Rotational Kinematics

Q: What is angular displacement, and what standard unit must be used for calculus-based rotational kinematics?

Angular displacement is the net change in angular position, \(\Delta\theta = \theta(t_2) - \theta(t_1)\), and it must always be expressed in radians (\(\text{rad}\)) .

Q: How are average and instantaneous angular velocity defined mathematically?

Average angular velocity is \(\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}\), while instantaneous angular velocity is the first time derivative of angular position: \(\omega = \frac{d\theta}{dt}\).

Q: How is instantaneous angular acceleration defined in terms of angular velocity and angular position?

\(\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}\), measured in \(\text{rad/s}^2\).

Q: How do the relative signs of angular velocity \(\omega\) and angular acceleration \(\alpha\) determine whether an object is speeding up or slowing down?

If \(\omega\) and \(\alpha\) share the same sign, the object is speeding up ; if they have opposite signs, the object is slowing down .

Q: How are angular velocity and position determined when angular acceleration \(\alpha(t)\) is not constant?

By integrating with respect to time: \(\omega(t) = \omega_0 + \int_{0}^{t} \alpha(t') \, dt'\) and \(\theta(t) = \theta_0 + \int_{0}^{t} \omega(t') \, dt'\).

Q: How can angular acceleration be related to angular velocity and angular position when time is not explicitly involved?

Using the chain rule: \(\alpha = \omega \frac{d\omega}{d\theta}\), which separates into \(\alpha \, d\theta = \omega \, d\omega\).

Q: What kinematic equations describe rotational motion when angular acceleration \(\alpha\) is constant?

\(\omega = \omega_0 + \alpha t\), \(\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\), and \(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\).

Q: What physical quantities are represented by the slope and concavity of an angular position vs. time (\(\theta\) vs. \(t\)) graph?

The slope equals instantaneous angular velocity (\(\omega = \frac{d\theta}{dt}\)), and the concavity indicates the sign of angular acceleration (\(\alpha\)).

Q: What do the slope and area under the curve represent on an angular velocity vs. time (\(\omega\) vs. \(t\)) graph?

The slope equals instantaneous angular acceleration (\(\alpha = \frac{d\omega}{dt}\)), and the area under the curve equals angular displacement (\(\Delta\theta = \int \omega \, dt\)).

Q: What physical quantity is given by the area under an angular acceleration vs. time (\(\alpha\) vs. \(t\)) graph?

The area under an \(\alpha\) vs. \(t\) curve represents the change in angular velocity: \(\Delta\omega = \int \alpha \, dt = \omega(t_2) - \omega(t_1)\).

AP Physics C: Mechanics · AP (Advanced Placement) · 9 min read · Updated: 22 Oct 2026

Unit 5: Torque and Rotational Dynamics — Rotational Kinematics

Welcome to Rotational Kinematics! If you felt comfortable with linear kinematics back in Unit 1, you are in luck. Rotational kinematics is almost an exact mirror of straight-line motion. Instead of measuring how far an object moves along a straight track in meters, we track how far it rotates around a fixed axis in radians. Don't worry if calculus-based physics feels intimidating at times; by taking it step by step, you will see that every rule, derivative, and integral from 1D motion translates cleanly into rotational motion.

Cross-reference note: This chapter focuses purely on the description of rotational motion (angles, speeds, and accelerations). The direct geometric bridge to translational quantities is explored in Connecting Linear and Rotational Motion, while the causes of rotational motion are studied in Torque and Newton's Second Law in Rotational Form.

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1. The Core Rotational Variables

To analyze a rotating rigid body, we define three fundamental rotational variables. In AP Physics C: Mechanics, we work with the magnitudes and algebraic signs of these quantities, defining directions simply as clockwise (CW) or counterclockwise (CCW) with respect to the axis of rotation.

Angular Position and Displacement (\(\theta\) and \(\Delta\theta\))

Just as \(x\) represents position along an axis, \(\theta\) represents the angular position of a reference line on a rotating object relative to a chosen fixed direction:

Units: Always measured in radians (\(\text{rad}\)) for calculus operations. While full rotations (\(\text{rev}\)) or degrees (\(^\circ\)) exist in common language, calculus relationships like derivatives of trigonometric functions require pure radian measure.

