AP Physics 1 Unit 5 Notes: Angular Motion Foundations

Rotational kinematics

The description of how rotational motion changes with time (angular position, angular velocity, angular acceleration) for an object rotating about an axis.

Axis of rotation

The line about which an object rotates; rotational kinematics must specify what is rotating about which axis.

Angular position (θ)

An angle that specifies an object’s rotational “where” (orientation) relative to a chosen reference (zero) direction.

Angular displacement ($Δθ$)

The change in angular position: $Δθ = θ_f - θ_i$; its sign depends on the chosen sign convention.

Sign convention (CCW positive)

A consistent choice for positive rotation direction, typically counterclockwise positive and clockwise negative in AP Physics 1.

Radian

The angle measure used in rotational formulas; defined so that 1 rad subtends an arc length equal to the radius.

Radians–arc length relation

Relationship between angle (in radians), arc length, and radius: $θ = \frac{s}{r}$ (equivalently $s = rθ$).

Degree–radian conversion

Conversion used to switch between units: $2π$ rad = $360°$ (so formulas requiring radians must not use degrees directly).

Angular velocity ($ω$)

Rate of change of angular position: $ω = \frac{Δθ}{Δt}$ (signed; units rad/s).

Angular acceleration ($α$)

Rate of change of angular velocity: $α = \frac{Δω}{Δt}$ (signed; units rad/s²).

Constant-angular-acceleration model

A rotational motion case where $α$ is constant, allowing use of rotational kinematics equations analogous to linear constant-acceleration equations.

Rotational kinematics equation: ($ω_f = ω_i + αt$)

For constant α, the final angular velocity equals initial angular velocity plus α times time.

Rotational kinematics equation: ($Δθ = ω_it + (1/2)αt²$)

For constant α, angular displacement equals initial angular velocity times time plus one-half α times time squared.

Rotational kinematics equation: ($ω_f² = ω_i² + 2αΔθ$)

For constant α, relates angular speeds and angular displacement without needing time.

Rotational kinematics equation: ($Δθ = \frac{(ω_i + ω_f)}{2}t$)

For constant α, angular displacement equals average angular velocity times time.

$θ$ vs. t graph interpretation

On an angular position–time graph, the slope equals angular velocity $ω$; increasing steepness indicates increasing $ω$.

$ω$ vs. t graph interpretation

On an angular velocity–time graph, slope equals angular acceleration α and the area under the curve equals angular displacement Δθ.

$α$ vs. t graph interpretation

On an angular acceleration–time graph, the area under the curve equals the change in angular velocity $Δω$.

Rigid-body rotation (about a fixed axis)

For a rigid object rotating about a fixed axis, every point has the same θ, ω, and α; linear (tangential) quantities vary with radius.

Tangential distance / arc length ($Δs$)

Linear distance traveled along the circular path: $Δs = rΔθ$.

Tangential speed (v)

Linear speed along the circular path related to angular speed by $v = rω$; increases with radius for the same $ω$.

Tangential acceleration (a_t)

Acceleration that changes the speed (magnitude) of the velocity along the tangent: $a_t = rα$.

Centripetal acceleration (a_c)

Inward (radial) acceleration due to change in direction of velocity in circular motion: $a_c = \frac{v^2}{r} = rω^2$.

Total acceleration in circular motion

If both tangential and centripetal accelerations exist, they are perpendicular, so magnitude a = $\sqrt{a_t^2 + a_c^2}$.

Rolling without slipping

Rolling where the contact point is instantaneously at rest relative to the ground, giving the kinematic constraints $v_{cm} = rω$ and (if speeding up/slowing down) $a_{cm} = rα$.