Disc Rotation: Uniformly Accelerated Motion

In this tutorial we derive and illustrate the fundamentals of a disc undergoing uniformly accelerated rotation about a fixed axis. We’ll look at kinematic relations, the role of torque, and how to compute the moment of inertia step by step.

Kinematics: Angular Displacement under Constant Angular Acceleration

When a disc experiences a constant angular acceleration α (rad/s²), its angular velocity ω increases linearly over time:

alpha = d(omega}/dt

For a small time increment dt, the additional angular displacement dφ is

Formula for angular displacement under constant angular acceleration
Figure 1. Disk Rotation: Angular Displacement under Constant Angular Acceleration

Step-by-Step Update Rules

Implementing the motion numerically, at each step we update:

This mirrors the translational kinematic formula s = s + v₀·t + ½·a·t².

Discrete update for disc rotation simulation
Figure 2. Iterative update rules for time and angular displacement.

Torque: Generating Angular Acceleration

To produce angular acceleration, a torque must act on the disc. Torque about the rotation axis is given by

M = F_t * r  (N·m)

where Ft is the tangential component of the force at radius r.

Diagram showing force applied at radius creating torque on disc
Figure 3. A tangential force Ft acting at radius r produces torque.

Moment of Inertia: Measure of Rotational Inertia

The rotational analogue of mass in linear motion is the moment of inertia I. For a rigid body made up of point masses mi at radii ri:

Definition of moment of inertia for discrete mass elements
Figure 4. \(I = \sum_i m_i\,r_i^2\)

Summing over all mass elements gives the total I, in units of kg·m².

Summation for rigid body moment of inertia
Figure 5. Summing individual contributions yields the body’s moment of inertia.

For a uniform solid disc of mass m and radius r, the closed-form result is

Moment of inertia of a solid disc formula
Figure 6. I_disc = mr2

Equation of Motion & Angular Momentum

Newton’s second law for rotation relates torque and angular acceleration:

Newton’s second law for rotation: torque equals moment of inertia times angular acceleration
Figure 7. Newton’s second law for rotation: torque equals moment of inertia times angular acceleration

The angular momentum L of the rotating disc is

Angular momentum of a rotating rigid body
Figure 8. Angular momentum of a rotating rigid body