The angular displacement is the net change in angle:

\(\Delta\theta = \theta(t_2) - \theta(t_1)\)

Angular Velocity (\(\omega\))

Angular velocity measures the rate at which angular position changes with respect to time.

The average angular velocity over a finite time interval \(\Delta t\) is:

\(\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}\)

The instantaneous angular velocity is the first time derivative of angular position:

\(\omega = \frac{d\theta}{dt}\)

Units: Radians per second (\(\text{rad/s}\)).

Angular Acceleration (\(\alpha\))

Angular acceleration measures how rapidly the angular velocity is changing over time.

The average angular acceleration over a time interval \(\Delta t\) is:

\(\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}\)

The instantaneous angular acceleration is the first time derivative of angular velocity, or the second time derivative of angular position:

\(\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}\)

Units: Radians per second squared (\(\text{rad/s}^2\)).

Sign Conventions and Boundaries

In AP Physics C: Mechanics, vector directions of rotation (such as the right-hand rule vector pointing along the axle) are not assessed. Instead:

• Rotation direction is described strictly as counterclockwise (CCW) or clockwise (CW).

• Conventionally, CCW is treated as positive (\(+\)) and CW as negative (\(-\)), though you may define either direction as positive as long as you stay consistent throughout the problem.

• Speeding up vs. Slowing down: Just like linear motion, if \(\omega\) and \(\alpha\) share the same sign, the object is speeding up. If \(\omega\) and \(\alpha\) have opposite signs, the object is slowing down.

Key Takeaway: \(\theta \xrightarrow{\frac{d}{dt}} \omega \xrightarrow{\frac{d}{dt}} \alpha\). Derivatives take you from position to acceleration; integrals take you back.

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2. Linear vs. Rotational Analogy Table

Here is an easy memory aid: whenever you see a rotational problem, picture its linear twin.

• Linear Position: \(x\) \(\longleftrightarrow\) Angular Position: \(\theta\)
• Linear Velocity: \(v = \frac{dx}{dt}\) \(\longleftrightarrow\) Angular Velocity: \(\omega = \frac{d\theta}{dt}\)
• Linear Acceleration: \(a = \frac{dv}{dt}\) \(\longleftrightarrow\) Angular Acceleration: \(\alpha = \frac{d\omega}{dt}\)
• Displacement: \(\Delta x = \int v \, dt\) \(\longleftrightarrow\) Angular Displacement: \(\Delta\theta = \int \omega \, dt\)
• Velocity Change: \(\Delta v = \int a \, dt\) \(\longleftrightarrow\) Angular Velocity Change: \(\Delta\omega = \int \alpha \, dt\)

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3. Calculus-Based Kinematics: Non-Constant Acceleration

In many AP Physics C scenarios, the angular acceleration is not constant; it may vary with time, such as \(\alpha(t) = b t^2\). When acceleration is variable, constant acceleration equations cannot be used. You must use integration and differentiation.

Going from \(\alpha(t)\) to \(\omega(t)\) and \(\theta(t)\)

Since \(\alpha = \frac{d\omega}{dt}\), we integrate with respect to time:

\(d\omega = \alpha(t) \, dt \implies \int_{\omega_0}^{\omega(t)} d\omega = \int_{0}^{t} \alpha(t') \, dt'\)

\(\omega(t) = \omega_0 + \int_{0}^{t} \alpha(t') \, dt'\)

Next, since \(\omega = \frac{d\theta}{dt}\), integrate once more to find position:

\(d\theta = \omega(t) \, dt \implies \int_{\theta_0}^{\theta(t)} d\theta = \int_{0}^{t} \omega(t') \, dt'\)

\(\theta(t) = \theta_0 + \int_{0}^{t} \omega(t') \, dt'\)

