Labda (thin spherical shell) made of a material with high density. What is its moment of inertia about an axis passing through its center?

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Labda (thin spherical shell) made of a material with high density. What is its moment of inertia about an axis passing through its center?
I. Solution. We consider the ball as a spherical shell whose thickness can be neglected and whose surface mass per unit area is $\mu$ (Fig. 1).

![](https://cdn.mathpix.com/cropped/2024_05_02_a139e0ed59c16ed03e6ag-1.jpg?height=385&width=442&top_left_y=257&top_left_x=825)

Fig. 1

The surface area of the spherical shell corresponding to $\Delta r$ is

$$
\Delta A=2 \pi R \Delta r
$$

its mass is

$$
\Delta m=2 \pi R \Delta r \mu
$$

and its moment of inertia about the $r$-axis for small $\Delta r$ is

$$
\Delta \Theta \approx \Delta m l^{2}=2 \pi R \Delta r \mu\left(R^{2}-r^{2}\right)
$$

Thus, the total moment of inertia of the ball is

$$
\Theta=\int_{-R}^{R} 2 \pi R \mu\left(R^{2}-r^{2}\right) \mathrm{d} r=4 \pi R \mu \int_{0}^{R}\left(R^{2}-r^{2}\right) \mathrm{d} r=\frac{8}{3} \pi R^{4} \mu
$$

Considering that the mass of the spherical shell is $m=4 R^{2} \pi \mu$,

$$
\Theta=\frac{2}{3} m R^{2}
$$

Nagy Pál (Debrecen, Fazekas M. Gym., IV. grade)

II. Solution. Let the density of the ball be $\varrho$, its inner and outer radii be $r$ and $R$ respectively.

![](https://cdn.mathpix.com/cropped/2024_05_02_a139e0ed59c16ed03e6ag-1.jpg?height=333&width=371&top_left_y=1541&top_left_x=866)

Fig. 2

The moment of inertia of the ball is the difference between the moments of inertia of a solid sphere of radius $R$ and a solid sphere of radius $r$, both with density $\varrho$. Using the fact that the moment of inertia of a solid sphere is $(2 / 5) m r^{2}$,

$$
\begin{gathered}
\Theta=\Theta_{R}-\Theta_{r}=(2 / 5) m_{R} R^{2}-(2 / 5) m_{r} r^{2}= \\
=(2 / 5) \cdot(4 / 3) R^{3} \pi \varrho R^{2}-(2 / 5) \cdot(4 / 3) r^{3} \pi \varrho r^{2}= \\
=(2 / 5) \cdot(4 / 3) \pi \varrho\left(R^{5}-r^{5}\right)
\end{gathered}
$$

Express the density in terms of the mass of the ball:

$$
\varrho=\frac{m}{(4 / 3) \pi\left(R^{3}-r^{3}\right)}, \quad \text { so } \quad \Theta=(2 / 5) m \frac{R^{5}-r^{5}}{R^{3}-r^{3}}
$$

The moment of inertia of a spherical shell with negligible thickness is obtained by taking the limit as $r \rightarrow R$:

$$
\Theta=\lim _{r \rightarrow R} \frac{2}{5} m \frac{R^{5}-r^{5}}{R^{3}-r^{3}}=\lim _{r \rightarrow R} \frac{2}{5} m \frac{(R-r)\left(R^{4}+R^{3} r+R^{2} r^{2}+R r^{3}+r^{4}\right)}{(R-r)\left(R^{2}+R r+r^{2}\right)}, \Theta=\frac{2}{3} m R^{2}
$$

Márkus László (Sopron, Széchenyi I. Gym., IV. grade)

Note. For the calculation of the moment of inertia, see also the article by Bodó Zalán in the 2nd issue of the 22nd volume of K.M.L.