Rotational Kinematics

Q: How can I study Rotational Kinematics effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Q: Is this Rotational Kinematics study guide free?

Yes — all study notes, flashcards, and practice problems for Rotational Kinematics on Study Mondo are free to access. No account is needed.

Q: What course covers Rotational Kinematics?

Rotational Kinematics is part of the AP Physics 1 course on Study Mondo, specifically in the Torque & Rotational Motion section. You can explore the full course for more related topics and practice resources.

Q: Are there practice problems for Rotational Kinematics?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Angular displacement, velocity, acceleration, and rotational kinematic equations

Written and reviewed by the Study Mondo Education TeamLast updated February 14, 2026

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🌀 Rotational Kinematics

Angular Quantities

Just as linear motion has position, velocity, and acceleration, rotational motion has corresponding angular quantities.

Angular Position $\theta$

Angle of rotation from reference line:

Units: radians (rad)
$2\pi$ rad = 360° = one full rotation
Conversion: $\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$

Angular Displacement $\Delta\theta$

Change in angular position: $\Delta\theta = \theta_f - \theta_i$

Units: radians (rad)

Positive: counterclockwise rotation Negative: clockwise rotation

Angular Velocity $\omega$

Rate of change of angular position:

Average: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$

Instantaneous: $\omega = \frac{d\theta}{dt}$

Units: rad/s (radians per second)

Angular Acceleration $\alpha$

Rate of change of angular velocity:

Average: $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$

Instantaneous: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$

Units: rad/s² (radians per second squared)

Analogy with Linear Motion

Linear Motion	Rotational Motion
Position $x$	Angular position $\theta$
Velocity $v$	Angular velocity $\omega$
Acceleration $a$	Angular acceleration $\alpha$
Mass $m$	Moment of inertia $I$
Force $F$	Torque $\tau$

Relationship Between Linear and Angular

For a point at distance $r$ from axis of rotation:

Arc Length

$s = r\theta$ (where $\theta$ is in radians)

Linear Velocity

$v = r\omega$

Tangential velocity - velocity tangent to circular path

Tangential Acceleration

$a_t = r\alpha$

Component of acceleration tangent to circle (changes speed)

Centripetal Acceleration

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

Component of acceleration toward center (changes direction)

Rotational Kinematic Equations

For constant angular acceleration $\alpha$ :

The Big Four Equations

1. Angular velocity: $\omega_f = \omega_i + \alpha t$

2. Angular displacement: $\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$

3. Velocity-displacement: $\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$

4. Average velocity: $\theta = \theta_i + \frac{1}{2}(\omega_i + \omega_f)t$

💡 These are EXACTLY analogous to linear kinematic equations! Just replace $x \to \theta$ , $v \to \omega$ , $a \to \alpha$ .

Comparison: Linear vs. Rotational Kinematics

Linear (constant $a$ )	Rotational (constant $\alpha$ )
$v_f = v_i + at$	$\omega_f = \omega_i + \alpha t$
$x = x_i + v_i t + \frac{1}{2}at^2$	$\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$
$v_f^2 = v_i^2 + 2a\Delta x$	$\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$
$x = x_i + \frac{1}{2}(v_i + v_f)t$	$\theta = \theta_i + \frac{1}{2}(\omega_i + \omega_f)t$

Period and Frequency

For uniform circular motion (constant $\omega$ ):

Period $T$

Time for one complete rotation: $T = \frac{2\pi}{\omega}$

Units: seconds

Frequency $f$

Rotations per second: $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Units: Hz (hertz) = rev/s

Relationships

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

Rolling Motion

For an object rolling without slipping:

Constraint condition: $v_{cm} = r\omega$

where:

$v_{cm}$ = velocity of center of mass
$r$ = radius
$\omega$ = angular velocity

No slipping means:

Point of contact is instantaneously at rest
Distance traveled = arc length: $s = r\theta$

Problem-Solving Strategy

For Rotational Kinematics:

Identify knowns: $\theta_i$ , $\omega_i$ , $\omega_f$ , $\alpha$ , $t$ , $\Delta\theta$
Identify unknown: What are you solving for?
Choose equation: Pick the one with known quantities and unknown
Solve algebraically
Check units: Should be rad, rad/s, or rad/s²
Check reasonableness: Does answer make sense?