The Chain Rule Link: Eliminating Time

Occasionally, \(\alpha\) is given as a function of angle \(\theta\) rather than time \(t\). Use the calculus chain rule:

\(\alpha = \frac{d\omega}{dt} = \frac{d\omega}{d\theta} \frac{d\theta}{dt} = \omega \frac{d\omega}{d\theta}\)

Rearranging and separating variables yields:

\(\alpha \, d\theta = \omega \, d\omega \implies \int_{\theta_0}^{\theta} \alpha \, d\theta = \int_{\omega_0}^{\omega} \omega \, d\omega\)

Step-by-Step Example (Calculus Analysis):
A flywheel rotates with an angular acceleration given by \(\alpha(t) = 6t - 4\) (in \(\text{rad/s}^2\)). At \(t = 0\), its initial angular velocity is \(\omega_0 = 2\text{ rad/s}\) and \(\theta_0 = 0\text{ rad}\). Find \(\omega(t)\) and \(\theta(t)\).

1. Integrate \(\alpha(t)\) to get \(\omega(t)\):
\(\omega(t) = \omega_0 + \int_{0}^{t} (6t' - 4) \, dt' = 2 + \left[ 3(t')^2 - 4t' \right]_{0}^{t} = 3t^2 - 4t + 2\)

2. Integrate \(\omega(t)\) to get \(\theta(t)\):
\(\theta(t) = \theta_0 + \int_{0}^{t} (3(t')^2 - 4t' + 2) \, dt' = 0 + \left[ (t')^3 - 2(t')^2 + 2t' \right]_{0}^{t} = t^3 - 2t^2 + 2t\)

Key Takeaway: Check whether \(\alpha\) is constant before picking a method. If \(\alpha\) changes with time, set up definite integrals using initial conditions as your lower limits.

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4. Motion with Constant Angular Acceleration

When \(\alpha\) is strictly constant (\(\alpha = \text{constant}\)), evaluating the integrals above leads directly to the standard rotational kinematic formulas.

• Velocity-Time Relation:
\(\omega = \omega_0 + \alpha t\)

• Position-Time Relation:
\(\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\)

• Velocity-Displacement Relation (Time-Independent):
\(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\)

• Displacement via Average Velocity:
\(\theta - \theta_0 = \frac{1}{2}(\omega_0 + \omega)t\)

Common Pitfall Alert!

Do not plug numbers into these equations if \(\alpha\) is a function of time! A very common AP exam error is attempting to use \(\omega^2 = \omega_0^2 + 2\alpha \Delta\theta\) when \(\alpha(t)\) varies. Always look closely at the problem statement to confirm whether \(\alpha\) is fixed.

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5. Graphical Representation of Rotational Motion

On both Multiple-Choice and Free-Response questions (specifically Translation Between Representations), you will be asked to interpret or sketch graphs of \(\theta(t)\), \(\omega(t)\), and \(\alpha(t)\).

Angular Position vs. Time (\(\theta\) vs. \(t\))

• Slope: The slope of the tangent line equals the instantaneous angular velocity: \(\text{Slope} = \frac{d\theta}{dt} = \omega\).
• Curvature (Concavity): Concave up means \(\alpha > 0\); concave down means \(\alpha < 0\); zero concavity (a straight line) means \(\alpha = 0\).

Angular Velocity vs. Time (\(\omega\) vs. \(t\))

• Slope: The slope of the tangent line equals the instantaneous angular acceleration: \(\text{Slope} = \frac{d\omega}{dt} = \alpha\).
• Area under the curve: The definite integral (area between the curve and the time axis) represents the angular displacement: \(\text{Area} = \int_{t_1}^{t_2} \omega \, dt = \Delta\theta\). (Area above the time axis is positive displacement; area below is negative displacement).

Angular Acceleration vs. Time (\(\alpha\) vs. \(t\))

• Area under the curve: The area represents the change in angular velocity: \(\text{Area} = \int_{t_1}^{t_2} \alpha \, dt = \Delta\omega = \omega(t_2) - \omega(t_1)\).

