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AP Physics 1 Unit 6 Review: Energy & Momentum

  • 5–8% of AP multiple choice
  • 6 topics
  • AP exam review verified for 2027

Unit 6 covers rotational kinetic energy, work done by torques, angular momentum, angular impulse, conservation of angular momentum, rolling motion, and satellite orbits. The core skill is applying conservation laws to systems that spin, roll, or orbit.

Fiveable's AP Physics 1 Unit 6 review covers 6 topics, with guided practice questions for the unit.

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AP Physics 1 unit 6 topics

6.1

Rotational Kinetic Energy

Spinning systems store energy described by K = 1/2 I omega squared. Total kinetic energy for a rolling or rotating-and-translating object is the sum of rotational and translational parts.

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6.2

Torque and Work

Torque does work W = tau delta theta when it acts through an angular displacement. Variable-torque work is the area under a torque-versus-angular-position graph.

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6.3

Angular Momentum and Angular Impulse

Angular momentum is L = I omega for rigid systems or L = rmv sin theta for point objects. Angular impulse tau delta t equals the change in angular momentum, mirroring the linear impulse-momentum theorem.

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6.4

Conservation of Angular Momentum

When net external torque is zero, total angular momentum is conserved. System selection determines whether angular momentum changes, and nonrigid systems can change angular speed by redistributing mass.

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6.5

Rolling

Rolling without slipping links translational and rotational motion via v_cm = r omega. Total kinetic energy is the sum of both parts. Slipping breaks this link and kinetic friction dissipates energy.

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6.6

Motion of Orbiting Satellites

Satellite orbits are governed by conservation of energy and angular momentum. Circular orbits have constant speed and energy; elliptical orbits have constant total energy and angular momentum but varying KE and PE.

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Practice AP Physics 1 unit 6 questions

Try AP-style questions on unit 6, then use the review notes below for anything you miss.

Example stimulus-based MCQs

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setup_diagram

Stimulus-based practice question

A block of mass mm hangs from a string wrapped around a massive pulley of radius RR, as shown in the figure. Released from rest, the block falls a distance hh. A student claims the work done by the string's torque on the pulley equals mghmgh.

Question

Which choice best evaluates the student's claim?

The claim is correct because the string tension equals the block's weight, mgmg.

The claim is correct because all lost gravitational potential energy goes to the pulley.

The claim is incorrect because T<mgT<mg, so the pulley does less than mghmgh of work.

The claim is incorrect because pulley work depends on fall distance, not inertia.

graph

Stimulus-based practice question

A grinding wheel of rotational inertia 2.0 kg⋅m22.0 \text{ kg}\cdot\text{m}^2 is initially at rest. A motor applies a net torque to the wheel, as shown in the graph of torque τ\tau versus time tt.

Question

Which of the following is most nearly the angular speed of the wheel at t=4.0 st = 4.0 \text{ s}?

5.0 rad/s5.0 \text{ rad/s}

10 rad/s10 \text{ rad/s}

20 rad/s20 \text{ rad/s}

40 rad/s40 \text{ rad/s}

Example FRQs

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FRQ

Constant torque effects on rotational motion systems

3. Students are investigating how a constant torque changes the angular momentum and rotational kinetic energy of a rotating system.

A.

Describe an experimental procedure to collect data that would allow the students to determine II. Include any steps necessary to reduce experimental uncertainty.

B.

Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine II.

Figure 1. Disk–pulley rotational inertia apparatus with hanging mass, drop height h, and photogate measurement of angular speed ω at the bottom of the drop.

Figure 1

Drop distance, hh (m)

Angular speed at bottom, ω\omega (rad/s)

0.30

15.0

0.40

17.2

0.50

19.8

0.60

21.3

0.80

25.2

C.

The students perform trials using the setup in Figure 1 with the same hanging mass m=0.200 kgm = 0.200\ \text{kg} and pulley radius r=0.020 mr = 0.020\ \text{m}, but they vary the drop distance hh. For each trial, the mass is released from rest and the students measure the angular speed ω\omega of the disk at the instant the mass has fallen the distance hh. Table 1 shows the measured values of hh and ω\omega.

