Q.Why does a solid sphere have smaller moment of inertia than a hollow cylinder of same mass and radius, about an axis passing through their axes of symmetry?

Concept: Rotational Inertia Comparison

The moment of inertia ($I$) of an object depends on its mass and how that mass is distributed relative to the axis of rotation. Objects with more mass concentrated closer to the axis have a smaller moment of inertia.

1. For a hollow cylinder of mass $M$ and radius $R$ rotating about its axis of symmetry, all its mass is located at the maximum distance $R$ from the axis. Its moment of inertia is $I_{cylinder} = MR^2$.
2. For a solid sphere of mass $M$ and radius $R$ rotating about an axis passing through its center (its axis of symmetry), its mass is distributed throughout its volume, from the center ($r=0$) to the surface ($r=R$). Its moment of inertia is $I_{sphere} = \frac{2}{5}MR^2$. …

The moment of inertia, often called rotational inertia, is a fundamental concept in rotational dynamics. It quantifies an object's resistance to changes in its rotational motion, much like mass quantifies resistance to changes in linear motion. The key factors determining moment of inertia are the object's total mass and, crucially, how that mass is distributed relative to the axis of rotation. The further the mass is, on average, from the axis, the greater the moment of inertia.

To understand why a solid sphere has a smaller moment of inertia than a hollow cylinder of the same mass and radius about their axes of symmetry, we need to compare how mass is distributed in each object.

1. Recall the Moment of Inertia Formulas: For a solid sphere of mass $M$ and radius $R$, rotating about an axis passing through its center (which is its axis of symmetry), the moment of inertia is given by:

For a thin-walled hollow cylinder of mass $M$ and radius $R$, rotating about its central axis (its axis of symmetry), the moment of inertia is given by:

> [!IMPORTANT]
> These formulas are standard results derived from integration, considering the mass distribution.

2. Compare the Coefficients:

Let's look at the coefficients multiplying $MR^2$ in both formulas:

* For the solid sphere: $\frac{2}{5} = 0.4$

* For the hollow cylinder: $1$

Clearly, $0.4 < 1$. This mathematical comparison directly shows that $I_{sphere} < I_{hollow\_cylinder}$ for the same mass $M$ and radius $R$.

3. Understand the Physical Reason (Mass Distribution): The difference in these coefficients arises from how the mass is distributed in each object relative to the axis of rotation.
   • Hollow Cylinder: In a thin-walled hollow cylinder, all of its mass $M$ is concentrated at the maximum radius $R$ from the axis of rotation. There is no mass closer to the axis than $R$. …

Concept: Moment of Inertia Depends on How Mass Is Distributed Relative to the Axis

Mass located farther from the axis contributes more to $I$ (weighted by $r^2$); mass located closer contributes less.

Step 1: Moment of inertia of the hollow cylinder

For a thin-walled hollow cylinder of mass $M$, radius $R$, spinning about its central axis, every mass element sits at exactly $r=R$:

Step 2: Moment of inertia of the solid sphere

For a solid sphere of mass $M$, radius $R$, spinning about a diameter, the mass is spread throughout the volume, from $r=0$ at the centre out to $r=R$ at the surface:

Step 3: Compare

$\frac{2}{5}MR^2 < MR^2 \quad\text{since } \frac25 = 0.4 < 1$ …