🌀 Physics Calculator & Formula Reference
Rotational Kinematics Calculator
This calculator solves all five rotational kinematics equations and all five angular kinematic equations for any unknown variable, converts between angular and linear motion, calculates rotational dynamics (τ = Iα), and includes a complete formula reference table with worked examples for every equation.
🧮 Rotational Kinematics 5-in-1 Solver
Click each variable to mark it as KNOWN (blue) or leave blank for UNKNOWN (green). Enter values for known variables. The calculator selects the right equation automatically.
ω₀
Initial ω
KNOWN
ω
Final ω
—
α
Angular Accel
KNOWN
θ
Displacement
—
t
Time
KNOWN
ω = ω₀ + αt
Equation 1 — solving for ω (final angular velocity)
Final Angular Velocity ω
—
rad/s
📋 Step-by-Step Working
⚖ Rotational Dynamics Calculator — τ = Iα
τ = Iα
τ = torque (N·m) | I = moment of inertia (kg·m²) | α = angular acceleration (rad/s²)
Don't know I? Calculate moment of inertia for your shape →
Angular Acceleration α
—
rad/s²
📋 Step-by-Step
⇆ Angular & Linear Motion Converter
v = ωr | aₙ = αr | aₒ = ω²r
📋 Working
🔄 Period & Frequency Calculator
T = 2π/ω | f = 1/T | ω = 2πf
Enter any one value to calculate the other two.
This rotational kinematics calculator solves the five rotational kinematics equations and all angular kinematic equations for any unknown variable, converts between angular and linear motion quantities, calculates rotational dynamics using τ = Iα, and includes a complete formula reference table with worked examples for every equation. Whether you need to find angular velocity, angular displacement, or angular acceleration, the solver picks the right equation automatically.
Linear Motion vs Rotational Motion — Complete Analogy
Every linear motion equation has an exact rotational equivalent. The mathematical structure is identical — only the variable names change. Replace x→θ, v→ω, a→α, m→I, F→τ and every linear formula becomes a rotational formula. The radius r connects the two worlds: v = ωr, a = αr, τ = F×r.
| Quantity | 💧 Linear Motion | Sym | 🌀 Rotational Motion | Relationship |
|---|---|---|---|---|
| Position / Angle | Distance x (m) | ⇄ | Angular displacement θ (rad) | θ = x/r |
| Velocity | Linear velocity v (m/s) | ⇄ | Angular velocity ω (rad/s) | ω = v/r |
| Acceleration | Linear acceleration a (m/s²) | ⇄ | Angular acceleration α (rad/s²) | α = a/r |
| Inertia | Mass m (kg) | ⇄ | Moment of inertia I (kg·m²) | — |
| Force / Torque | Force F (N) | ⇄ | Torque τ (N·m) | τ = F×r |
| Newton's 2nd Law | F = ma | ⇄ | τ = Iα | Same form |
| Kinetic Energy | KE = ½mv² | ⇄ | KE = ½Iω² | Same form |
| Momentum | p = mv (kg·m/s) | ⇄ | L = Iω (kg·m²/s) | Same form |
| Power | P = Fv (W) | ⇄ | P = τω (W) | Same form |
| Work | W = F·x (J) | ⇄ | W = τ·θ (J) | Same form |
Key insight: Every linear motion equation has an exact rotational equivalent. Just replace x→θ, v→ω, a→α, m→I, F→τ. The radius r is the bridge between the two: v = ωr, a = αr, τ = F×r.
The Five Kinematic Equations — Linear vs Rotational
| # | 💧 Linear Kinematic Equation | 🌀 Rotational Kinematic Equation | Variable eliminated |
|---|---|---|---|
| 1 | v = v₀ + at | ω = ω₀ + αt | x / θ |
| 2 | x = v₀t + ½at² | θ = ω₀t + ½αt² | v / ω |
| 3 | v² = v₀² + 2ax | ω² = ω₀² + 2αθ | t |
| 4 | x = (v₀+v)t/2 | θ = (ω₀+ω)t/2 | a / α |
| 5 | x = vt - ½at² | θ = ωt - ½αt² | v₀ / ω₀ |
Rotational Kinematics Formulas — All Five Equations Explained
These five rotational kinematics formulas are the foundation of all rotational motion problems with constant angular acceleration. Each equation eliminates one variable, giving you flexibility to solve from whatever three quantities you know.
