Question 20.10: A star is modelled as a sphere of radius a whose density is ...
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.
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).