Quick Summary Table for Graphs:

• \(\theta\) vs. \(t\) graph → Slope is \(\omega\)
• \(\omega\) vs. \(t\) graph → Slope is \(\alpha\), Area is \(\Delta\theta\)
• \(\alpha\) vs. \(t\) graph → Area is \(\Delta\omega\)

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6. Worked AP-Style Practice Problems

Example A: Mathematical Routine (FRQ Skill 2.A & 2.B)

A grinding wheel starts from rest at \(t = 0\) and accelerates with a constant angular acceleration of \(\alpha = 4.0\text{ rad/s}^2\) for \(5.0\text{ s}\). It then rotates at constant angular speed for \(10\text{ s}\). Finally, a braking torque brings it to rest with a constant angular acceleration in an angular displacement of \(50\text{ rad}\). Calculate the total angular displacement \(\Delta\theta_{\text{total}}\) over the entire motion.

Solution:
Divide the motion into three distinct stages:

Stage 1: Speeding up (\(t_1 = 5.0\text{ s}\), \(\omega_0 = 0\), \(\alpha_1 = 4.0\text{ rad/s}^2\))
\(\omega_1 = \omega_0 + \alpha_1 t_1 = 0 + (4.0)(5.0) = 20\text{ rad/s}\)
\(\Delta\theta_1 = \omega_0 t_1 + \frac{1}{2}\alpha_1 t_1^2 = 0 + \frac{1}{2}(4.0)(5.0)^2 = 50\text{ rad}\)

Stage 2: Constant speed (\(t_2 = 10\text{ s}\), \(\omega = 20\text{ rad/s}\), \(\alpha_2 = 0\))
\(\Delta\theta_2 = \omega t_2 = (20)(10) = 200\text{ rad}\)

Stage 3: Slowing to rest
The problem directly gives the displacement during braking:
\(\Delta\theta_3 = 50\text{ rad}\)

Total Displacement:
\(\Delta\theta_{\text{total}} = \Delta\theta_1 + \Delta\theta_2 + \Delta\theta_3 = 50\text{ rad} + 200\text{ rad} + 50\text{ rad} = 300\text{ rad}\)

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Example B: Derivation with Variable Acceleration (FRQ Skill 2.A)

A turntable rotates such that its angular velocity is described by \(\omega(t) = \omega_{\text{max}} \left(1 - e^{-kt}\right)\), where \(\omega_{\text{max}}\) and \(k\) are positive constants. The turntable is at \(\theta = 0\) when \(t = 0\). Derive symbolic expressions for:
(a) The angular acceleration \(\alpha(t)\) as a function of time.
(b) The angular position \(\theta(t)\) as a function of time.

Solution:

Part (a): Differentiate \(\omega(t)\) with respect to time
\(\alpha(t) = \frac{d\omega}{dt} = \frac{d}{dt}\left[ \omega_{\text{max}} - \omega_{\text{max}} e^{-kt} \right]\)
The derivative of the constant \(\omega_{\text{max}}\) is \(0\). Applying the chain rule to the exponential term:
\(\alpha(t) = -\omega_{\text{max}} (-k e^{-kt}) = k \omega_{\text{max}} e^{-kt}\)

Part (b): Integrate \(\omega(t)\) with respect to time
\(\theta(t) = \theta_0 + \int_{0}^{t} \omega(t') \, dt'\)
Since \(\theta_0 = 0\):
\(\theta(t) = \int_{0}^{t} \omega_{\text{max}} \left(1 - e^{-kt'}\right) \, dt' = \omega_{\text{max}} \left[ t' - \left( -\frac{1}{k} e^{-kt'} \right) \right]_{0}^{t}\)
\(\theta(t) = \omega_{\text{max}} \left[ t' + \frac{1}{k}e^{-kt'} \right]_{0}^{t}\)
Evaluate at the limits \(t\) and \(0\):
\(\theta(t) = \omega_{\text{max}} \left( \left( t + \frac{1}{k}e^{-kt} \right) - \left( 0 + \frac{1}{k}e^{0} \right) \right)\)
Because \(e^0 = 1\):
\(\theta(t) = \omega_{\text{max}} \left( t + \frac{1}{k}e^{-kt} - \frac{1}{k} \right) = \omega_{\text{max}} t - \frac{\omega_{\text{max}}}{k}\left(1 - e^{-kt}\right)\)

