Moment Of Inertia Of A Disc

The Moment of Inertia (MI) is a measure of an object’s resistance to rotational motion about a specific axis. The moment of inertia depends on the mass of the object and how that mass is distributed relative to the axis of rotation. A disc is a flat, circular object with uniform mass distribution across its area.

Moment of Inertia Of A Disc About Different Axes

Considering a solid disc of Mass M, Radius R, and negligible thickness. The following are the cases to calculate its moment of inertia:

Moment of Inertia About the Central Axis (Perpendicular to the Plane)

Considering a disc rotating about an axis perpendicular to its plane and passing through its centre (central axis), the small element of mass of the disc is at a distance r from the axis. The general definition of moment of inertia is:

I = ∫ r² dm

where, dm - an infinitesimal mass element at distance r from the axis.

By polar coordinates and integrating the disc, the derivation of the moment of inertia of a solid disc is:

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where,

M - mass

R - radius

Moment of Inertia About a Diameter (In-Plane Axis)

If the disc rotates about an axis lying in its plane and passing through its centre (like a diameter), the moment of inertia is different. Using calculus or the perpendicular axis theorem, which states:

I_{central axis} = I_x + I_y

where, I_x and I_y are the moment of inertia about two perpendicular axes lying in the plane of the disc,

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Moment of Inertia About Any Parallel Axis

If the axis of rotation is parallel to the central axis but passes through a point that is not the centre of mass, the Parallel Axis Theorem is applied:

I = I_CM + Md²

where,

I_CM - MI about the central axis through the centre of mass

d - distance between the central axis and the new axis

Formulae at a Glance:

Shape	Axis of Rotation	Moment of Inertia
Solid Disc	Perpendicular to the plane, through CM
Solid Disc	Along the diameter, in-plane
Solid Disc	Parallel axis through a point	I_CM + Md²

Applications

Flywheels – Store rotational energy and stabilise engines.
Vehicle Wheels – Provide smooth rotation and transfer torque efficiently.
Pulleys – Facilitate lifting and transmission of mechanical power.
Rotors in Motors – Convert electrical energy into mechanical rotation.
CDs/DVDs – Spin uniformly for reading and writing data.
Gears – Transfer rotational motion between mechanical components.
Grinding Wheels – Rotate to shape or polish materials in machining.
Clutch Plates – Transmit torque in vehicles and machinery.
Rotating Platforms – Enable smooth rotation in stages or machinery.
Equipment - Measurement instruments like gyroscopes depend on precise MI values of discs for stability.

Summing Up

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Frequently Asked Questions

Q1. Why is the moment of inertia higher about the central axis than along a diameter?

More mass lies farther from the axis in the perpendicular rotation, increasing resistance to rotation. For this reason, the moment of inertia is more about the central axis.

Q2. How is the moment of inertia important in rotational kinetic energy?

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Q3. A solid disc of mass M=4 kg and radius R=0.5 m rotates about an axis perpendicular to its plane through its centre. Find its moment of inertia.

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