AP Physics 1 Unit 5 Notes: Angular Motion Foundations

Rotational Kinematics

Rotational kinematics is the “how does it move?” description of spinning motion. Just like linear kinematics describes where an object is, how fast it’s moving, and how its velocity changes in time (position, velocity, acceleration), rotational kinematics does the same for objects rotating about an axis—using angular versions of those quantities.

This matters because many real systems rotate: wheels, pulleys, fans, gears, planets, and even a door swinging on hinges. In Unit 5, you’ll eventually connect this motion-description to “why it changes” (torque and rotational dynamics). But you cannot do rotational dynamics well unless you’re fluent in the kinematics vocabulary and relationships.

A key theme: rotational kinematics mirrors linear kinematics almost perfectly if (1) you use radians and (2) you are clear about what is rotating about what axis.

Angular position and displacement

Angular position tells you “where” something is in its rotation. We measure it with an angle, typically written as $\theta$, referenced to some chosen zero direction.

Angular displacement is the change in that angle:

$\Delta\theta = \theta_f - \theta_i$

The sign of $\Delta\theta$ depends on a sign convention. In AP Physics 1, you’ll almost always take counterclockwise as positive and clockwise as negative, as long as you stick to it consistently.

Radians (why you must use them)

Angles can be measured in degrees, but the core rotational formulas use radians. A radian is defined by arc length: an angle of 1 radian subtends an arc equal in length to the radius.

The crucial relationship is:

$\theta = \frac{s}{r}$

• $s$ is arc length along a circle
• $r$ is the radius
• $\theta$ is in radians

Rearranged:

$s = r\theta$

This is the rotational analog of “distance along the circle.” If you use degrees in this equation without converting to radians, you will get the wrong answer.

A useful conversion (often given or assumed known) is:

$2\pi\ \text{rad} = 360^\circ$

Angular velocity

Angular velocity describes how quickly the angular position changes:

$\omega = \frac{\Delta\theta}{\Delta t}$

Instantaneously (conceptually, at a moment), it’s the derivative, but AP Physics 1 usually uses average values or constant acceleration situations.

• Units: rad per second
• Direction/sign: positive for counterclockwise, negative for clockwise (with the usual convention)

A common misconception is to treat angular velocity as “speed” that must be positive. In physics, $\omega$ is a signed quantity in 1D rotational problems, because direction of rotation matters.

Angular acceleration

Angular acceleration describes how quickly angular velocity changes:

$\alpha = \frac{\Delta\omega}{\Delta t}$

• Units: rad per second squared
• Sign indicates whether $\omega$ is increasing or decreasing in the chosen positive direction.

A subtle but important point: if an object is rotating clockwise (negative $\omega$) and it is slowing down, its angular acceleration is counterclockwise (positive $\alpha$). Students often incorrectly assign $\alpha$ the same “direction” as the rotation rather than the direction of change of $\omega$.

Constant-angular-acceleration equations (rotational kinematics equations)

When angular acceleration is constant, rotational motion uses the same mathematical forms as linear constant-acceleration kinematics. Replace:

• $x$ with $\theta$
• $v$ with $\omega$
• $a$ with $\alpha$

Key equations:

$\omega_f = \omega_i + \alpha t$

$\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2$

$\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$

$\Delta\theta = \frac{\omega_i + \omega_f}{2}t$

These work only when $\alpha$ is constant and the rotation is about a fixed axis.

Notation reference (so you don’t get tripped up)

Different teachers and problems may use slightly different symbols:

If you ever see $f$ and $i$ subscripts, they mean final and initial.

Reading rotational kinematics graphs

Graphs are a frequent way AP Physics asks conceptual questions.

• A $\theta$ vs $t$ graph: slope is $\omega$. If the curve is getting steeper, $\omega$ is increasing.
• An $\omega$ vs $t$ graph: slope is $\alpha$ and area under the curve is $\Delta\theta$.
• An $\alpha$ vs $t$ graph: area under the curve is $\Delta\omega$.

Students often mix up “slope gives rate-of-change” and “area gives accumulation.” A safe habit is to say out loud what each axis means, then use the same reasoning you use for linear graphs.

Worked example 1: spinning up a turntable

A turntable starts from rest and reaches an angular velocity of 6.0 rad per second in 2.0 s with constant angular acceleration.

1) Find the angular acceleration.

Use:

$\omega_f = \omega_i + \alpha t$

Given $\omega_i = 0$, $\omega_f = 6.0$, $t = 2.0$:

$6.0 = 0 + \alpha(2.0)$

$\alpha = 3.0\ \text{rad/s}^2$

2) Find the angular displacement during that time.

Use:

$\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2$

$\Delta\theta = 0 + \frac{1}{2}(3.0)(2.0)^2$

$\Delta\theta = 6.0\ \text{rad}$

Interpretation: 6.0 rad is a bit less than one full revolution (since one revolution is $2\pi$ rad).

