**Rotational Inertia**

From the very definition of inertia, rotational inertia is the tendency of a rotating object to resist a change in its state of motion. It depends on the mass and radius of the object. Mathematically, rotational motion can be expressed as

\(I=\frac 12 mR^2\)

where I is the rotational inertia, m is the mass and R is the radius of the object.

If the mass is outside the rotating object, rotational inertia is measured as \(I=mR^2\).

**Example 1.**

What is the rotational inertia of a spinning disc with a mass \(0.09-kg\) and radius \(0.06 m\)?

Solution.

To solve for rotational inertia, the equation \(I=\frac 12 mR^2\) must be evaluated.

\(I=\frac 12 (0.09\;kg)(0.06\;m)^2=0.000162\;kg\cdot m^2 \text{or}\; 1.62 \times 10^{-4}\;kg\cdot m^2\)

The rotational inertia of the object is \(1.62 \times 10^{-4}\;kg\cdot m^2\).

**Example 2.**

A unicycle’s wheel has a mass of \(3.17\;kg\). If the rotational inertia of the wheel is \(0.15\;kg\cdot m^2\) , what is the radius of the wheel?

Solution:

Derive the formula to solve the radius from the equation \(I=\frac 12 mR^2\).

\(I=\frac 12 mR^2 \implies mR^2=2I \implies R^2=\frac {2I}{m} \implies R=\sqrt {\frac{2I}{m}}\).

\(R=\sqrt{\frac{2(0.15\;kg\cdot m^2)}{3.17\;kg}}=0.31\;m\)

Hence, the radius of the wheel is \(0.31\;m\).