Tip: If you know linear quantities ( $v$ , $a_t$ , $s$ ), convert using:

$\omega = v/r$
$\alpha = a_t/r$
$\theta = s/r$

⚠️ Common Mistakes

Mistake 1: Degrees vs. Radians

MUST use radians in equations! Convert degrees to radians first.

Mistake 2: Confusing $a_t$ and $a_c$

$a_t = r\alpha$ : tangential (changes speed)
$a_c = \omega^2 r$ : centripetal (changes direction)
Total: $a = \sqrt{a_t^2 + a_c^2}$

Mistake 3: Wrong Sign for $\alpha$

Speeding up in positive direction: $\alpha > 0$
Slowing down in positive direction: $\alpha < 0$

Mistake 4: Forgetting Initial Conditions

$\omega_i$ and $\theta_i$ are not always zero!

Special Cases

Starting from Rest

$\omega_i = 0$ :

$\omega_f = \alpha t$
$\theta = \frac{1}{2}\alpha t^2$
$\omega_f^2 = 2\alpha\theta$

Uniform Rotation

$\alpha = 0$ (constant $\omega$ ):

$\omega_f = \omega_i = \omega$
$\theta = \omega t$
Period: $T = \frac{2\pi}{\omega}$

Coming to Rest

$\omega_f = 0$ :

$0 = \omega_i + \alpha t$ → $t = -\frac{\omega_i}{\alpha}$
$\omega_i^2 = -2\alpha\Delta\theta$ → $\Delta\theta = -\frac{\omega_i^2}{2\alpha}$

Applications

Wheels and Gears

Angular velocity determines linear speed
Gear ratios change angular velocities
$v = r\omega$ connects the two

Rotating Machinery

Turbines, engines, motors
Angular acceleration during startup
Constant $\omega$ during normal operation

Sports

Figure skating spins (angular velocity)
Gymnastics rotations
Diving somersaults

Astronomy

Planetary rotation (Earth: $T \approx 24$ hr)
Orbital motion
Galaxy rotation

Key Formulas Summary

Quantity	Formula	Units
Angular velocity	$\omega = \frac{d\theta}{dt}$	rad/s
Angular acceleration	$\alpha = \frac{d\omega}{dt}$	rad/s²
Linear velocity	$v = r\omega$	m/s
Tangential acceleration	$a_t = r\alpha$	m/s²
Centripetal acceleration	$a_c = \omega^2 r$	m/s²
Period	$T = \frac{2\pi}{\omega}$	s
Frequency	$f = \frac{\omega}{2\pi}$	Hz

Kinematic equations (constant $\alpha$ ):

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

📚 Practice Problems

1Problem 1easy

❓ Question:

A wheel starts from rest and accelerates uniformly at 2 rad/s² for 5 seconds. Find: (a) the final angular velocity, (b) the angular displacement during this time, and (c) the number of revolutions completed.

💡 Show Solution

Given Information:

Initial angular velocity: $\omega_i = 0$ rad/s (starts from rest)
Angular acceleration: $\alpha = 2$ rad/s²
Time: $t = 5$ s

(a) Find final angular velocity

Step 1: Use first kinematic equation

$\omega_f = \omega_i + \alpha t$

$\omega_f = 0 + (2)(5)$

$\omega_f = 10 \text{ rad/s}$

Answer (a): Final angular velocity = 10 rad/s

(b) Find angular displacement

Step 2: Use displacement equation

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

$\theta = 0(5) + \frac{1}{2}(2)(5)^2$

$\theta = 0 + \frac{1}{2}(2)(25)$

$\theta = 25 \text{ rad}$

Alternative: Use average velocity

$\theta = \frac{1}{2}(\omega_i + \omega_f)t = \frac{1}{2}(0 + 10)(5) = 25 \text{ rad}$

Both methods agree! ✓

Answer (b): Angular displacement = 25 rad

(c) Find number of revolutions

Step 3: Convert radians to revolutions

$\text{Revolutions} = \frac{\theta}{2\pi} = \frac{25}{2\pi}$

$\text{Revolutions} = \frac{25}{6.28} = 3.98 \text{ rev}$

Answer (c): Number of revolutions ≈ 4.0 revolutions

Summary: The wheel accelerates from rest to 10 rad/s, turning through 25 radians (about 4 complete rotations) in 5 seconds.