The students correctly determine that the relationship among ω\omega and hh is
ω2=2mghI+mr2\omega^2 = \frac{2mgh}{I + mr^2}

The students create a graph with hh plotted on the horizontal axis.

i.

Indicate what measured or calculated quantity could be plotted on the vertical axis to yield a linear graph whose slope can be used to calculate an experimental value for the rotational inertia II of the disk–pulley system.

Vertical axis: Horizontal axis: hh

ii.

On the blank grid provided, create a graph of the quantities indicated in part C(i) that can be used to determine II.

•

Use Table 2 to record the data points or calculated quantities that you will plot.

•

Clearly label the vertical axis, including units as appropriate.

•

Plot the points you recorded in Table 2.

iii.

Draw a straight best-fit line for the data graphed in part C(ii).

D.

Using the best-fit line that you drew in part C(iii), calculate an experimental value for the rotational inertia II of the disk–pulley system. Using the setup in Figure 1 and the same mass m=0.200 kgm = 0.200\ \text{kg} and radius r=0.020 mr = 0.020\ \text{m}, the students use their best-fit line from part C(iii) to determine the slope SS of the linear graph. The students find S=760 s−2 m−1S = 760\ \text{s}^{-2}\ \text{m}^{-1}. Assume g=9.8 m/s2g = 9.8\ \text{m/s}^2.

Figure 2. Modified setup with an in-line force sensor measuring string tension T while a mass m falls, driving the disk–pulley system.

Figure 2
FRQ

Rotating disk-ring system angular momentum behavior

1. A rigid system consists of a uniform solid disk and a thin ring that are rigidly attached and rotate together about a vertical, frictionless axle through their common center, as shown in Figure 1.

Figure 1. Top view of a rigid disk–ring system rotating about a vertical, frictionless axle; a tangential force F is applied at the ring’s outer rim. Counterclockwise is defined as the positive rotation direction.

Figure 1

Figure 2. Axes for plotting net torque τ_net versus time t from t = 0 to t = t₁.

Figure 2
A.
i.

On the axes shown in Figure 2, sketch a graph of the net torque τnet\tau_{\text{net}} exerted on the disk-ring system as a function of time tt from t=0t=0 to t=t1t=t_1.

ii.

Derive an expression for the magnitude of the angular impulse JτJ_\tau delivered to the disk-ring system from t=0t=0 to t=t1t=t_1 in terms of FF, RR, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

iii.

Derive an expression for the rotational work WrotW_{\text{rot}} done on the disk-ring system by the applied force from t=0t=0 to t=t1t=t_1 in terms of FF, RR, and θ1\theta_1. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

B.

Indicate whether the magnitude of the angular momentum of the disk-ring system about the axle increases, decreases, or remains constant during 0<t<t10 < t < t_1. The student now considers angular momentum for the disk-ring system about the axle during the interval 0<t<t10 < t < t_1.

Increases
Decreases
Remains constant
Justify your response.

FRQ

Rotational dynamics of horizontal bar torques

2. A rigid, uniform bar is mounted to a frictionless pivot at one end and can rotate in a horizontal plane, as shown in Figure 1.

Figure 1. Uniform bar of length L pivoted at one end with a constant perpendicular force applied at the free end.

Figure 1
A.

Identify all forces that exert a torque about the pivot on the bar, and identify the direction (clockwise or counterclockwise) of each torque. If a force produces zero torque about the pivot, state that explicitly. Consider the bar about the pivot at the left end. The force FF is applied at the free end and is always perpendicular to the bar.

Figure 2. Angular velocity ω as a function of time t for the first 2.0 s.

Figure 2
B.

Starting with the relationship between torque and angular acceleration, derive an expression for the magnitude of the applied force FF in terms of MM, LL, and the slope of the ω(t)\omega(t) graph in Figure 2. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The bar rotates in a horizontal plane, and Figure 2 shows ω(t)\omega(t) for the first 2.0 s. The rotational inertia of a uniform bar about an axis through one end and perpendicular to the bar is I=13ML2I = \frac{1}{3}ML^2.

