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The rotational analog of Newton's second law is Στ = Iα: the net torque on a rigid body about a fixed axis equals its moment of inertia times its angular acceleration. This equation governs all rotational dynamics, from spinning tops to rolling cylinders. For rolling-without-slipping problems, linear and rotational equations couple through the constraint a = αr.
Draw a free-body diagram, compute all torques about the rotation axis, set Στ = Iα. For rolling objects, write both ΣF = ma (linear) and Στ = Iα (rotational) and connect them via the no-slip condition a = αr.
Newton's second law for linear motion is the central equation of classical mechanics: the net force on an object equals its mass times its linear acceleration (ΣF = ma). You've now built all the ingredients to write the exact rotational analog. Your study of torque established that torque is the rotational cause of angular acceleration — it's the "twisting force" that depends on both the force applied and how far from the axis it acts. Your study of moment of inertia established that I is the rotational analog of mass — it measures how a body's mass is distributed relative to the rotation axis, and therefore how resistant the body is to changes in its rotational motion. Put these together: Στ = Iα. Net torque drives angular acceleration, with moment of inertia as the proportionality constant.
The analogy table is worth internalizing explicitly: force F ↔ torque τ; mass m ↔ moment of inertia I; linear acceleration a ↔ angular acceleration α; linear momentum p = mv ↔ angular momentum L = Iω. Every theorem you know about linear dynamics has a rotational counterpart with this substitution. The equation Στ = Iα is not a new law — it is the rotational expression of the same underlying physics as F = ma. This is why your work on rotational kinematics (relating θ, ω, α) maps exactly onto the kinematic equations for linear motion.
The cross product (from your prerequisites) reveals why torque is a vector. Torque τ = r × F depends not just on the magnitudes of the position vector r and the force F, but on the angle between them: τ = rF sin θ. A force applied directly toward or away from the rotation axis (θ = 0° or 180°) produces zero torque — it cannot cause rotation. A force applied perpendicular to r (θ = 90°) produces maximum torque. The direction of τ = r × F, given by the right-hand rule, tells you which axis the torque rotates around and in which sense. For 2D problems — a disk spinning in a plane, a door opening — you only need the magnitude, but the vector nature of torque is essential for 3D problems like gyroscopes and precession.
Rolling without slipping is the signature problem that combines linear and rotational dynamics. When a cylinder rolls down a ramp, friction at the contact point produces a torque that angularly accelerates the cylinder as it linearly accelerates down the slope. Write two equations: ΣF = ma (net linear force = mass × linear acceleration) and Στ = Iα (net torque about the center = moment of inertia × angular acceleration). The no-slip constraint connects them: a = αr, meaning the linear acceleration of the center equals the angular acceleration times the radius. Together, these three relationships uniquely determine both a and α. The fraction of total kinetic energy stored in rotation depends on I — which depends on how mass is distributed. A hollow cylinder (all mass at radius r, so I = mr²) stores more energy in rotation than a solid cylinder (I = ½mr²), which is why the solid cylinder reaches the bottom of a ramp faster: less of its energy is "tied up" in spinning.