a^{3}.$

ii) The moment of inertia of a hollow shell of mass m about a diameter is $\frac{2}{3}mr²$, so the moment of inertia of an elementary shell is

$\frac{2}{3} × \frac{4πρ_{0}}{a²} (a²r² – r^{4}) δr × r²$   (from ①)

⇒        moment of inertia of sphere = $\int_{0}^{a}{\frac{8πρ_{0}}{3a^{2}} (a²r^{4} – r^{6})dr}$

$= \frac{8πρ_{0}}{3a^{2}} \left[a²\frac{r^{5}}{5} – \frac{r^{7}}{7}\right]^{a}_{0}$

$= \frac{16πρ_{0}}{105}a^{5}$

$= \frac{2}{7}Ma^{2}$         (from ②).