Rotational Kinematics and Dynamics

TL;DR

Rotational motion follows the same logical structure as linear motion, just with different variables: angular position, velocity, and acceleration replace distance, velocity, and acceleration. Torque plays the role of force, and moment of inertia replaces mass. Everything works the same way once you make these substitutions.

1. The Mental Model

You already know how things move in straight lines. Rotational motion is not fundamentally different—it's the same physics, just happening around an axis. Instead of tracking how far something travels, you track how many degrees or radians it's rotated. Instead of asking "what force makes it accelerate?", you ask "what torque makes it spin faster?". The moment of inertia tells you how hard it is to spin something up, just like mass tells you how hard it is to push something. Once you see that the equations have the same structure, the mystery vanishes and rotation becomes just another language for describing motion.

2. The Core Material

Angular Position, Velocity, and Acceleration

In linear motion, you work with position x, velocity v = dx/dt, and acceleration a = dv/dt. In rotation, you use:

• Angular position θ (theta), measured in radians. One full rotation = 2π rad.
• Angular velocity ω (omega) = dθ/dt. This is how fast the angle is changing, in rad/s.
• Angular acceleration α (alpha) = dω/dt. This is how fast the rotation is speeding up or slowing down, in rad/s².

If something rotates at constant angular velocity (like a record player at steady speed), then ω is constant and α = 0. If you spin up a wheel from rest, ω increases and α > 0.

The kinematic equations you know work exactly the same way, with these substitutions:

Notice the structure is identical. If you can solve a linear kinematics problem, you can solve a rotational one.

Moment of Inertia

Mass resists changes in linear motion. Moment of inertia I resists changes in rotational motion. But I is not just "the rotational mass"—it depends on how mass is distributed relative to the axis of rotation.

For a point mass m at distance r from the axis:
$$I = mr^2$$

For a solid object, you sum (or integrate) over all the mass:
$$I = \int r^2 \, dm$$

This is why a figure skater spins faster when they pull their arms in—the same mass is now closer to the axis (r decreases), so I decreases, and by conservation of angular momentum, ω increases.

Common moments of inertia (about the centre of mass):
- Solid sphere of mass M and radius R: I = (2/5)MR²
- Solid cylinder or disk: I = (1/2)MR²
- Thin hoop: I = MR²
- Thin rod (rotating about centre): I = (1/12)ML²

The higher the moment of inertia, the harder it is to change the rotation.

Torque and Rotational Dynamics

Force causes linear acceleration. Torque τ (tau) causes angular acceleration. Torque is the rotational equivalent of force.

$$\tau = r F \sin(\theta)$$

where r is the distance from the axis, F is the force applied, and θ is the angle between the force and the position vector. Practically: if you push perpendicular to a lever (θ = 90°), you get maximum torque. If you push along the lever, you get zero torque.

The rotational equivalent of Newton's second law is:
$$\tau_{net} = I \alpha$$

This is clean and powerful: just like F = ma, torque equals moment of inertia times angular acceleration.

Angular Momentum and Conservation

Linear momentum p = mv. Angular momentum is:
$$L = I \omega$$

In the absence of external torque, angular momentum is conserved. A spinning object with no external torque will keep spinning forever at the same ω. This is why gyroscopes stay upright and why a spinning skater's spin rate changes when their arms move.

graph TD
    A["Torque τ applied<br/>to object"] --> B["Angular acceleration α<br/>using τ = Iα"]
    B --> C["Angular velocity ω<br/>changes over time"]
    C --> D["Angular position θ<br/>keeps updating"]
    E["Moment of inertia I<br/>depends on mass distribution"] --> B
    F["External torque τ_ext"] --> G["Angular momentum L<br/>changes"]
    H["No external torque"] --> I["Angular momentum L<br/>stays constant"]
    style A fill:#e1f5ff
    style I fill:#c8e6c9

3. Worked Example

A solid disk of mass 2 kg and radius 0.5 m starts at rest on a frictionless axle. You apply a constant torque of 3 N⋅m to spin it up. How long does it take to reach 10 rad/s?

Step 1: Find the moment of inertia.
For a solid disk: I = (1/2)MR² = (1/2)(2)(0.5)² = 0.25 kg⋅m²

Step 2: Apply τ = Iα to find angular acceleration.
3 = 0.25α
α = 12 rad/s²

Step 3: Use ω = ω₀ + αt with ω₀ = 0.
10 = 0 + 12t
t = 10/12 ≈ 0.83 seconds

Check: Does this make sense? A modest torque on a light disk produces a large angular acceleration, and we reach 10 rad/s in less than a second. Yes.

4. Key Takeaways

4.1 Most Important Concepts

1. Angular position, velocity, and acceleration are the rotational analogues of linear kinematics; the equations have identical form.
2. Moment of inertia measures resistance to angular acceleration and depends on how mass is distributed around the axis (closer mass = smaller I).
3. Torque is the rotational force: τ = rF sin(θ), and τ = Iα is Newton's second law for rotation.
4. Angular momentum L = Iω is conserved when no external torque acts, explaining why figure skaters spin faster when they pull their arms in.
5. The mathematics is parallel to linear motion—swap m for I, F for τ, v for ω, a for α, and everything works the same.
6. Rotational kinetic energy is (1/2)Iω², analogous to (1/2)mv².

4.2 Common Misconceptions

• Misconception: "Moment of inertia is just the rotational version of mass and acts the same way everywhere."
  Reality: Moment of inertia depends on the choice of axis. Spinning a rod about its centre gives a different I than spinning it about one end. Mass does not change; I does.

• Misconception: "Torque is just force applied at a distance."
  Reality: Only the component of force perpendicular to the lever arm contributes to torque. Pushing along the handle of a wrench does nothing; pushing perpendicular to it spins the nut.

• Misconception: "Angular acceleration is always proportional to torque."
  Reality: Angular acceleration depends on both torque and moment of inertia (α = τ/I). A large torque on a massive, extended object can produce a small angular acceleration.

• Misconception: "A spinning object slows down because it 'runs out' of rotation."
  Reality: A spinning object with no external torque maintains constant angular momentum forever. It only slows down if friction or other forces apply an opposing torque.

4.3 Compare & Contrast

5. Now Try It

Problem: A thin cylindrical hoop of mass 5 kg and radius 1 m is initially spinning at 4 rad/s. A constant frictional torque of 2 N⋅m opposes the rotation. How long does it take to stop?

1. Look up or calculate the moment of inertia for a thin hoop: I = MR².
2. Use τ = Iα to find the angular acceleration (note: the torque is negative because it opposes motion).
3. Use ω = ω₀ + αt with ω_final = 0 to solve for t.

Write out your working with numbers at each step. Show the moment of inertia calculation first, then the angular acceleration, then the time. Check that your answer makes physical sense: a 2 N⋅m torque on this hoop should slow it down gradually, not instantly.

Success looks like: You calculate I = 5 kg⋅m², find α = −0.4 rad/s², and determine that it takes 10 seconds to stop, with all working clearly shown.