Torque and Rotational Dynamics

Six topics on rotation, where every linear idea returns with an angle in front of it. Rotational kinematics mirrors $x$, $v$, $a$ with $\theta$, $\omega$, $\alpha$; the bridge equations $v = r\omega$ and $a_t = r\alpha$ tie a spinning rigid body to the motion of its points; torque $\vec{\tau} = \vec{r}\times\vec{F}$ is the rotational cause of angular acceleration; rotational inertia $I = \int r^2\,dm$ weights mass by how far it sits from the axis; and finally rotational equilibrium and Newton's second law in rotational form $\tau_{net} = I\alpha$.

Topics

5.1

Rotational Kinematics

Connecting Linear and Rotational Motion

Torque

Rotational Inertia

Rotational Equilibrium

Newton's Second Law in Rotational Form

Lesson

Diag

locked

Equations For every problem in this unit

Angular velocity

$\omega = \dfrac{d\theta}{dt}$

Angular acceleration

$\alpha = \dfrac{d\omega}{dt}$

Constant angular acceleration

$\omega = \omega_0 + \alpha t$

Angular position

$\theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2$

Linear and rotational link

$v = r\omega,\ \ a_t = r\alpha$

Torque, magnitude

$\tau = rF\sin\theta$

Torque, vector

$\vec{\tau} = \vec{r}\times\vec{F}$

Centripetal acceleration

$a_c = \omega^2 r = \dfrac{v^2}{r}$

Rotational inertia, discrete

$I = \sum m_i r_i^2$

Rotational inertia, continuous

$I = \int r^2\,dm$

Parallel-axis theorem

$I' = I_{cm} + Md^2$

Newton's second law

$\tau_{net} = I\alpha$

Unit 5 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items pulled from across all 6 topics, identifying which misconceptions still bite when the question does not tell you which topic it came from.