Figure 3. Blank axes for sketching θ(t) and v(t) from 0 to 2.0 s.

Figure 3
C.

Figure 3 provides blank axes for a graph of angular displacement θ\theta as a function of time for the first 2.0 s. Assume the bar starts at θ=0\theta = 0 when t=0t = 0.

i.

Sketch and label a curve on Figure 3 that represents θ(t)\theta(t) for the first 2.0 s, consistent with Figure 2.

ii.

Sketch and label on Figure 3 a second curve that represents the tangential speed v(t)v(t) of the free end of the bar for the first 2.0 s. Use the same horizontal time axis, and use a separate vertical scale on the right side labeled vv (m/s).

D.

Indicate whether the kinetic energy of the bar at t=1.0 st = 1.0\ \text{s} is greater than, less than, or equal to one-fourth of the kinetic energy of the bar at t=2.0 st = 2.0\ \text{s}. Use the information given: M=2.0 kgM = 2.0\ \text{kg}, L=0.80 mL = 0.80\ \text{m}, and Figure 2 shows ω=9.0 rad/s\omega = 9.0\ \text{rad/s} at t=2.0 st = 2.0\ \text{s}.

K1.0 s>14K2.0 sK_{1.0\,\text{s}} > \frac{1}{4}K_{2.0\,\text{s}}
K1.0 s<14K2.0 sK_{1.0\,\text{s}} < \frac{1}{4}K_{2.0\,\text{s}}
K1.0 s=14K2.0 sK_{1.0\,\text{s}} = \frac{1}{4}K_{2.0\,\text{s}}
Justify how your response is consistent with the representations in parts B and C.

AP Physics 1 unit 6 practice test

A timed, AP-style test built only from unit 6 questions — multiple choice plus free response, scored with AI feedback. Use the standard format or build your own.

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Unit 6 review notes

6.1

Rotational Kinetic Energy

A spinning rigid system has rotational kinetic energy given by K = 1/2 I omega squared, where I is the rotational inertia and omega is the angular velocity. This is a scalar quantity. When an object both translates and rotates, total kinetic energy is the sum of translational and rotational parts: K_tot = 1/2 M v_cm squared + 1/2 I_cm omega squared. A rigid system can have rotational kinetic energy even when its center of mass is stationary, because individual points within the system still have linear speed.

  • K = 1/2 I omega squared: Rotational kinetic energy of a rigid system; depends on both how mass is distributed (I) and how fast it spins (omega).
  • Total KE decomposition: K_tot = K_trans + K_rot = 1/2 M v_cm squared + 1/2 I_cm omega squared; applies to any object with both translational and rotational motion.
  • Rotational inertia I: Measures how mass is distributed relative to the rotation axis; larger I means more energy stored at the same omega.
  • Scalar quantity: Rotational kinetic energy has magnitude only, no direction, just like translational kinetic energy.

A solid disk and a thin ring have the same mass and radius. Which stores more rotational kinetic energy at the same angular velocity, and why?

QuantityTranslational formRotational form
Kinetic energyK = 1/2 m v squaredK = 1/2 I omega squared
Inertiamass mrotational inertia I
Speedv (linear)omega (angular)
6.2

Torque and Work

A torque transfers energy into or out of a rotating system when it acts through an angular displacement. The work done by a constant torque is W = tau delta theta, where theta is measured in radians. For a variable torque, work equals the area under the torque-versus-angular-position graph. This is the rotational analog of W = F delta x from Unit 3.

  • W = tau delta theta: Work done by a torque acting through angular displacement delta theta; theta must be in radians.
  • Area under torque-angle graph: For variable torque, the work done equals the area under the curve of torque versus angular position.
  • Energy transfer direction: A torque in the same direction as angular displacement does positive work (adds energy); opposite direction does negative work (removes energy).
  • Work-energy theorem for rotation: Net work done by all torques equals the change in rotational kinetic energy of the system.

