The moment of inertia of a hollow sphere of mass M and inner and outer radii R and 2 R about the

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The moment of inertia of a hollow sphere of mass M and inner...

The moment of inertia of a hollow sphere of mass $\mathrm{M}$ and inner and outer radii $\mathrm{R}$ and $2 \mathrm{R}$ about the axis passing through its centre and perpendicular to its plane is $\{\mathrm{A}\}(3 / 2) \mathrm{MR}^{2}$ \{B $\}(13 / 32) \mathrm{MR}^{2}$ $\{\mathrm{C}\}(31 / 35) \mathrm{MR}^{2}$ $\{\mathrm{D}\}(62 / 35) \mathrm{MR}^{2}$

The moment of inertia of a hollow sphere of mass $\mathrm{M}$ and inner and outer radii $\mathrm{R}$ and $2 \mathrm{R}$ about the axis passing through its centre and perpendicular to its plane is
$\{\mathrm{A}\}(3 / 2) \mathrm{MR}^{2}$
\{B $\}(13 / 32) \mathrm{MR}^{2}$
$\{\mathrm{C}\}(31 / 35) \mathrm{MR}^{2}$
$\{\mathrm{D}\}(62 / 35) \mathrm{MR}^{2}$

Key Concepts

Integration in Spherical Coordinates

When dealing with symmetric objects like spheres or spherical shells, using spherical coordinates simplifies the integration process. This method takes advantage of the symmetry of the object, making it easier to compute the moment of inertia by integrating over radial and angular components.

Hollow Spherical Shells

A hollow spherical shell refers to a spherical object that has mass distributed between an inner and an outer radius. The analysis of such bodies often involves subtracting the contribution of the inner void from that of the full sphere, thereby obtaining an expression for the moment of inertia that accounts for the hollow region.

Moment of Inertia

This is a measure of an object's resistance to changes in its rotational motion, defined as the sum (or integral) of each mass element multiplied by the square of its distance from the axis of rotation. It serves as a rotational analogue to mass in linear motion and is central in analyzing and predicting rotational dynamics.

Mass Distribution and Density

For objects with complex shapes or non-uniform mass distributions, the mass density—mass per unit volume—is a key consideration. In calculating the moment of inertia, knowing how mass is distributed throughout the volume allows for setting up the correct integrals, particularly when the object has regions with different radii or materials.

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