Torque from Moment of Inertia and Angular Acceleration
Torque from Moment of Inertia and Angular Acceleration
Newton's second law has a direct rotational counterpart: instead of force equals mass times acceleration, torque equals moment of inertia times angular acceleration. This relationship is the starting point for sizing a motor to spin up a load — a fan, a flywheel, a robotic arm joint — within a required time.Motor and drivetrain designers use this formula constantly: knowing the load's moment of inertia and the desired spin-up time (which sets the required angular acceleration) directly determines the torque rating the motor must deliver, which in turn drives motor selection and gearbox ratio.
The rotational form of Newton's second law is T = Iα. where I_shaft is the moment of inertia of the rotating load and alpha_ang is the required angular acceleration, giving the required torque T_torque.
Inputs
I_shaft := 1.5 kg·m^2=1.5 kg·m^2
alpha_ang := 6 s^-2=6 1/s^2
Multiply the moment of inertia by the required angular acceleration — exactly the rotational analog of F = ma — to find the torque the drive must supply.
T_torque := I_shaft·alpha_ang=9 Nm
Results
A moment of inertia of 1.5 kg·m² accelerating at 6 rad/s² needs a modest torque, well within the range of a small electric motor or gearmotor. If the required spin-up time were shortened, the needed angular acceleration — and therefore the required torque — would increase proportionally, which is the direct trade-off between motor size and acceleration performance in drivetrain design.
Assumptions & limitations
- Rigid body with constant moment of inertia during acceleration.
- No friction or other resisting torques included (this is the torque for acceleration alone).
- Constant angular acceleration over the interval considered.