A torque of 4 N m acts on a wheel through an angular displacement of pi/2 radians. How much work is done on the wheel?

ConceptLinear versionRotational version
Work formulaW = F delta xW = tau delta theta
Variable-force workArea under F-x graphArea under tau-theta graph
Work-energy theoremW_net = delta KE_transW_net = delta KE_rot
6.3

Angular Momentum and Angular Impulse

Angular momentum quantifies how much rotational motion a system has. For a rigid system rotating about an axis, L = I omega. For a point object moving about a reference point, L = rmv sin theta, where theta is the angle between the radial distance and the velocity. The choice of reference axis affects the calculated value of L. Angular impulse equals tau times delta t and produces a change in angular momentum: delta L = tau delta t. This is the rotational impulse-momentum theorem. On a torque-versus-time graph, the area under the curve equals the angular impulse delivered.

  • L = I omega: Angular momentum of a rigid system; depends on rotational inertia and angular velocity.
  • L = rmv sin theta: Angular momentum of a point object about a reference point; theta is the angle between r and v.
  • Angular impulse = tau delta t: The product of net torque and time interval; equals the change in angular momentum of the system.
  • delta L = tau delta t: Rotational impulse-momentum theorem; angular impulse equals change in angular momentum.
  • Torque-time graph: The area under a torque-versus-time graph gives the angular impulse delivered to the system.

A net torque of 3 N m acts on a flywheel for 2 seconds. What is the change in angular momentum of the flywheel?

QuantityLinear formRotational form
Momentump = mvL = I omega
ImpulseJ = F delta tangular impulse = tau delta t
Impulse-momentum theoremdelta p = F delta tdelta L = tau delta t
Graph interpretationArea under F-t = impulseArea under tau-t = angular impulse
6.4

Conservation of Angular Momentum

The total angular momentum of a system is the sum of the angular momenta of its parts. If the net external torque on a chosen system is zero, total angular momentum is constant. If net external torque is nonzero, angular momentum is transferred between the system and its surroundings. A nonrigid system can change its angular speed without any external torque if it redistributes mass, changing its moment of inertia. The classic example is a spinning skater who pulls her arms in, decreasing I and increasing omega to keep L constant. System selection determines whether angular momentum is conserved.

  • Conservation condition: Angular momentum is conserved when net external torque on the chosen system is zero.
  • System selection: Choosing which objects to include in your system determines whether angular momentum changes or stays constant.
  • Nonrigid system: A system that can change shape, redistributing mass and changing I, so omega changes even with no external torque.
  • L_initial = L_final: When angular momentum is conserved: I_1 omega_1 = I_2 omega_2.
  • Newton's third law connection: Angular impulses between two interacting objects are equal and opposite, so internal torques cannot change total system angular momentum.

A student sits on a frictionless rotating stool holding weights at arm's length. She pulls the weights to her chest. Describe what happens to her angular velocity and why.

ConditionAngular momentumExample
Net external torque = 0Conserved (constant)Skater pulling arms in
Net external torque nonzeroChanges; transferred to/from environmentTorque applied by braking force
Internal torques onlyNo change to total LTwo parts of a system pushing on each other
6.5

Rolling Motion

Rolling combines translational motion of the center of mass with rotation about that center. When rolling without slipping, the no-slip condition links the two motions: v_cm = r omega, a_cm = r alpha, and delta x_cm = r delta theta. Total kinetic energy is K_tot = 1/2 M v_cm squared + 1/2 I omega squared. Static friction enables rolling without slipping but does no work and dissipates no energy. When slipping occurs, kinetic friction acts, the no-slip condition breaks down, and energy is dissipated as heat.

