Unit 6: Energy and Momentum of Rotating Systems

What Unit 6 Covers and Why It Matters

Unit 6, Energy and Momentum of Rotating Systems, takes the conservation laws from linear motion and reapplies them to spinning objects. Everything you learned about kinetic energy in Unit 3 and momentum in Unit 4 has a rotational twin here: kinetic energy becomes rotational kinetic energy, impulse becomes angular impulse, and conservation of momentum becomes conservation of angular momentum.

At 5 to 8 percent of the multiple-choice section, Unit 6 is one of the two lightest units in AP Physics 1 (Unit 7, Oscillations, is the other). Do not let the small weight fool you: free-response questions often mix rotation with energy or momentum conservation, so this unit's tools show up beyond its own weighting.

You need Unit 5 first. Angular velocity, torque, and rotational inertia are assumed everywhere in Unit 6, so if those feel shaky, review rotational kinematics and how to calculate torque before starting.

Topics 6.1 and 6.2: Rotational Kinetic Energy and Work

A spinning object has kinetic energy even if it stays in place. Rotational kinetic energy is $K_{rot} = \frac{1}{2} I \omega^2$, the exact analog of $K = \frac{1}{2} m v^2$ with rotational inertia $I$ replacing mass and angular speed $\omega$ replacing speed. An object that spins while its center of mass moves carries both terms:

Topic 6.2 connects torque to energy. A constant net torque acting through an angular displacement does work $W = \tau \Delta\theta$ (with $\Delta\theta$ in radians), and that work changes the object's rotational kinetic energy. This is the rotational version of the work-energy theorem.

On problems, treat rotational kinetic energy as one more account in your energy bookkeeping. Every strategy from conservation of energy carries over; you just add a $\frac{1}{2} I \omega^2$ term wherever something spins.

Topic 6.3: Angular Momentum and Angular Impulse

Angular momentum measures how much rotation a system has. For a rigid object spinning about a fixed axis, $L = I\omega$, which is printed on the AP equation sheet.

An object moving in a straight line can also have angular momentum about a point: $L = m v r_{\perp}$, where $r_{\perp}$ is the perpendicular distance from the point to the object's line of motion. This matters in problems where a ball strikes a rod or a child jumps onto a merry-go-round, because the incoming object brings angular momentum with it.

Angular impulse mirrors linear impulse: a net external torque applied over a time interval changes angular momentum, $\tau \Delta t = \Delta L$. Compare that with the impulse-momentum theorem, where force times time changes linear momentum. Same logic, rotational symbols. Expect conceptual questions about which quantity changes when the same force is applied at different distances from the axis, so keep torque calculations sharp.

Topic 6.4: Conservation of Angular Momentum

If the net external torque on a system is zero, its total angular momentum stays constant. That single sentence powers some of the most recognizable problems in the course.

The classic: a figure skater spinning with arms extended pulls them in. Her rotational inertia $I$ drops, so her angular speed $\omega$ rises to keep $L = I\omega$ constant. Note the subtlety exams like to probe: her rotational kinetic energy increases, because she does internal work pulling her arms inward. Angular momentum is conserved; kinetic energy is not.

Rotational collisions work like linear ones. When a child jumps onto a spinning merry-go-round, angular momentum about the axle is conserved, but kinetic energy drops, just as in a perfectly inelastic collision. The bookkeeping mirrors conservation of momentum: write the total $L$ of the system before, set it equal to the total $L$ after, and solve for the unknown.

Topic 6.5: Rolling

Rolling without slipping links translation and rotation: the contact point is momentarily at rest, so $v_{cm} = r\omega$. That one constraint lets you write total kinetic energy in terms of a single variable.

The signature problem is the downhill race. Release a hoop, a disk, and a solid sphere from rest at the top of a ramp. The sphere wins, the hoop loses, and mass and radius are irrelevant. Each object turns all of its gravitational potential energy into kinetic energy, but shapes with larger rotational inertia relative to $mr^2$ park a larger fraction of that energy in rotation, leaving less for forward motion.

Two facts worth memorizing: static friction is what makes rolling without slipping possible, and it does no work in that case because the contact point never slides. So mechanical energy is conserved on the way down even though friction acts. Work one full rolling problem by hand before exam day; the worked example below is a good template.

Topic 6.6: Motion of Orbiting Satellites

Unit 6 closes with satellites, which tie together gravitation, circular motion, energy, and angular momentum. For a circular orbit, gravity supplies the centripetal force. Setting $\frac{GMm}{r^2} = \frac{mv^2}{r}$ and solving gives orbital speed $v = \sqrt{GM/r}$, using $G = 6.67 \times 10^{-11}$ N m^2/kg^2. Bigger orbit means slower satellite; the satellite's own mass cancels out.

Elliptical orbits are where the conservation laws shine. Gravity points from the satellite toward the planet's center, so it exerts zero torque about that center, and the satellite's angular momentum is conserved: it moves fastest at the closest point in its orbit and slowest at the farthest point. Total mechanical energy is also conserved, trading kinetic energy for gravitational potential energy as the satellite climbs.

Expect qualitative comparison questions here more than heavy computation. Review centripetal force from Unit 2 first, since every circular-orbit equation starts there.

How Unit 6 Is Tested and Where to Practice

On the multiple-choice section (42 questions in 85 minutes), Unit 6 leans on ranking and comparison tasks: which shape reaches the bottom of the ramp first, which skater configuration spins faster, which orbit has the greater speed. In free response, rotation usually arrives blended with energy or momentum. A calculator is allowed on both sections of the exam.

A practice sequence that works:

1. Redo the ramp race with a hoop, disk, and sphere until the energy split is automatic.
2. Drill $L = I\omega$ bookkeeping on skater and merry-go-round scenarios, before and after.
3. Rebuild the circular-orbit speed equation from Newton's second law instead of memorizing it.

For tools, the torque calculator checks the Unit 5 skills this unit assumes, and the kinetic energy calculator handles the translational term in rolling problems. Keep the AP Physics 1 formula sheet open while you practice so you know exactly which equations you get on exam day.