A system's angular velocity changes precisely when the net torque exerted on it is not equal to zero. This is the direct fulfillment of Lesson 5.5's corollary — unbalanced torque doesn't just theoretically imply changing angular velocity, it fully determines exactly how much that velocity changes.
The rate at which a rigid system's angular velocity changes is directly proportional to the net torque exerted on it, and points in that same direction. Angular acceleration is inversely proportional to the system's rotational inertia:
Explore the proportionality directly below.
A single object with adjustable net torque and rotational inertia. Watch angular acceleration respond — directly proportional to torque, inversely proportional to I.
Double the torque with I fixed, and α doubles. Double I with torque fixed, and α is cut in half — exactly the same proportionality pattern as F=ma back in Lesson 2.5.
A solid disk with rotational inertia 0.4 kg·m² is initially at rest. A constant net torque of 2.4 N·m is applied. Find the disk's angular velocity after 5 seconds.
Lesson 2.5 always assumed an ideal, massless pulley — tension was simply the same on both sides of the string. Now that pulleys can have real rotational inertia, that assumption breaks down entirely. Some of the tension's torque goes into spinning the pulley itself, and any friction at the axle contributes its own opposing torque:
Combine this with the tangential relationship from Lesson 5.2 (a = rα) and Newton's second law for the hanging mass, and the whole system can be solved together. Explore this directly below — the full pulley system from this unit's header art.
The pulley system from this unit's header art — but now with a real, massive pulley and friction at the axle, exactly matching Iα = Tr − τ(f). Adjust everything and watch the full system respond.
Notice tension no longer just equals mg minus ma the way it did with Unit 2's massless pulleys — some of gravity's pull now goes into spinning up the pulley itself.
A pulley with rotational inertia 0.05 kg·m² and radius 0.08 m has negligible friction. A 3 kg mass hangs from a string wrapped around it. Find the system's acceleration and the string's tension.