Topic 5.6 Newton's Second Law in Rotational Form: relate the net torque on a rigid body to its angular acceleration and rotational inertia through tau_net = I*alpha.

What this topic is asking

The College Board (Topic 5.6) wants you to relate the net torque on a rigid body to its angular acceleration and rotational inertia through the rotational form of Newton's second law, $\tau_{net} = I\alpha$. This is the rotational twin of $F_{net} = ma$: net torque causes angular acceleration, and rotational inertia mediates the relationship just as mass does for linear motion.

The rotational second law

This law completes the rotational picture begun in Topic 5.1. Where torque (Topic 5.3) is the rotational analogue of force and rotational inertia (Topic 5.4) is the analogue of mass, this law ties them to angular acceleration exactly as $F_{net} = ma$ ties force and mass to linear acceleration.

The two proportionalities

These proportionalities let you reason qualitatively. A larger torque spins an object up more quickly; a heavier or more spread-out object (larger $I$) resists that spin-up. A flywheel with large rotational inertia barely speeds up under a modest torque, while a light spindle responds instantly.

Solving rotational dynamics problems

The solution routine parallels translational dynamics step for step. Identify the forces and where they act, compute the torque each produces about the axis ($\tau = rF\sin\theta$), add them with signs to get the net torque, find the rotational inertia about that axis, and apply $\alpha = \tau_{net}/I$. From the angular acceleration you can then use the rotational kinematic equations (Topic 5.1) to find angular velocities and displacements over time, just as you used linear kinematics after finding a linear acceleration. The analogy with $F_{net} = ma$ is the organizing idea for the whole of rotational dynamics and is worth stating explicitly: translation and rotation obey the same logical structure, with torque, rotational inertia and angular acceleration as the rotational counterparts of force, mass and linear acceleration. This unifying parallel means every problem-solving habit from Unit 2, draw the diagram, find the net cause, divide by the resistance to get the effect, transfers directly. Many AP problems combine the two worlds: a falling mass on a string wrapped around a pulley exerts a torque that angularly accelerates the pulley while the mass linearly accelerates, and you solve by writing $F_{net} = ma$ for the mass and $\tau_{net} = I\alpha$ for the pulley, linked by $a = r\alpha$ from Topic 5.2. Recognizing $\tau_{net} = I\alpha$ as "the rotational $F = ma$" is the single most useful insight for the dynamics half of Unit 5.

Try this

Q1. A net torque of $6.0$ N$\cdot$m acts on a body of rotational inertia $1.5$ kg$\cdot$m squared. Calculate its angular acceleration. [2 points]

• Cue. $\alpha = \tau_{net}/I = 6.0/1.5 = 4.0$ rad/s squared.

Q2. A body of rotational inertia $0.50$ kg$\cdot$m squared angularly accelerates at $8.0$ rad/s squared. Calculate the net torque on it. [1 point]

• Cue. $\tau_{net} = I\alpha = (0.50)(8.0) = 4.0$ N$\cdot$m.