ω = ω₀ + αt
Equation 1 — No θ | Use when: know ω₀, α, t | Solve for: ω
Variables: ω (final, rad/s) · ω₀ (initial, rad/s) · α (angular accel, rad/s²) · t (time, s)
Example: ω₀ = 50 rad/s, α = -2 rad/s², t = 3 s → find ω
- Apply ω = ω₀ + αt
- ω = 50 + (-2) × 3 = 50 - 6 = 44 rad/s
- Convert: 44 rad/s = 7.003 rev/s = 420.2 RPM = 2521 °/s
θ = ω₀t + ½αt²
Equation 2 — No ω | Use when: know ω₀, α, t | Solve for: θ
Variables: θ (angular displacement, rad) · ω₀ · α · t
Example: ω₀ = 0, α = 3 rad/s², t = 4 s → find θ
- θ = ω₀t + ½αt² = 0×4 + ½×3×4²
- θ = 0 + ½×3×16 = 0 + 24 = 24 radians
- Convert: 24 rad = 3.820 revolutions = 1375 degrees
ω² = ω₀² + 2αθ
Equation 3 — No t | Use when: know ω₀, α, θ (no time) | Solve for: ω or θ
Variables: ω · ω₀ · α · θ — eliminates time t
Example: ω₀ = 10 rad/s, ω = 30 rad/s, α = 2 rad/s² → find θ
- 30² = 10² + 2×2×θ
- 900 = 100 + 4θ
- θ = 800/4 = 200 rad = 31.83 revolutions
θ = (ω₀ + ω)t / 2
Equation 4 — No α | Use when: know ω₀, ω, t (no acceleration) | Solve for: θ
Variables: θ · ω₀ · ω · t — average angular velocity method
Example: ω₀ = 10 rad/s, ω = 20 rad/s, t = 5 s → find θ
- θ = (ω₀ + ω)t/2 = (10 + 20) × 5 / 2
- θ = 30 × 5 / 2 = 150/2 = 75 radians = 11.94 revolutions
θ = ωt − ½αt²
Equation 5 — No ω₀ | Use when: know final ω, α, t | Solve for: θ
Variables: θ · ω (final) · α · t — alternative form using final angular velocity
Example: ω = 30 rad/s, α = -2 rad/s², t = 5 s → find θ
- θ = ωt - ½αt² = 30×5 - ½×(-2)×25
- θ = 150 - (-25) = 150 + 25 = 175 radians
Rotational Dynamics — τ = Iα Formula and Applications
Newton's second law for rotation states that the net torque equals the moment of inertia times the angular acceleration. Just as F = ma governs linear motion, τ = Iα governs rotational motion. Torque τ plays the same role as force F; moment of inertia I plays the same role as mass m.
τ = Iα
τ
Torque (N·m)
= F × r
Torque (N·m)
= F × r
I
Moment of Inertia
(kg·m²)
Moment of Inertia
(kg·m²)
α
Angular Acceleration
(rad/s²)
Angular Acceleration
(rad/s²)
Need moment of inertia I? Use our Moment of Inertia Calculator for all standard shapes →
Example 1 — Find α
- Grinding wheel: I = 0.5 kg·m²
- Motor torque: τ = 10 N·m
- τ = Iα → α = τ/I
- α = 10/0.5 = 20 rad/s²
Example 2 — Find τ
- Rotating disk: I = 2 kg·m²
- Need: α = 5 rad/s²
- τ = Iα = 2 × 5
- τ = 10 N·m
Example 3 — Force applied
- F = 50 N at r = 0.2 m
- τ = F×r = 50×0.2 = 10 N·m
- I = 1 kg·m²
- α = τ/I = 10/1 = 10 rad/s²
Converting Between Angular and Linear Motion
For any point on a rotating object at distance r from the rotation axis, the angular and linear quantities are linked by the radius r. The formula v = ωr is the most important connection — it converts between angular velocity and tangential (linear) speed.
v = ωr
Tangential speed (m/s)
aₙ = αr
Tangential acceleration (m/s²)
aₒ = ω²r = v²/r
Centripetal acceleration (m/s²)
Fₒ = mω²r = mv²/r
Centripetal force (N)
Example 1 — Point on spinning disk at r = 0.3 m, ω = 20 rad/s
- v = ωr = 20 × 0.3 = 6 m/s
- aₒ = ω²r = 20² × 0.3 = 400 × 0.3 = 120 m/s²
Example 2 — Centrifuge at 10,000 RPM, r = 0.05 m
- ω = 10,000 × 2π/60 = 1047.2 rad/s
- aₒ = ω²r = 1047.2² × 0.05 = 1,096,638 × 0.05 = 54,832 m/s² ˜ 5,590 g's
Example 3 — String breaks at 500 N, m = 0.5 kg, r = 1 m
- Fₒ = mω²r ? 500 = 0.5 × ω² × 1
- ω² = 1000 ? ω = √1000 = 31.62 rad/s = 301.9 RPM
Period, Frequency, and Angular Velocity — Rotational Rate Formulas
The period T, frequency f, and angular velocity ω all describe rotation rate and are all equivalent — just different ways of expressing how fast something is spinning.
