The rotational motion calculator covers all three sections of rotational mechanics: angular kinematics (ω, α, θ, t), moment of inertia for 8 common shapes, and rotational dynamics (torque, angular momentum, rotational KE). Switch between metric and imperial inputs for mass and dimensions.
Section 1: Angular Kinematics
Solve any variable: ω (final), ω₀ (initial), α (angular acceleration), t (time), θ (angular displacement)
Section 2: Moment of Inertia
Select a shape and enter dimensions to compute I = moment of inertia (kg·m²)
Section 3: Rotational Dynamics
Calculate torque, angular momentum, and rotational kinetic energy from I and ω
How to Use the Rotational Motion Calculator
Rotational motion mirrors linear motion but uses angular quantities. This calculator has three independent sections: kinematics equations, moment of inertia by shape, and rotational dynamics. You can use them individually or chain them together (e.g., calculate I in Section 2, then paste it into Section 3).
Section 1: Angular Kinematics
The four angular kinematic equations relate ω (final angular velocity), ω₀ (initial), α (angular acceleration), t (time), and θ (angular displacement). Select which variable to solve for, enter the other known values, and click calculate. These are the rotational analogs of the linear kinematic equations v = u + at, s = ut + ½at², etc.
Section 2: Moment of Inertia by Shape
Select the shape from the dropdown. For a spinning wheel (solid disk or cylinder), use the solid cylinder formula: I = ½mr². Example: a disk of mass 2 kg and radius 0.5 m has I = ½ × 2 × 0.5² = 0.25 kg·m². For a thin rod rotating about its center, I = (1/12)mL². The same rod rotating about one end has I = (1/3)mL² — four times larger — which is why it's much harder to start rotating about the end.
Section 3: Rotational Dynamics
Enter I and ω (and optionally α) to calculate torque (τ = Iα), angular momentum (L = Iω), and rotational kinetic energy (KE = ½Iω²). For the 2 kg disk spinning at 10 rad/s: L = 0.25 × 10 = 2.5 kg·m²/s, KE = ½ × 0.25 × 100 = 12.5 J.
Angular Unit Conversions
Angular velocity can be displayed in rad/s, RPM, or degrees/second. 1 rad/s = 9.549 RPM = 57.296 deg/s. The ISS orbits at about 0.00105 rad/s = 0.063 RPM = 0.0601 deg/s.
FAQ
What is angular velocity and how is it related to RPM?
Angular velocity (ω) measures how fast an object rotates in radians per second. 1 full rotation = 2π radians, so ω = 2π × f where f is frequency in Hz. To convert from RPM: ω = RPM × 2π / 60. A motor spinning at 1,800 RPM has ω = 1,800 × 2π/60 ≈ 188.5 rad/s.
What is moment of inertia?
Moment of inertia (I) is the rotational analog of mass — it measures how hard it is to change an object's rotation. I depends on both mass and how that mass is distributed relative to the rotation axis. A hollow cylinder has higher I than a solid cylinder of the same mass because more mass is far from the axis.
What is the formula for rotational kinetic energy?
Rotational kinetic energy is KE = ½Iω², where I is moment of inertia (kg·m²) and ω is angular velocity (rad/s). For a rolling object, total KE = ½mv² (translational) + ½Iω² (rotational). A solid sphere rolling without slipping has 5/7 of its total energy as translational KE.
Is this tool free?
Yes, completely free with no signup required. All calculations run locally in your browser.
What is angular momentum?
Angular momentum L = I × ω (kg·m²/s). Like linear momentum, angular momentum is conserved when no external torque acts on a system. This explains why a spinning ice skater spins faster when pulling in their arms — reducing I while conserving L requires ω to increase.
What is the difference between torque and angular acceleration?
Torque (τ) is the rotational equivalent of force — it causes angular acceleration. Newton's second law for rotation is τ = I × α, where α is angular acceleration (rad/s²). Just as F=ma relates force, mass, and linear acceleration, τ=Iα relates torque, moment of inertia, and angular acceleration.