Moment of inertia

What just happened?

From the same ramp the sphere rolled down faster than the hoop. Both convert the same potential energy $mgh$, but the hoop must "hide" a larger part of it in rotational motion — so less is left for speeding up downhill.

What is moment of inertia?

It is a measure of a body’s resistance to changes in rotational motion — the rotational counterpart of mass. It depends not only on the mass, but also on its distribution about the axis: the farther the mass is from the axis, the larger $I$. For uniform bodies $I = c\,mR^2$.

The hoop has all its mass on the rim (far from the axis) → the largest $c=1$. The sphere also has mass spread near the center → the smallest $c=\tfrac25$.

Energy balance when rolling:

Rotational energy $E_{rot}=\tfrac12 I\omega^2$. Substituting $I=c\,mR^2$ and $\omega=v/R$ we get:

Mass and radius cancelled out — the speed at the bottom depends only on the shape $c$. A smaller $c$ ⇒ a larger $v$. That is why the order is always: sphere, then cylinder, and the hoop last.

Why you need this for the exam:

You solve rolling problems with a single equation $mgh=\tfrac12 mv^2+\tfrac12 I\omega^2$ with the substitution $\omega=v/R$. Trick: write out $I=c\,mR^2$ for the given body, and $m$ and $R$ will cancel out by themselves.

Remember: moment of inertia is the resistance to spinning and depends on mass distribution — the farther the mass from the axis, the larger $I$ and the slower the body rolls down.