T = 2π/ω = 1/f
Period (seconds per revolution)
f = ω/(2π) = 1/T
Frequency (Hz = rev/s)
ω = 2πf = 2π/T
Angular velocity (rad/s)
ω = RPM × 2π/60
Convert RPM to rad/s
Example 1 — Washing machine at 1200 RPM
- f = 1200/60 = 20 Hz
- T = 1/20 = 0.05 s
- ω = 2π × 20 = 125.66 rad/s
Example 2 — Earth orbit T = 365.25 days
- T = 365.25 × 86400 = 31,557,600 s
- ω = 2π/T = 6.2832/31,557,600 = 1.991 × 10-7 rad/s
- f = 1/T = 3.170 × 10-8 Hz
Example 3 — ω = 50 rad/s
- f = 50/(2π) = 7.958 Hz
- T = 1/7.958 = 0.1257 s = 125.7 ms
- RPM = 50 × 60/(2π) = 477.5 RPM
Worked Examples
1. Find final angular velocity using ω = ω₀ + αt
- A flywheel starts at ω₀ = 2 rad/s, accelerates at α = 3 rad/s² for t = 4 s.
- Apply equation 1: ω = ω₀ + αt = 2 + 3×4 = 2 + 12 = 14 rad/s
- Convert: 14 rad/s = 2.228 rev/s = 133.7 RPM = 802 °/s
2. Find angular displacement using θ = ω₀t + ½αt²
- Same flywheel as above: ω₀ = 2 rad/s, α = 3 rad/s², t = 4 s.
- θ = 2×4 + ½×3×4² = 8 + ½×3×16 = 8 + 24 = 32 radians
- 32 rad = 5.093 revolutions = 1833 degrees
3. Find angular acceleration using ω² = ω₀² + 2αθ
- Disk speeds from ω₀ = 5 to ω = 15 rad/s over θ = 40 rad.
- 15² = 5² + 2α×40 ? 225 = 25 + 80α ? 80α = 200
- α = 200/80 = 2.5 rad/s²
4. Find angular velocity from tangential velocity and radius
- A point on a wheel moves at v = 12 m/s; wheel radius r = 0.4 m.
- v = ωr ? ω = v/r = 12/0.4 = 30 rad/s
- Convert: 30 rad/s = 286.5 RPM = 4.775 rev/s
5. Calculate centripetal acceleration for a rotating object
- Object at r = 0.5 m rotates at ω = 50 rad/s.
- aₒ = ω²r = 50² × 0.5 = 2500 × 0.5 = 1250 m/s²
- This equals approximately 127.4 g's (where g = 9.81 m/s²).
6. Convert RPM to rad/s
- Convert 1200 RPM to rad/s.
- ω = RPM × 2π/60 = 1200 × 2π/60 = 1200 × 0.10472
- ω = 125.66 rad/s = 20 rev/s = 20 Hz
7. Find torque for a given angular acceleration
- A motor must achieve α = 8 rad/s² on a shaft with I = 0.25 kg·m².
- τ = Iα = 0.25 × 8 = 2 N·m
- If the motor arm is r = 0.1 m, required force F = τ/r = 2/0.1 = 20 N.
8. Calculate period and frequency from angular velocity
- A turbine spins at ω = 200 rad/s.
- f = ω/(2π) = 200/6.2832 = 31.83 Hz
- T = 1/f = 1/31.83 = 0.03142 s = 31.42 ms
- RPM = f × 60 = 1909.9 RPM
9. Find final ω without knowing time (using ω² = ω₀² + 2αθ)
- Wheel starts at ω₀ = 10 rad/s, α = 4 rad/s², rotates θ = 50 rad.
- ω² = 10² + 2×4×50 = 100 + 400 = 500
- ω = √500 = 22.36 rad/s = 213.6 RPM
10. Convert between linear and rotational kinetic energy
- A solid disk (m = 5 kg, R = 0.2 m) spins at ω = 30 rad/s. I = ½mR² = ½×5×0.04 = 0.1 kg·m².
