Moment of Inertia of a Circle

In rotational motion, the moment of inertia (MI) is similar to mass in linear motion. Mass resists a change in linear velocity, and the moment of inertia resists a change in angular velocity. It depends on the object’s mass and distribution of mass relative to the axis of rotation. For a circular object, such as a disk or wheel, the MI is not the same for all axes; it changes according to the axis passing, whether it passes through the centre or along the diameter.

Moment of Inertia of a Solid Circular Disk

Considering a circular disk of radius R and mass M, two cases of inertia are calculated. The following are the two cases –

Moment of Inertia about an Axis Perpendicular to the Plane of the Disk through its Centre

The axis of the disk is vertical through the centre, and the disk is divided into thin concentric rings.

Derivation –

Here,

r – Radius of a ring

dr – Thickness of ring

2πr dr – Area of ring

If the mass per unit area is σ, then:

formula

Mass of ring: dm = σ · 2πr dr

Moment of inertia of this ring about the vertical central axis:

dI = r² dm

Substituting dm:

dI = r² · σ · 2πr dr

dI = 2πσ r³ dr

formula

Moment of Inertia about a Diameter of the Disk (Lying in the Plane)

The perpendicular axis theorem is used for calculation:

I_z = I_x + I_y

Here:

I_z – MI about the perpendicular axis through the centre
I_x and I_y – MI about diameters (both equal for a circle)

formula

Moment of Inertia of a Thin Circular Ring

Considering a thin circular ring of radius R and mass M, in two cases moment of inertia is calculated.

The following are the two cases –

Moment of Inertia About a Perpendicular Axis through the Centre

I = MR²

All the mass is at the same distance R from the axis, so integration is not needed for the calculation.

Moment of Inertia about a Diameter in the Plane

formula

Formulae for the Following Shapes

Shape	Axis	Moment of Inertia
Solid disk	Perpendicular through the centre
Solid disk	Diameter through centre
Thin ring	Perpendicular through the centre
Thin ring	Diameter through centre

Applications

Rotating Machinery – Flywheels, turbines, and pulleys use the MI formula to determine rotational acceleration.
Automotive Engineering – Brake discs and steering wheels depend on MI for stability and performance.
Sports Equipment – Frisbees, gym weights, and discus require balanced MI for the desired motion.
Aerospace – Propeller and rotor blade design considers MI to manage torque and rotational energy.

Summing Up

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Frequently Asked Questions

Q1. Why does a thin ring have a greater moment of inertia than a solid disk of the same mass and radius?

In a ring, all the mass is at the maximum distance from the axis, so it resists rotation more compared to a disk, where mass is distributed closer to the centre.

Q2. What does the moment of inertia depend on?

The moment of inertia of a shape or object depends on its mass, shape, size, and the distribution of mass relative to the axis of rotation.

Q3. Does the distribution of mass affect rotational speed?

Yes, a higher moment of inertia for the same torque results in slower angular acceleration.