Unit 5 Rotation: Understanding Rotational Kinematics (AP Physics C: Mechanics)

Rotational kinematics

The set of concepts/equations used to describe how an object rotates: angular displacement (θ), angular velocity (ω), and angular acceleration (α).

Angular position (θ)

An object’s rotational position measured as an angle from a chosen reference line; SI angle unit is the radian (rad).

Radian (rad)

Angle unit defined by θ = s/r (arc length divided by radius); treated as dimensionless but labeled “rad” to indicate an angle.

Arc length (s)

Distance traveled along a circular path; related to angular displacement by s = rθ (θ must be in radians).

Counterclockwise-positive sign convention

Common rotation sign convention in AP Physics: CCW angles/ω/α are positive, CW are negative; must be used consistently.

Angular velocity (ω)

Rate of change of angular position: ω = dθ/dt (units rad/s); can be positive or negative depending on direction.

Average angular velocity (ω_avg)

Angular displacement per time interval: ω_avg = Δθ/Δt.

Instantaneous angular velocity

Angular velocity at a moment in time: ω = dθ/dt (the slope of a θ vs t graph).

Right-hand rule (for ω direction)

In 3D, curl right-hand fingers with rotation direction; thumb points along the angular velocity vector (axis direction).

Angular acceleration (α)

Rate of change of angular velocity: α = dω/dt = d²θ/dt² (units rad/s²).

Average angular acceleration (α_avg)

Change in angular velocity per time interval: α_avg = Δω/Δt.

Constant angular acceleration

A situation where α is constant; then ω changes linearly with time and θ changes quadratically with time.

Constant-α kinematics: ω = ω₀ + αt

Relates angular velocity and time when α is constant; ω₀ is angular velocity at t = 0.

Constant-α kinematics: θ = θ₀ + ω₀t + (1/2)αt²

Angular position as a function of time for constant α; θ₀ is the initial angular position.

Constant-α kinematics: ω² = ω₀² + 2α(θ − θ₀)

Time-eliminated rotational kinematics equation valid only when α is constant.

Average ω under constant α

If α is constant, ωavg = (ω₀ + ω)/2, so Δθ = ωavg t = ((ω₀ + ω)/2)t.

Slope of θ vs t graph

The slope of a θ(t) graph equals angular velocity ω.

Area under ω vs t graph

The area under an ω(t) graph over a time interval equals angular displacement Δθ.

Tangential speed (v)

Linear speed of a point at radius r on a rotating object: v = rω (units m/s).

Tangential acceleration (a_t)

Linear acceleration that changes speed along the tangent: a_t = rα (units m/s²).

Centripetal (radial) acceleration (a_c)

Inward acceleration that changes direction of velocity in circular motion: a_c = rω² = v²/r.

Total acceleration in circular motion

Vector sum of perpendicular components: a = √(at² + ac²).

Rigid body rotation about a fixed axis

Model where all points share the same θ, ω, and α, but have different linear speeds/accelerations depending on radius r.

Degrees-to-radians conversion factor

To use s = rθ, v = rω, etc., convert degrees to radians: radians = degrees × (π/180).

Direction reversal condition

An object reverses rotation when angular velocity changes sign; the turning point occurs when ω = 0.