2Problem 2medium

❓ Question:

A wheel starts from rest and accelerates uniformly to 120 rpm in 8.0 seconds. (a) What is the angular acceleration in rad/s²? (b) How many revolutions does it make during this time? (c) What is the final angular velocity in rad/s?

💡 Show Solution

Solution:

Given: ω₀ = 0, ω_f = 120 rpm, t = 8.0 s

(a) Angular acceleration: Convert to rad/s: ω_f = 120 rev/min × (2π rad/rev) × (1 min/60 s) = 4π rad/s

α = (ω_f - ω₀)/t = (4π - 0)/8.0 = 1.57 rad/s² or π/2 rad/s²

(b) Number of revolutions: θ = ω₀t + ½αt² = 0 + ½(π/2)(8.0)² θ = ½(π/2)(64) = 16π rad

Convert to revolutions: 16π rad × (1 rev/2π rad) = 8.0 rev

Or 120 rpm as given.

3Problem 3medium

❓ Question:

A car tire with radius 0.3 m is rotating at 10 rev/s. The car brakes, and the tire comes to rest in 4 seconds with constant angular acceleration. Find: (a) the angular acceleration, and (b) the linear distance traveled during braking.

💡 Show Solution

Given Information:

Radius: $r = 0.3$ m
Initial angular velocity: $\omega_i = 10$ rev/s
Final angular velocity: $\omega_f = 0$ rad/s (comes to rest)
Time: $t = 4$ s

Step 0: Convert units

$\omega_i = 10 \text{ rev/s} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} = 20\pi \text{ rad/s} \approx 62.8 \text{ rad/s}$

(a) Find angular acceleration

Step 1: Use first kinematic equation

$\omega_f = \omega_i + \alpha t$

$0 = 62.8 + \alpha(4)$

$4\alpha = -62.8$

$\alpha = -15.7 \text{ rad/s}^2$

Answer (a): Angular acceleration = −15.7 rad/s² (negative because it's slowing down)

(b) Find linear distance traveled

Step 2: Find angular displacement

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

$\theta = (62.8)(4) + \frac{1}{2}(-15.7)(4)^2$

$\theta = 251.2 + \frac{1}{2}(-15.7)(16)$

$\theta = 251.2 - 125.6$

$\theta = 125.6 \text{ rad}$

Alternative: Use average velocity

$\theta = \frac{1}{2}(\omega_i + \omega_f)t = \frac{1}{2}(62.8 + 0)(4) = 125.6 \text{ rad}$

Both methods agree! ✓

Step 3: Convert to linear distance

$s = r\theta = (0.3)(125.6)$

$s = 37.7 \text{ m}$

Answer (b): Linear distance traveled = 37.7 m (about 38 meters)

Check: This is reasonable for a car braking from moderate speed over 4 seconds.

Note: Number of revolutions = $\frac{125.6}{2\pi} \approx 20$ revolutions during braking.

4Problem 4medium

❓ Question:

A merry-go-round with radius 2.0 m rotates at 0.50 rev/s. A child stands at the outer edge. (a) What is the child's angular velocity? (b) What is the child's tangential (linear) speed? (c) What is the child's centripetal acceleration?

💡 Show Solution

Solution:

Given: r = 2.0 m, f = 0.50 rev/s

(a) Angular velocity: ω = 2πf = 2π(0.50) = π rad/s or 3.14 rad/s

(b) Tangential speed: v = rω = 2.0(π) = 2π m/s or 6.28 m/s

Or: a_c = rω² = 2.0(π)² = 2π² = 19.7 m/s² ✓

5Problem 5hard

❓ Question:

A disk of radius 0.5 m starts from rest and rotates with constant angular acceleration. After 10 seconds, a point on the rim of the disk has a tangential speed of 15 m/s. Find: (a) the angular acceleration, (b) the angular displacement in those 10 seconds, and (c) the magnitude of the total acceleration of a point on the rim at t = 10 s.

