Rigid-body rotation and rotating machinery

Define angle, angular velocity and inertia

Let θ be angular position in radians,ω its rate of change in rad/s, and α the rate of change of ω in rad/s². One revolution equals 2π rad. A radian relates arc length to radius; it is not a distance in metres. Use a stated positive rotation direction just as a translating model uses a positive axis.

The mass moment of inertia I sums how mass is distributed relative to the rotation axis. Its units are kg·m². Moving mass farther from that axis generally raises I, even if total mass stays the same. This I is different from the mm⁴ second moment of area used for beam bending. A numerical value is meaningful only with its axis and included rotating parts.

Use the net torque for angular acceleration

For a rigid body rotating about a fixed axis with constant I, net external torque about that axis equals Iα. A driving torque and a resisting torque combine with signs. Bearing forces can support the axis without contributing a turning moment about it in a chosen idealisation. Real bearings can also introduce resisting torque, which must be represented when relevant.

If net torque stays constant,α is constant. The angular versions of constant-acceleration relations give ω=ω0+αt and angular displacement ω0t+αt²/2. These results rely on the fixed-axis, rigid-body assumptions. A shaft that twists appreciably or a rotating assembly whose mass moves relative to the axis can need a richer model.

Separate stored energy from continuous transfer

Rotational kinetic energy is Iω²/2 for this model. A constant torque acting through angle Δθ does work τΔθ. Instantaneous power is τω. A rotor at constant speed can transmit substantial power while its kinetic energy remains unchanged, because input and output torques balance.

A point at radius r has tangential speed rω, tangential acceleration rα and inward radial acceleration rω². Constant angular speed eliminates tangential acceleration, not radial acceleration. Radial force is needed to keep the point moving around the axis. None of these kinematic quantities alone establishes the stresses or balance quality of real rotating machinery.

Close a constructed rotor’s work account

Let I=0.020 kg·m², input torque 0.100 N·m and resisting torque 0.020 N·m, constant over the interval. Net torque 0.080 N·m gives α=0.080/0.020=4 rad/s². From rest over 2 s, the rotor reaches 8 rad/s and turns 8 rad.

Input work is 0.100×8=0.80 J. The resisting interaction transfers 0.020×8=0.16 J away from the rotor, leaving 0.64 J kinetic gain, matching½×0.020×8². At radius 0.10 m, the final tangential speed is 0.80 m/s and inward acceleration 6.4 m/s². The latter exists alongside a tangential acceleration 0.40 m/s²; they point in different directions.

Try the reasoning

1. Why must I specify an axis?

   Show the answer and reasoning

   Each mass element’s distance from that axis affects I; another axis can give another value.

2. Can a constant-speed rotor transfer power?

   Show the answer and reasoning

   Yes. Balanced input/output torques can transfer power without changing stored rotational kinetic energy.

3. Is radial acceleration zero at constant angular speed?

   Show the answer and reasoning

   No. Direction of velocity still changes, requiring inward acceleration rω².

What to carry forward

Fixed-axis motion, torque and energy fit together when the inertia, axis and other transfers are explicit.