AP Physics C: Mechanics · Unit 5: Torque and Rotational Dynamics · Lesson 5.1

Deep Dive: Rotational Kinematics

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

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Angular Displacement and Rigid Systems

Angular displacement measures the angle, in radians, through which a point on a rigid system rotates about a specified axis:

Δθ = θ − θ₀

A rigid system is one that holds its overall shape, but where different points on it move in different directions during rotation. This is a genuinely new idea: a rigid system cannot be modeled as a single object the way you've done since Unit 1, because "where is it" and "which way is it moving" no longer have single answers for the system as a whole.

🔑As with sign conventions for linear motion, one rotation direction — clockwise or counterclockwise — is chosen as mathematically positive, with the other negative. And if a system's rotation about an axis can be adequately described just by tracking its center of mass (Earth's own rotation is negligible when studying Earth's orbit around the Sun, for instance), the system can still be treated as a simple object after all.

Explore rigid-system rotation directly below, matching this unit's header art.

A rigid disk rotating at constant angular velocity, matching this unit's header art. Adjust ω and t — angular displacement accumulates exactly like linear displacement did back in Unit 1, just measured in radians around an axis instead of meters along a line.

ω (rad/s)1.2

t (s)2.0

Counterclockwise (positive direction)

Δθ = ωt = (1.2)(2.0) = 2.40 rad (0.38 full revolutions)

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Angular Velocity and Angular Acceleration

Angular velocity is the rate at which angular position changes with respect to time:

ω = dθ/dt

Angular acceleration is the rate at which angular velocity itself changes with respect to time:

α = dω/dt

💡These definitions are exact derivatives, exactly like v = dx/dt and a = dv/dt from Lesson 1.2 — the calculus hasn't changed at all, only the symbols and the physical quantity being tracked.

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The Rotational Kinematics Equations

Angular displacement, angular velocity, and angular acceleration around a single axis are directly analogous to their linear counterparts, and satisfy exactly the same mathematical relationships. For constant angular acceleration:

θ = θ₀ + ω₀t + ½αt²

ω = ω₀ + αt

ω² = ω₀² + 2α(θ − θ₀)

🔑Every technique you already have for reading θ(t), ω(t), and α(t) graphs — slopes giving derivatives, areas giving integrals — carries over completely unchanged from Lesson 1.3.

The full linear-to-angular dictionary, side by side:

Quantity	Linear	Angular
Displacement	x	θ
Velocity	v = dx/dt	ω = dθ/dt
Acceleration	a = dv/dt	α = dω/dt
Constant-a equation 1	x = x₀ + v₀t + ½at²	θ = θ₀ + ω₀t + ½αt²
Constant-a equation 2	v = v₀ + at	ω = ω₀ + αt
Constant-a equation 3	v² = v₀² + 2a(x−x₀)	ω² = ω₀² + 2α(θ−θ₀)

Explore the full stacked-graph relationship directly below — the same tool structure from Lesson 1.2, now spinning.

The same stacked-graph relationship from Lesson 1.2's position/velocity/acceleration tool — now for angular quantities. Adjust initial angular velocity and constant angular acceleration, and scrub the time cursor to see how all three graphs stay in sync.

ω₀ (rad/s)1.0

α (rad/s²)0.60

time cursor3.0s

θ = 5.70 radω = 2.80 rad/sα = 0.60 rad/s²

ExampleGuided Example — Applying the Rotational Kinematics Equations

A merry-go-round starts at rest and speeds up with a constant angular acceleration of 0.4 rad/s². Find its angular velocity and total angular displacement after 6 seconds.

Step 1 — Identify known quantities

ω₀ = 0 (starts at rest), α = 0.4 rad/s², t = 6 s

← Back to Lesson 5.1Ready for 5.2? Connecting Linear and Rotational Motion ties angular quantities back to the linear speeds of individual points on a rigid system.