Torque for Angular Acceleration

Because I = Σmᵢrᵢ², where rᵢ is the perpendicular distance from each mass element to the axis. If you change the axis, the distances change, so I changes. A thin rod has a small I about its long axis (most mass is close to the axis) but a much larger I about an axis through its center perpendicular to its length (most mass is far from the axis). The parallel-axis theorem relates the two: I = I_cm + m·d², where I_cm is the moment about the center of mass and d is the distance between the axes. This is why the choice of axis must always be specified when quoting I.

For a composite body, sum the moments of inertia of its parts about the same axis. Use the parallel-axis theorem to shift each part's I from its own center of mass to the common axis: I = Σ(I_cm,i + mᵢ · dᵢ²). For CAD models, most software can compute I directly from the geometry and density. For hand calculations, break the body into simple shapes (cylinders, plates, rods), look up their formulas, shift to the common axis, and add. If the body has a complex shape that cannot be decomposed easily, use a physical pendulum test or a torsion pendulum to measure I experimentally.

Rotational power is P = T · ω, the product of torque and angular velocity. A motor that produces 10 N·m at 100 rad/s delivers 1,000 W (1 kW) of mechanical power. A motor that produces 100 N·m at 10 rad/s delivers the same power. This is why gearboxes are so useful: they trade speed for torque without changing power (except for losses). The peak power of a motor occurs at the point where T · ω is maximized, which is usually at about half the no-load speed. For a given power rating, you can have high torque and low speed, or low torque and high speed, but not both.

Motor sizing involves four steps: (1) estimate the load inertia I_load about the joint axis, including the link, payload, and any gearing reflected to the motor side; (2) decide the required angular acceleration α for the motion profile (often from a trapezoidal velocity profile and the time available); (3) compute the acceleration torque T_acc = I · α, then add friction torque T_fric, gravity torque T_grav (which depends on the arm angle), and any external load torque; (4) add a safety factor of 1.5–2.0 for peak torque, and check that the motor's continuous torque rating exceeds the RMS torque over the duty cycle to avoid overheating. The required power is then P = T · ω at the operating point.

Torque is the cause of change in rotation; angular momentum is the quantity of rotation. The relationship is τ = dL/dt, the rotational analog of F = dp/dt. Angular momentum is L = I · ω and is conserved when the net external torque is zero. This conservation law explains why a spinning figure skater speeds up when pulling her arms in (I decreases, so ω increases to keep L constant), why a gyroscope maintains its orientation, and why a satellite can change its attitude by spinning internal flywheels. Torque, in contrast, is what you apply to change L — it is a force-like quantity, not a conserved one.

Yes. Torque is a vector, and its sign indicates the direction of rotation (usually positive for counterclockwise, negative for clockwise, by the right-hand rule). A negative torque produces a negative angular acceleration, which either slows down a positive rotation or speeds up a negative one. In motor sizing, both signs matter: a motor must produce positive torque to accelerate in one direction and negative torque (or a brake) to decelerate. The magnitude |T| determines how quickly the motion changes; the sign determines which way. In this calculator, we use magnitudes throughout for simplicity, but in a full dynamic analysis the signs must be tracked.