💡 Show Solution

Given Information:

Radius: $r = 0.5$ m
Initial angular velocity: $\omega_i = 0$ rad/s (starts from rest)
Time: $t = 10$ s
Final tangential speed: $v_f = 15$ m/s

(a) Find angular acceleration

Step 1: Find final angular velocity

$v = r\omega$

$\omega_f = \frac{v_f}{r} = \frac{15}{0.5} = 30 \text{ rad/s}$

Step 2: Calculate angular acceleration

$\omega_f = \omega_i + \alpha t$

$30 = 0 + \alpha(10)$

$\alpha = 3 \text{ rad/s}^2$

Answer (a): Angular acceleration = 3 rad/s²

(b) Find angular displacement

Step 3: Use displacement equation

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

$\theta = 0(10) + \frac{1}{2}(3)(10)^2$

$\theta = \frac{1}{2}(3)(100)$

$\theta = 150 \text{ rad}$

Alternative: Use average velocity

$\theta = \frac{1}{2}(\omega_i + \omega_f)t = \frac{1}{2}(0 + 30)(10) = 150 \text{ rad}$

Answer (b): Angular displacement = 150 rad

(This is $\frac{150}{2\pi} \approx 23.9$ revolutions)

(c) Find total acceleration at t = 10 s

Step 4: Calculate tangential acceleration

$a_t = r\alpha = (0.5)(3) = 1.5 \text{ m/s}^2$

Step 5: Calculate centripetal acceleration

$a_c = \omega^2 r = (30)^2(0.5)$

$a_c = 900(0.5) = 450 \text{ m/s}^2$

Step 6: Find magnitude of total acceleration

Tangential and centripetal accelerations are perpendicular:

$a_{total} = \sqrt{a_t^2 + a_c^2}$

$a_{total} = \sqrt{(1.5)^2 + (450)^2}$

$a_{total} = \sqrt{2.25 + 202,500}$

$a_{total} = \sqrt{202,502.25}$

$a_{total} \approx 450 \text{ m/s}^2$

Answer (c): Total acceleration ≈ 450 m/s²

Note: The centripetal acceleration (450 m/s²) is MUCH larger than the tangential acceleration (1.5 m/s²), so the total acceleration is essentially just the centripetal acceleration. This makes sense at high rotational speeds!

Direction: The total acceleration points slightly inward from the purely radial direction (mostly toward center, with small tangential component).

Explain using:

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❓ Frequently Asked Questions

What is Rotational Kinematics?▾

Angular displacement, velocity, acceleration, and rotational kinematic equations

How can I study Rotational Kinematics effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Rotational Kinematics study guide free?▾

Yes — all study notes, flashcards, and practice problems for Rotational Kinematics on Study Mondo are free to access. No account is needed.

What course covers Rotational Kinematics?▾

Rotational Kinematics is part of the AP Physics 1 course on Study Mondo, specifically in the Torque & Rotational Motion section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Rotational Kinematics?▾

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Rotational Kinematics

Try the Interactive Version!

🌀 Rotational Kinematics

Angular Quantities

Angular Position θ\thetaθ

Angular Displacement Δθ\Delta\thetaΔθ

Angular Velocity ω\omegaω

Angular Acceleration α\alphaα

Analogy with Linear Motion

Relationship Between Linear and Angular

Arc Length

Linear Velocity

Tangential Acceleration

Centripetal Acceleration

Rotational Kinematic Equations

The Big Four Equations

Comparison: Linear vs. Rotational Kinematics

Period and Frequency

Period TTT

Frequency fff

Relationships

Rolling Motion

Problem-Solving Strategy

For Rotational Kinematics:

⚠️ Common Mistakes

Mistake 1: Degrees vs. Radians

Mistake 2: Confusing ata_tat​ and aca_cac​

Mistake 3: Wrong Sign for α\alphaα

Mistake 4: Forgetting Initial Conditions

Special Cases

Starting from Rest

Uniform Rotation

Coming to Rest

Applications

Wheels and Gears

Rotating Machinery

Sports

Astronomy

Key Formulas Summary

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2medium

❓ Question:

3Problem 3medium

❓ Question:

4Problem 4medium

❓ Question:

5Problem 5hard

❓ Question:

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📌 Related Topics in Torque & Rotational Motion

❓ Frequently Asked Questions

Angular Position $\theta$

Angular Displacement $\Delta\theta$

Angular Velocity $\omega$

Angular Acceleration $\alpha$

Period $T$

Frequency $f$

Mistake 2: Confusing $a_t$ and $a_c$

Mistake 3: Wrong Sign for $\alpha$