Worked example 2: stopping rotation with opposite angular acceleration

A wheel rotates counterclockwise with $\omega_i = 10$ rad per second. It experiences a constant angular acceleration of $\alpha = -2.0$ rad per second squared (clockwise).

1) How long until it stops?

Stopping means $\omega_f = 0$. Use:

$\omega_f = \omega_i + \alpha t$

$0 = 10 + (-2.0)t$

$t = 5.0\ \text{s}$

2) How many radians does it rotate before stopping?

Use:

$\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$

$0^2 = 10^2 + 2(-2.0)\Delta\theta$

$0 = 100 - 4\Delta\theta$

$\Delta\theta = 25\ \text{rad}$

Notice the displacement is positive because it is still rotating counterclockwise while it slows down.

What can go wrong (common conceptual pitfalls)

One of the biggest sources of error is confusing “direction of rotation” with “direction of angular acceleration.” Remember:

• $\omega$ tells the direction you’re rotating.
• $\alpha$ tells the direction $\omega$ is changing.

Another common issue is using degrees inside equations like $s = r\theta$ or the constant-$\alpha$ kinematics equations. Those equations assume radians.

Finally, students sometimes think a larger radius means a different $\omega$ for the same rigid object. In rigid-body rotation about a fixed axis, every point has the same $\omega$ and $\alpha$, no matter the radius. What changes with radius is the linear (tangential) speed.

Exam Focus

• Typical question patterns:

  • Use constant-angular-acceleration equations to find $\omega_f$, $\alpha$, $\Delta\theta$, or $t$, sometimes with “comes to rest” language.
  • Interpret a $\omega$ vs $t$ graph: slope as $\alpha$ and area as $\Delta\theta$.
  • Convert between revolutions, degrees, and radians before using kinematics equations.

• Common mistakes:

  • Plugging degrees into equations that require radians; convert first using $2\pi\ \text{rad} = 360^\circ$.
  • Getting signs wrong: treating $\omega$ as always positive or assuming $\alpha$ must match the rotation direction.
  • Mixing up what “area under” and “slope of” a graph represent.

Connecting Linear and Rotational Motion

Rotational kinematics becomes powerful when you connect it to linear motion. Real objects are not just abstract angles—they have points at different radii moving through space. The key idea is that for a rigid object rotating about a fixed axis, every point shares the same angular motion (same $\theta$, $\omega$, $\alpha$), but points farther from the axis travel longer paths in the same time.

That is exactly why a point on the rim of a wheel moves faster than a point near the hub, even though the entire wheel has one angular velocity.

Arc length and tangential distance

If a point is at radius $r$ from the axis and the object rotates through angular displacement $\Delta\theta$, the point travels tangential distance (arc length):

$\Delta s = r\Delta\theta$

This equation is the bridge from angle to distance. It’s also the reason radians are “built in” to rotational physics: the geometry is simplest in radians.

Tangential speed and angular speed

Differentiate the arc-length relationship in time (conceptually) and you get the speed relationship:

$v = r\omega$

• $v$ is the tangential speed (linear speed along the circular path)
• $r$ is radius
• $\omega$ is angular velocity

This matters because AP problems often give you a wheel’s radius and angular speed and ask for the speed of a point on the rim (or vice versa). It also shows why “same wheel, bigger radius” means bigger linear speed for the outer edge.

A frequent misunderstanding is thinking $v$ is the same everywhere on the wheel. It is not: $v$ increases with $r$.

Tangential acceleration and angular acceleration

If angular velocity changes, points on the object speed up or slow down tangentially. The tangential acceleration magnitude is:

$a_t = r\alpha$

• $a_t$ points along the tangent (the direction of motion)
• Its sign/direction depends on whether the object is speeding up or slowing down in that direction

This is the linear counterpart of changing angular speed.

Centripetal acceleration and angular speed

Even if angular speed is constant, a point moving in a circle is constantly changing direction, so it has centripetal acceleration (radially inward toward the center):

$a_c = \frac{v^2}{r}$

Using $v = r\omega$, you can rewrite centripetal acceleration in angular form:

$a_c = r\omega^2$

Two-acceleration reality check: in many rotation problems, both accelerations can exist at once.

• $a_t$ changes the speed (magnitude of velocity).
• $a_c$ changes the direction of the velocity.

Students often mistakenly add these as if they were in the same direction. They are perpendicular (tangent vs radial), so the total acceleration magnitude is:

$a = \sqrt{a_t^2 + a_c^2}$

Period and frequency (useful rotational “timing” quantities)

For steady rotation, you often describe “how fast it spins” with:

• Period $T$: time for one full revolution
• Frequency $f$: revolutions per second

Their relationship is:

$f = \frac{1}{T}$

Angular speed relates to period and frequency:

$\omega = \frac{2\pi}{T}$

$\omega = 2\pi f$

These are especially useful when a problem gives “revolutions per minute” or a rotational rate and you need $\omega$ in rad per second.

