Angular Position, Velocity, Acceleration
omega = d theta/dt, alpha = d omega/dt.
Example: theta(t)=3t^2: omega=6t, alpha=6.
Adaptive
Learn Rotational Mechanics
Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~18 min
Adaptive Checks
16 questions
Transfer Probes
8
Lesson Notes
Rotational mechanics extends Newtonian dynamics to spinning and rolling objects using calculus. Torque drives angular acceleration through tau = I alpha.
Moment of inertia is computed by integrating r^2 dm. Angular momentum L = I omega is conserved when no external torque acts.
Rolling without slipping couples translation and rotation through v = R omega.
You'll be able to:
- Relate angular position, velocity, and acceleration using calculus
- Calculate torque and apply the rotational second law
- Derive moment of inertia using integration
- Apply the parallel axis theorem
- Apply conservation of angular momentum
One step at a time.
Interactive Exploration
Adjust the controls and watch the concepts respond in real time.
Key Concepts
Torque
tau = r x F. Magnitude rF sin(theta).
Example: 20N at 0.5m from pivot: tau=10 N m.
Moment of Inertia
I = integral r^2 dm.
Example: Rod about end: I=ML^2/3.
Rotational Second Law
tau_net = I alpha.
Example: I=0.5, tau=4: alpha=8 rad/s^2.
Parallel Axis Theorem
I = I_cm + Md^2.
Example: Disk at rim: I=3MR^2/2.
Angular Momentum
L = I omega. Conserved when net torque = 0.
Example: Skater pulls arms in: omega increases.
Rotational Kinetic Energy
KE_rot = (1/2)I omega^2.
Example: Rolling sphere: KE = (7/10)mv^2.
Rolling Without Slipping
v_cm = R omega.
Example: R=0.3m, omega=10: v=3 m/s.
Explore your way
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
Concept Map
See how the key ideas connect. Nodes color in as you practice.
Worked Example
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
Adaptive Practice
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
- Math Lens cues for what to look for and what to ignore.
- Progressive hints (direction, rule, then apply).
- Targeted feedback when a common misconception appears.
Ad Placeholder
Teach It Back
The best way to know if you understand something: explain it in your own words.
Keep Practicing
More ways to strengthen what you just learned.