Torque and Rotational Dynamics

Six topics on rotation. Rotational kinematics is the geometry of spin: how angle, angular velocity, and angular acceleration relate over time. Connecting linear and rotational motion links a point on a rotating body to the body's spin through the radius. Torque is the rotational analogue of force; rotational inertia the analogue of mass. Rotational equilibrium generalizes Newton's first law to spinning systems, and Newton's second law in rotational form predicts angular acceleration from the net torque and the rotational inertia.

Topics

5.1

Rotational Kinematics

Connecting Linear and Rotational Motion

Torque

Rotational Inertia

Rotational Equilibrium and Newton's First Law

Newton's Second Law in Rotational Form

Lesson

Diagnostic

after diagnostic

Equations For every problem in this unit

Angular velocity (constant $\alpha$)

$\omega = \omega_0 + \alpha t$

Angular displacement (constant $\alpha$)

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

Squared form (constant $\alpha$)

$\omega^2 = \omega_0^2 + 2 \alpha \theta$

Linear-rotational speed link

$v = r \omega$

Tangential acceleration

$a_t = r \alpha$

Centripetal acceleration

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

Torque magnitude

$\tau = r F \sin\theta$

Newton's 2nd law (rotational)

$\Sigma \tau = I \alpha$

Rotational kinetic energy

$K_{\text{rot}} = \tfrac{1}{2} I \omega^2$

Angular momentum

$L = I \omega$

Rotational impulse

$\Delta L = \tau \, \Delta t$

Point mass at radius $r$

$I = m r^2$

Hoop or thin cylindrical shell (about axis)

$I = M R^2$

Solid disk or cylinder (about axis)

$I = \tfrac{1}{2} M R^2$

Solid sphere (about diameter)

$I = \tfrac{2}{5} M R^2$

Thin spherical shell (about diameter)

$I = \tfrac{2}{3} M R^2$

Thin rod (axis through center, perpendicular)

$I = \tfrac{1}{12} M L^2$

Thin rod (axis through end, perpendicular)

$I = \tfrac{1}{3} M L^2$

Parallel-axis theorem

$I = I_{\text{cm}} + M d^2$

Pythagorean

$a^2 + b^2 = c^2$

Sine, opp / hyp

$\sin\theta = \dfrac{a}{c}$

Cosine, adj / hyp

$\cos\theta = \dfrac{b}{c}$

Tangent, opp / adj

$\tan\theta = \dfrac{a}{b}$

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. Identifies which misconceptions still bite when you cannot see which topic the question came from.

Begin assessment →

Check what stuck.

Units 1 through 5, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 42-question section, your choice. Even coverage means this is a retention check rather than a score estimate.

Begin assessment →

Rotational inertia lab

Collect your own data, choose the transform that straightens the plot, and read a moment of inertia off the slope. Results record to your class dashboard.

Open the lab →