Result

Rotational form of Newton's second law

τ_{net} = I α

Statement

Net torque produces angular acceleration in proportion to moment of inertia, exactly as net force produces linear acceleration in proportion to mass.

Symbols and units

$τ_{net}$	net torque	N·m
$I$	moment of inertia	kg·m²
$α$	angular acceleration	rad/s²

When it applies

Rigid body rotation about a fixed axis, with I computed about that same axis.

Don't use when

I and the net torque must be computed about the same axis — mixing axes gives a meaningless result.

Derivation

For a particle at radius r, tangential force gives tangential acceleration a_t = rα; multiplying by r and using F=ma converts this to a torque relation, then summing over every mass element gives I:

τ_{net} = i \sum m_{i} r_{i}^{2} α = I α

Worked examples

Angular acceleration of a disk

A disk with I = 0.20 kg·m² experiences a net torque of 3.0 N·m:

α = \frac{τ _{net}}{I} = \frac{3.0}{0.20} = 15 rad/s^{2}

Watch out for

I must be computed about the same axis the torque and angular acceleration are measured about.

Connects to

Newton's second law Moment of inertia