# Moment of Inertia of a Circle

on . Posted in Classical Mechanics

The moment of inertia of a circle, also known as the rotational inertia, depends on the mass distribution within the circle and its axis of rotation.  The formula assumes that the axis of rotation is perpendicular to the plane of the circle and passes through its center.  If the axis of rotation is different or the mass distribution within the circle is not uniform, the moment of inertia will have a different value.

### Moment of Inertia of a Circle formula, Hollow Plane

$$I_z = m \; r^2$$     (Moment of Inertia of a Circle, Hollow Plane)

$$m = Iz \;/\; r^2$$

$$r = \sqrt{ Iz \;/\; m }$$

### Solve for m

 moment of inertia, Iz radius, r

### Solve for r

 moment of inertia, Iz mass, m

Symbol English Metric
$$I$$ = moment of inertia $$lbm \;/\; ft^2-sec$$  $$kg \;/\; m^2$$
$$m$$ = mass $$lbm$$ $$kg$$
$$r$$ = radius $$in$$ $$mm$$

### Moment of Inertia of a Circle formulas, Solid Plane

$$I_z = \frac{1}{2}\; m \; r^2$$

$$I_z = \frac{1}{2}\; \pi \; r^4$$

$$I_x = I_y = \frac{1}{4} \;m \; r^4$$

$$I_x = I_y = \frac{1}{4}\; \pi \; r^4$$

$$I_x = I_y = \frac{1}{64}\; d^4$$

Symbol English Metric
$$I$$ = moment of inertia $$lbm \;/\; ft^2-sec$$   $$kg \;/\; m^2$$
$$d$$ = diameter $$in$$ $$mm$$
$$m$$ = mass $$lbm$$ $$kg$$
$$\pi$$ = Pi $$dimensionless$$
$$r$$ = radius $$in$$ $$mm$$