- KE₁₂ₗ = ½Iω² = ½×0.1×900 = 45 J
- At the rim, v = ωR = 30×0.2 = 6 m/s. KEₖ₍ₙ = ½mv² = ½×5×36 = 90 J (rim point only).
Frequently Asked Questions
What are the rotational kinematics equations? ▼
There are five rotational kinematic equations for constant angular acceleration: (1) ω = ω₀ + αt, (2) θ = ω₀t + ½αt², (3) ω² = ω₀² + 2αθ, (4) θ = (ω₀+ω)t/2, (5) θ = ωt - ½αt². Each relates a different subset of the five variables: θ, ω₀, ω, α, t.
What is angular acceleration? ▼
Angular acceleration (α) is the rate of change of angular velocity with respect to time, measured in rad/s². It is the rotational analog of linear acceleration. When a spinning object speeds up or slows down, it has angular acceleration. It relates to torque and moment of inertia by τ = Iα.
How do you find angular velocity? ▼
Angular velocity (ω) can be found from: ω = ω₀ + αt (knowing initial velocity, acceleration, time); ω = √(ω₀² + 2αθ) (knowing initial velocity, acceleration, displacement); ω = 2πf (from frequency); ω = v/r (from linear velocity at radius r). It is measured in rad/s.
What is the relationship between linear and rotational motion? ▼
Every linear quantity has a rotational equivalent: x↔θ, v↔ω, a↔α, m↔I, F↔τ. They connect through radius r: v = ωr, a = αr, τ = Fr. Newton's second law F = ma becomes τ = Iα in rotational form. Both share the same kinematic equation structure.
How do you convert RPM to rad/s? ▼
ω (rad/s) = RPM × 2π / 60. For example, 1200 RPM = 1200 × 2π/60 = 125.66 rad/s. The factor 2π converts revolutions to radians; dividing by 60 converts minutes to seconds.
What is the difference between tangential and centripetal acceleration? ▼
Tangential acceleration aₙ = αr changes the speed of rotation (acts along the direction of motion). Centripetal acceleration aₒ = ω²r changes the direction of motion (always points toward the center). Both exist simultaneously in non-uniform circular motion. Centripetal acceleration never changes speed — it only changes direction.
What is torque and how does it relate to angular acceleration? ▼
Torque (τ) is the rotational equivalent of force — it causes angular acceleration. The relationship is τ = Iα. A larger torque produces more angular acceleration for the same object. Torque equals force times the perpendicular arm: τ = F × r (units: N·m).
What is the period of rotation? ▼
The period T is the time for one complete revolution (2π radians). It relates to frequency by T = 1/f and to angular velocity by T = 2π/ω. For ω = 10 rad/s: T = 2π/10 = 0.6283 s. Period is measured in seconds; frequency in Hz (revolutions per second).
How do you find the moment of inertia for rotational dynamics? ▼
Moment of inertia I depends on mass and its distribution about the axis. Common values: solid disk I = ½mR²; solid sphere I = (2/5)mR²; thin rod about center I = (1/12)mL². Use our Moment of Inertia Calculator for all standard shapes. Then apply τ = Iα.
What is the analogy between linear and rotational motion? ▼
Linear and rotational motion share identical mathematical structure. F = ma becomes τ = Iα; KE = ½mv² becomes KE = ½Iω²; p = mv becomes L = Iω; P = Fv becomes P = τω. The bridge between them is the radius r: v = ωr, a = αr, τ = Fr.
Related Calculators
📋 Five Kinematic Equations
Eq1
ω = ω₀ + αt
no θ
Eq2
θ = ω₀t + ½αt²
no ω
Eq3
ω²=ω₀²+2αθ
no t
Eq4
θ=(ω₀+ω)t/2
no α
Eq5
θ=ωt−½αt²
no ω₀
⇄ Key Conversions
RPM
×2π/60 = rad/s
rev
1 rev = 2π rad = 360°
v=ωr
linear ? angular speed
aₒ
ω²r = v²/r (centripetal)
T
2π/ω = 1/f (period)
⚖ Dynamics
τ=Iα
Newton 2 for rotation
L=Iω
Angular momentum
kg·m²/s
KE
½Iω²
J
P
τω (rotational power)
W
⇆ Linear ↔ Rotational
| x | ↔ | θ |
| v | ↔ | ω |
| a | ↔ | α |
| m | ↔ | I |
| F | ↔ | τ |
| F=ma | ↔ | τ=Iα |