Mistake Master

Newton's Second Law for Rotation

Topic 5.5 was about when a body's rotation doesn't change: $\sum\tau = 0$. Topic 5.6 is what happens when the torques don't cancel. The body angularly accelerates, and the size of $\alpha$ is set by the rotational version of Newton's second law: $\sum\tau = I\alpha$. Same shape as $\sum F = m a$, with rotational inertia $I$ in place of mass.

ANGULAR VELOCITY CHANGES WHEN the net torque on the object or system is not equal to zero. αsys F1 F2 r1 r2 AXIS OF ROTATION τ1 = r1F1 sin θ1 τ2 = −r2F2 sin θ2 αsys = Στ Isys = τnet Isys
Fig. 5.6.1   When the net torque on a rigid body is not zero, its angular velocity changes. Here F1 (upward at the left) and F2 (downward at the right) produce torques in opposite rotational senses about the axis; because their magnitudes differ, they do not cancel. The net torque divided by the system's moment of inertia is the resulting angular acceleration: $\alpha_\text{sys} = \Sigma \tau / I_\text{sys}$.
Angular Acceleration Lab · Open the sandbox →

The lesson closes by knocking down three traps. Confusing constant net torque with constant $\omega$ (constant $\tau$ actually gives constant $\alpha$, so $\omega$ grows steadily). Reading "same mass" as "same $I$", since distribution matters as much as total mass. And at the turnaround of a swing, concluding $\alpha = 0$ from $\omega = 0$, which $\sum\tau = I\alpha$ never says.

The work

3 ways in · any order
Lesson
Newton's Second Law for Rotation

Build $\sum\tau = I\alpha$ as Newton's second law for rotation. Two worked examples: a rod released from horizontal, and a pulley with a hanging mass. Ten-scenario applet hits the $\omega$ vs $\alpha$ confusion, the zero-$\omega$-means-zero-$\alpha$ trap, and the mass-distribution effect on $\alpha$.

Skill check · 10 scenarios
Diagnostic
10-item topic check

Ten multiple-choice items mapped to the misconception traps active in this topic. Each wrong answer is tied to a specific failure mode, so a miss tells you exactly which trap fired.

Not started · 10 items · ~15 min
Targeted Practice
Drill a single misconception

Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the misconception and you move on.

Take the diagnostic to identify your misconceptions