  • No-slip condition: v_cm = r omega; links the translational speed of the center of mass to the angular velocity when rolling without slipping.
  • K_tot = K_trans + K_rot: Total kinetic energy of a rolling object is the sum of translational and rotational kinetic energies.
  • Static friction in rolling: Provides the torque needed to maintain rolling without slipping; does no work and dissipates no energy.
  • Rolling while slipping: When slipping, v_cm and r omega are not equal; kinetic friction acts and dissipates energy from the system.
  • Effect of shape on rolling speed: Objects with more mass concentrated near the rim (larger I) roll more slowly down a ramp than objects with mass near the center.

A solid sphere and a hollow sphere of equal mass and radius roll from rest down the same ramp. Which reaches the bottom first, and what principle explains the difference?

Conditionv_cm vs r omegaFriction typeEnergy dissipated?
Rolling without slippingv_cm = r omegaStaticNo
Rolling while slippingv_cm not equal to r omegaKineticYes
6.6

Motion of Orbiting Satellites

When a satellite's mass is negligible compared to the central body, the central body's motion is ignored. Satellite orbits are governed by conservation of energy and angular momentum. In a circular orbit, total mechanical energy, gravitational potential energy, kinetic energy, and angular momentum are all constant. In an elliptical orbit, total mechanical energy and angular momentum remain constant, but kinetic energy and gravitational potential energy trade off as the satellite moves closer to or farther from the central body. Gravitational potential energy is defined as U_g = -G m_1 m_2 / r, with zero at infinite separation. Escape velocity is the speed at which total mechanical energy equals zero.

  • Circular orbit: Speed, kinetic energy, gravitational potential energy, and angular momentum are all constant throughout the orbit.
  • Elliptical orbit: Total mechanical energy and angular momentum are constant; kinetic energy and gravitational potential energy vary as the satellite moves.
  • U_g = -G m_1 m_2 / r: Gravitational potential energy; negative and defined as zero at infinite separation.
  • Escape velocity: v_esc = sqrt(2GM/r); the speed at which total mechanical energy of the satellite-planet system equals zero.
  • Conservation of angular momentum in orbits: A satellite moves fastest at closest approach (periapsis) and slowest at farthest point (apoapsis) to conserve L = rmv sin theta.

A satellite in an elliptical orbit is at its closest point to Earth. Compared to its farthest point, is its speed greater, smaller, or the same? Which conservation law explains this?

Orbit typeTotal mechanical energyAngular momentumKE and PE
CircularConstantConstantBoth constant
EllipticalConstantConstantBoth vary (trade off)

Key terms

TermDefinition
rigid systemA system that holds its shape but in which different points move in different directions during rotation; cannot be modeled as a single point object.
axis of rotationThe specified line about which a rigid system rotates; the choice of axis affects calculated values of angular momentum and rotational inertia.
scalarA physical quantity described by magnitude only; rotational kinetic energy is a scalar, just like translational kinetic energy.
impulse-momentum theoremIn rotational form: angular impulse (tau delta t) equals the change in angular momentum (delta L), directly paralleling the linear version.
elliptical orbitAn orbital path in which a satellite's distance from the central object varies; total mechanical energy and angular momentum are constant, but kinetic energy and gravitational potential energy each change.
escape velocityThe minimum speed needed for a satellite to escape a central object's gravity; at this speed, total mechanical energy of the system equals zero. v_esc = sqrt(2GM/r).
Mass independence of orbital periodThe orbital period of a satellite depends only on the orbital radius and the mass of the central body, not on the satellite's own mass.

How to review unit 6

Topics students miss most

Miss rates come from Fiveable multiple-choice practice.

Common unit 6 mistakes

Forgetting to include both KE terms for rolling objects

When a ball or cylinder rolls, total kinetic energy is K_trans + K_rot. Using only 1/2 mv squared ignores the rotational contribution and gives the wrong answer for speed or energy comparisons.

Using degrees instead of radians in rotational work

The formula W = tau delta theta requires angular displacement in radians. Converting to degrees before plugging in will produce an incorrect result.

Assuming angular momentum is always conserved

Angular momentum is conserved only when net external torque on the chosen system is zero. If an external torque acts, angular momentum changes. Always check your system boundary before applying conservation.