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7. Essential Review & Exam Checklist

• Always check units: Make sure angular values are expressed in radians, radians/sec, or radians/sec\(^2\). If an exam prompt mentions rotations or rpm (revolutions per minute), convert immediately: \(1\text{ rev} = 2\pi\text{ rad}\), and \(1\text{ rpm} = \frac{2\pi}{60}\text{ rad/s}\).
• Verify whether \(\alpha\) is constant: If \(\alpha\) is constant, use standard rotational kinematic formulas. If \(\alpha\) depends on time or angle, use calculus: \(\omega = \int \alpha \, dt\) or \(\alpha = \omega \frac{d\omega}{d\theta}\).
• Graph slopes and areas: Remember that slope on a \(\theta\)-\(t\) graph is \(\omega\), slope on an \(\omega\)-\(t\) graph is \(\alpha\), area under \(\omega\)-\(t\) is \(\Delta\theta\), and area under \(\alpha\)-\(t\) is \(\Delta\omega\).
• Direction conventions: Describe direction as clockwise or counterclockwise. Never worry about 3D vector arrows for \(\vec{\omega}\) or \(\vec{\alpha}\) on this exam; sign and rotation sense (CW/CCW) are all that is expected.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is angular displacement, and what standard unit must be used for calculus-based rotational kinematics?

Angular displacement is the net change in angular position, \(\Delta\theta = \theta(t_2) - \theta(t_1)\), and it must always be expressed in radians (\(\text{rad}\)).

How are average and instantaneous angular velocity defined mathematically?

Average angular velocity is \(\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}\), while instantaneous angular velocity is the first time derivative of angular position: \(\omega = \frac{d\theta}{dt}\).

How is instantaneous angular acceleration defined in terms of angular velocity and angular position?

\(\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}\), measured in \(\text{rad/s}^2\).

How do the relative signs of angular velocity \(\omega\) and angular acceleration \(\alpha\) determine whether an object is speeding up or slowing down?

If \(\omega\) and \(\alpha\) share the same sign, the object is speeding up; if they have opposite signs, the object is slowing down.

How are angular velocity and position determined when angular acceleration \(\alpha(t)\) is not constant?

By integrating with respect to time: \(\omega(t) = \omega_0 + \int_{0}^{t} \alpha(t') \, dt'\) and \(\theta(t) = \theta_0 + \int_{0}^{t} \omega(t') \, dt'\).

How can angular acceleration be related to angular velocity and angular position when time is not explicitly involved?

Using the chain rule: \(\alpha = \omega \frac{d\omega}{d\theta}\), which separates into \(\alpha \, d\theta = \omega \, d\omega\).

What kinematic equations describe rotational motion when angular acceleration \(\alpha\) is constant?

\(\omega = \omega_0 + \alpha t\), \(\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\), and \(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\).

What physical quantities are represented by the slope and concavity of an angular position vs. time (\(\theta\) vs. \(t\)) graph?

The slope equals instantaneous angular velocity (\(\omega = \frac{d\theta}{dt}\)), and the concavity indicates the sign of angular acceleration (\(\alpha\)).

What do the slope and area under the curve represent on an angular velocity vs. time (\(\omega\) vs. \(t\)) graph?

The slope equals instantaneous angular acceleration (\(\alpha = \frac{d\omega}{dt}\)), and the area under the curve equals angular displacement (\(\Delta\theta = \int \omega \, dt\)).

What physical quantity is given by the area under an angular acceleration vs. time (\(\alpha\) vs. \(t\)) graph?

The area under an \(\alpha\) vs. \(t\) curve represents the change in angular velocity: \(\Delta\omega = \int \alpha \, dt = \omega(t_2) - \omega(t_1)\).

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