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 $$0.31\;m$$