Confusing the two angular momentum formulas

L = I omega applies to a rigid system rotating about an axis. L = rmv sin theta applies to a point object about a reference point. Using the wrong formula, especially when the object moves in a straight line, leads to errors.

Treating elliptical and circular orbits the same

In a circular orbit, speed and both energy terms are constant. In an elliptical orbit, only total mechanical energy and angular momentum are constant; kinetic energy and gravitational potential energy both change as the satellite moves.

How this unit shows up on the AP exam

Conservation law reasoning across scenarios

AP Physics 1 questions in this unit frequently ask you to identify which quantities are conserved in a given scenario and explain why. You may need to compare a system before and after an interaction, such as a mass landing on a rotating disk or a satellite moving between orbital positions, and justify your answer using conservation of energy or angular momentum.

Graph interpretation for rotational quantities

Expect questions that present torque-versus-time or torque-versus-angular-position graphs and ask you to extract angular impulse or work from the area under the curve. You may also be asked to interpret the slope of an angular-momentum-versus-time graph as net torque.

Qualitative and quantitative system analysis

Free-response questions often ask you to both calculate a quantity (such as angular velocity after a collision or speed at a point in an elliptical orbit) and explain the physics in words. Connecting the formula to the underlying principle, such as why a skater spins faster or why a satellite speeds up at closest approach, is a key skill tested in this unit.

More ways to review

Topic study guides

Open the individual guides for Unit 6 when you want a closer review of one topic.

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Practice questions

Use AP-style practice after you review the notes so you can check what you understand.

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FRQ practice

Practice free-response reasoning and compare your answer with scoring guidance.

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Cram archive videos

Watch past review streams filtered to Unit 6 when you want a video walkthrough.

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Official unit cheatsheet

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Score calculator

Estimate your broader AP score goal after you review the course and exam format.

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Unit 6 printables

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Frequently Asked Questions

What's on the AP Physics 1 Unit 6 progress check (MCQ and FRQ)?

The AP Physics 1 Unit 6 progress check includes both MCQ and FRQ parts drawn from all six unit topics: rotational kinetic energy, torque and work, angular momentum, angular impulse, conservation of angular momentum, rolling, and orbiting satellites. MCQ questions typically ask you to apply or compare these concepts, while the FRQ part asks you to justify reasoning about energy and momentum in rotating systems. For matched practice that mirrors the progress check format, visit /ap-physics-1-revised/unit-6.

How do I practice AP Physics 1 Unit 6 FRQs?

AP Physics 1 Unit 6 FRQs most often come from torque and work, conservation of angular momentum, and rolling, since those topics require multi-step reasoning and written justification. Expect questions that ask you to calculate work done by a torque, explain why angular momentum is conserved, or analyze a rolling object's energy. Practice by writing out full solution steps and explaining your reasoning in words, not just equations. Find Unit 6 FRQ practice at /ap-physics-1-revised/unit-6.

Where can I find AP Physics 1 Unit 6 practice questions?

The best place to find AP Physics 1 Unit 6 practice questions, including multiple-choice and practice test sets, is /ap-physics-1-revised/unit-6. That page has MCQ and FRQ practice covering all six topics: rotational kinetic energy, torque, work, angular momentum, impulse, rolling, and orbiting satellites. Working through topic-by-topic MCQs before taking a full practice test helps you spot which concepts need more attention.

How should I study AP Physics 1 Unit 6?

Start with rotational kinetic energy and torque so you have a solid foundation before tackling angular momentum and angular impulse. Those two concepts build on each other the same way linear momentum and impulse do, so the comparison helps. Then work through conservation of angular momentum with concrete examples like a spinning skater pulling in their arms. Rolling is tricky because it mixes translational and rotational energy, so practice splitting the kinetic energy into both parts. Finish with orbiting satellites, connecting orbital motion back to energy conservation. Practice problems and topic guides for each step are at /ap-physics-1-revised/unit-6.

Ready to review Unit 6?Start with the notes, check the topic cards, and use the practice or resource links when they are available for this course.