Moment of inertia

Rotational inertia

Moment of inertia is the rotational analog of mass. It measures how difficult it is to change an object's rotational motion about a chosen axis. Unlike mass, moment of inertia depends on both the amount of mass and how far that mass is from the axis.

For point particles,

$I=sum_i m_i r_i^2$

where $r_i$ is the perpendicular distance from the rotation axis to the particle.

For a continuous object,

$I=int r^2,dm$

Mass farther from the axis contributes much more because of the $r^2$ factor.

Why distribution matters

A figure skater spinning with arms extended has a larger moment of inertia than with arms pulled in. The mass is the same, but the distribution changes. Pulling arms inward reduces $I$, allowing angular speed to increase if angular momentum is conserved.

Similarly, a hoop and a solid disk of the same mass and radius have different moments of inertia. The hoop's mass is concentrated farther from the center, so it has larger $I$.

Common moments of inertia

Some standard results are useful:

Thin hoop about central axis:

$I=MR^2$

Solid disk or cylinder about central axis:

I=rac{1}{2}MR^2

Solid sphere about diameter:

I=rac{2}{5}MR^2

Thin rod about center:

I=rac{1}{12}ML^2

Thin rod about end:

I=rac{1}{3}ML^2

These formulas depend on the axis. Always check the axis before using a formula.

Rotational kinetic energy

A rotating rigid body has kinetic energy:

K_{rot}=rac{1}{2}Iomega^2

This mirrors translational kinetic energy:

K_{trans}=rac{1}{2}Mv^2

For rolling motion, total kinetic energy is the sum of translational and rotational parts:

K=rac{1}{2}Mv_{cm}^2+rac{1}{2}I_{cm}omega^2

If rolling without slipping,

$v_{cm}=Romega$

Why rolling objects accelerate differently

On an incline, some gravitational potential energy becomes translational kinetic energy and some becomes rotational kinetic energy. Objects with larger moment of inertia relative to $MR^2$ put more energy into rotation and less into center-of-mass motion, so they roll down more slowly.

For example, a solid sphere generally accelerates faster down an incline than a hoop of the same mass and radius because the sphere has smaller moment of inertia.

Parallel-axis theorem

If you know the moment of inertia about an axis through the center of mass, you can find it about a parallel axis a distance $d$ away using

$I=I_{cm}+Md^2$

This is the parallel-axis theorem. It explains why rotating a rod about its end has larger moment of inertia than rotating it about its center.

Radius of gyration

Sometimes moment of inertia is written as

$I=Mk^2$

where $k$ is the radius of gyration. It represents the distance from the axis at which all mass could be concentrated to give the same moment of inertia.

The big idea

Moment of inertia measures rotational resistance and depends on mass distribution relative to the axis. It appears in rotational dynamics, energy, rolling motion, and angular momentum. Because distance enters as $r^2$, moving mass outward greatly increases rotational inertia.