Cylinder Moment of Inertia
Compute solid cylinder moment of inertia I = ½·m·r².
I = — kg·m²
Moment of inertia of a cylinder: I = (1/2)·M·R²
Spin a solid cylinder about its central (longitudinal) axis and you get I = (1/2)·M·R². A thin hollow cylinder (a shell) gives I = M·R². For a thick hollow cylinder with inner radius R₁ and outer R₂, it becomes I = (1/2)·M·(R₁² + R₂²). Switch to a transverse axis through the center, perpendicular to the length L, and you have I = (1/4)·M·R² + (1/12)·M·L². Think of moment of inertia as mass for rotation, with rotational kinetic energy given by E_rot = (1/2)·I·ω². Say you have a flywheel of M = 50 kg and R = 0.3 m: I = (1/2)·50·0.09 = 2.25 kg·m², and at ω = 100 rad/s it holds E_rot = (1/2)·2.25·10000 = 11.25 kJ.
Applications
Engine flywheels lean on it to store rotational energy and smooth out power delivery. So do industrial centrifuges, gears, driveshafts and cardan shafts, rotating machinery on the factory floor, and gyroscope design. Wherever someone has to know how much torque it takes to spin up a cylindrical part, or to brake it, this is the number they want.
FAQ
Why is the hollow cylinder's I larger than the solid one's at the same M and R? Its mass sits farther out from the rotation axis, and since I grows with r², mass that is farther away resists angular acceleration more.
Does the length L matter for the central axis? No. When the rotation is about the longitudinal axis, L never enters the formula. It only comes into play for the transverse axis.
How do I find I about an off-center axis? Reach for the parallel-axis theorem, I = I_cm + M·d², where d is the distance from the center of mass out to the new axis.
Related Tools
Moment of Inertia (Solid Cylinder)
Computes I=0.5·m·r² for a solid cylinder spinning about its central axis.
Moment of Inertia (Solid Sphere)
Computes I=0.4·m·r² for a solid sphere spinning about an axis through its center.
Sphere Moment of Inertia
Compute solid sphere (I = 2/5·m·r²) or hollow shell (I = 2/3·m·r²) moment of inertia.
Rectangular Section Inertia
Compute Ix and Iy of a rectangular section: Ix = b·h³/12, Iy = h·b³/12. Also section modulus W.
Torque for Angular Acceleration
Calculate the torque needed to angularly accelerate a rotating body, T = I·α, from the moment of inertia I and the desired angular acceleration α (rad/s²). The result, in N·m, is the rotational version of Newton's second law (F = m·a): the greater the assembly's inertia or the faster the intended acceleration, the more torque the motor must provide. It is fundamental in sizing drives that must accelerate and decelerate loads quickly — robots, positioners, spindles — where the acceleration torque adds to the friction and load torque. Enter the moment of inertia and the angular acceleration.
Rotational Braking Time
Calculate the time to brake (stop) a rotating system, t = (I·ω) ÷ T, from the moment of inertia I (kg·m²), the initial angular velocity ω (rad/s) and the braking torque T (N·m). When a brake applies a constant torque to a spinning system (a shaft, flywheel, machine rotor), it DECELERATES it to a stop. By Newton's second law for rotation (T = I·α, with α the angular deceleration), the stopping time is the initial angular momentum (I·ω) divided by the braking torque. This matters in several situations: EMERGENCY STOPPING of machines (safety codes require dangerous parts to stop within a maximum time after brake actuation — the shorter, the safer), sizing motor and shaft brakes, and clutches (the engagement time, where the clutch 'synchronizes' two shafts' speeds, follows the same physics). Systems with large moment of inertia (heavy flywheels, big rotors) take longer to stop with a given torque — so high-inertia machines need powerful brakes or more stopping time. The braking time, with the dissipated energy and power, completes a braking analysis. Enter the moment of inertia, the angular velocity and the braking torque.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.