Rolling without slipping (where linear and rotational truly lock together)

A particularly important connection in AP Physics 1 is rolling without slipping. This happens when a wheel rolls on a surface such that the point of contact is instantaneously at rest relative to the ground (no sliding). In that case, the translation of the wheel’s center and the rotation about its center are linked.

The key condition is:

$v_{cm} = r\omega$

• $v_{cm}$ is the speed of the center of mass of the rolling object
• $r$ is the wheel radius
• $\omega$ is the angular speed about the center

And for accelerations (if the rolling speed is changing):

$a_{cm} = r\alpha$

These look identical to the earlier relationships, but the meaning is special: $v_{cm}$ is not just the tangential speed of a rim point; it is the translational speed of the whole wheel’s center.

Why “point of contact is at rest” makes sense

For rolling without slipping, a point on the rim has speed $r\omega$ relative to the center, but that point’s velocity can cancel with the center’s forward motion at the bottom.

• Top of the wheel: rotation and translation add, so the top point moves fastest relative to the ground.
• Bottom of the wheel: rotation and translation subtract and can cancel, making the instantaneous velocity zero.

This is a conceptual favorite on exams: you may be asked which point has the greatest speed or whether the bottom point is moving.

A major misconception is thinking “every point on a rolling wheel has speed $v_{cm}$.” In reality, different points have different ground-relative speeds.

Worked example 1: converting angular speed to linear speed

A bike wheel of radius 0.35 m spins with angular speed 12 rad per second. Find the tangential speed of a point on the rim.

Use:

$v = r\omega$

$v = (0.35)(12)$

$v = 4.2\ \text{m/s}$

Interpretation: a point on the rim is moving 4.2 m/s relative to the wheel’s center. If the wheel is also translating, the ground-relative speed of that point depends on where it is on the wheel.

Worked example 2: rolling without slipping down a ramp (kinematics link only)

A solid cylinder rolls without slipping. At some instant, its center of mass has speed 3.0 m/s, and its radius is 0.20 m.

Find its angular speed.

For rolling without slipping:

$v_{cm} = r\omega$

$3.0 = 0.20\omega$

$\omega = 15\ \text{rad/s}$

Notice what you did not need here: torque, friction forces, or moment of inertia. This is purely the kinematic “no slip” constraint.

Worked example 3: combining tangential and centripetal acceleration

A point on the rim of a wheel of radius 0.50 m has angular speed 8.0 rad per second and angular acceleration 3.0 rad per second squared.

Find $a_t$ and $a_c$.

Tangential:

$a_t = r\alpha$

$a_t = (0.50)(3.0) = 1.5\ \text{m/s}^2$

Centripetal:

$a_c = r\omega^2$

$a_c = (0.50)(8.0)^2 = (0.50)(64) = 32\ \text{m/s}^2$

This illustrates a common physical reality: in fast rotation, centripetal acceleration can be much larger than tangential acceleration.

Direction conventions and the right-hand rule (when it appears)

In AP Physics 1, many rotational kinematics problems are treated as “1D rotation” with a sign convention (counterclockwise positive). But sometimes problems describe angular velocity as a vector pointing along the axis of rotation.

The right-hand rule connects the sense of rotation to a direction along the axis: curl the fingers of your right hand in the rotation direction, and your thumb points in the direction of the angular velocity vector. If you use the scalar sign convention instead, you’re essentially choosing one axis direction as positive.

What goes wrong here is mixing conventions: a student might assign counterclockwise positive in the plane of the page but then draw a vector pointing the opposite direction without noticing. The fix is to commit to one representation per problem and stay consistent.

Real-world connections that help you remember

• A door: the doorknob is far from the hinges because for the same angular speed of the door, the knob’s linear speed $v = r\omega$ is larger, so it sweeps through space quickly.
• A merry-go-round: people near the edge feel a stronger inward “pull” requirement because centripetal acceleration $a_c = r\omega^2$ grows with radius.
• Car tires: rolling without slipping links how fast the car moves to how fast the tire spins; a skid breaks that link.

Exam Focus

• Typical question patterns:

  • Convert between angular quantities and linear quantities using $v = r\omega$ and $a_t = r\alpha$; often the radius is a key piece of information.
  • Identify centripetal vs tangential acceleration (and sometimes compute both) using $a_c = \frac{v^2}{r}$ or $a_c = r\omega^2$.
  • Rolling without slipping constraints: use $v_{cm} = r\omega$ (and sometimes $a_{cm} = r\alpha$) to connect translation and rotation.

• Common mistakes:

  • Assuming the same linear speed for all points on a rotating rigid body; only $\omega$ is the same everywhere.
  • Confusing $a_t$ and $a_c$ or adding them as if they point the same way; they are perpendicular (tangent vs radial).
  • Using rolling-without-slipping equations in situations with slipping (skidding) where the contact point is not at rest relative to the ground.