Review AP Physics C: Mechanics Unit 7, Oscillations, with the main topics, key terms, practice focus, and exam connections in one place. Use this page as a full unit review before moving into individual guides or questions.
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Oscillations sets up a specific part of AP Physics C: Mechanics. Review the major topics in order, then check how each idea connects to examples, vocabulary, and AP-style questions.
Focus on Defining Simple Harmonic Motion (SHM), Frequency and Period of SHM, Representing and Analyzing SHM, Energy of Simple Harmonic Oscillators, Simple and Physical Pendulums. These are the ideas and resources most directly connected to this page.
This page helps connect Unit 7 content to later course ideas, vocabulary, and AP question types.
Practice explaining the idea in your own words, recognizing it in a prompt, and choosing evidence or calculations that fit the task.
The strongest review move is to connect terms, examples, and AP reasoning. For Unit 7, ask how each topic changes what you should notice in a question.
Describe simple harmonic motion.
Describe the frequency and period of an object exhibiting SHM.
Describe the displacement, velocity, and acceleration of an object exhibiting SHM.
Describe the mechanical energy of a system exhibiting SHM.
Describe the properties of a physical pendulum.
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Across 2.0k recent Unit 7 multiple-choice attempts.
Across 4 scored free-response attempts for this unit.
MCQ, FRQ, and SAQ activity included in this snapshot.
Review how Simple and Physical Pendulums shows up in practice questions, especially when prompts ask you to apply the idea in context.
Review how Representing and Analyzing SHM shows up in practice questions, especially when prompts ask you to apply the idea in context.
Review how Defining Simple Harmonic Motion (SHM) shows up in practice questions, especially when prompts ask you to apply the idea in context.
Review how Frequency and Period of SHM shows up in practice questions, especially when prompts ask you to apply the idea in context.
Describe simple harmonic motion.
Describe the frequency and period of an object exhibiting SHM.
Describe the displacement, velocity, and acceleration of an object exhibiting SHM.
Describe the mechanical energy of a system exhibiting SHM.
Describe the properties of a physical pendulum.
Try one image-based and one text-based AP-style MCQ, then practice a stimulus-based SAQ and one FRQ from each available type.
1. A uniform rigid rod of mass M = 0.80 kg and length L = 1.2 m is pivoted at one end and can oscillate as a physical pendulum in a vertical plane, as shown in Figure 1. The rod is displaced from its vertical equilibrium position by an angle θ₀ = 0.25 rad and released from rest at time t = 0. The moment of inertia of a uniform rod about an axis through one end is I = (1/3)ML².
Figure 1. Physical pendulum setup
Figure 2. Mechanical energy versus time
In Scenario 1, consider the rod oscillating with negligible friction.
Derive an expression for the angular frequency ω of the rod's oscillation. Express your answer in terms of L, g, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Calculate the numerical value of the period T of oscillation for the rod.
On the axes in Figure 2, sketch the total mechanical energy E of the rod-Earth system as a function of time t from t = 0 to t = 2.0 s. Explicitly label any maximum or minimum values with numerical quantities.
At the instant when the rod passes through its vertical equilibrium position for the first time after release, the angular speed of the rod is ω₁ = 0.52 rad/s. After one complete period of oscillation, the rod again passes through the vertical equilibrium position with angular speed ω₂. In Scenario 2, the rod oscillates in the viscous fluid with damping torque τ_damping = -bω_angular.
Derive an expression for the energy dissipated by the damping force during one complete period. Express your answer in terms of M, L, ω₁, ω₂, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
3. A student is given a uniform rigid rod of length L = 0.80 m and mass M = 0.50 kg. The rod can be pivoted about different points along its length to create a physical pendulum.
Students are asked to experimentally determine the moment of inertia of the rod about its center of mass, I_cm, using a linear graph. To determine I_cm, the students are permitted to use measurements from only a meterstick, a protractor, and the stopwatch.
Describe an experimental procedure using the described setup to collect data that would allow the students to determine an experimental value of I_cm using a linear graph. Include any steps necessary to reduce experimental uncertainty. In your description, state what quantities would be measured and how each measurement would be made.
Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine the value of I_cm. Include:
Figure 1
Figure 2
Figure 3
d (m) | T (s) |
|---|---|
0.10 | 1.52 |
0.20 | 1.37 |
0.30 | 1.32 |
0.50 | 1.36 |
0.60 | 1.45 |
The student conducts the experiment by pivoting the rod at various distances d from one end. For each pivot position, the rod is displaced by a small angle (less than 10°) and released from rest. The student measures the period T of oscillation for each position.
The student's measurements of d and T are shown in Table 1.
Note: For a physical pendulum, the period of small-angle oscillations is given by T = 2π√(I/(Mgh)), where I is the moment of inertia about the pivot point, M is the mass of the rod, g is the acceleration due to gravity (9.8 m/s²), and h is the perpendicular distance from the pivot to the center of mass.
Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine I_cm. The parallel axis theorem states that I = I_cm + Mh², where I is the moment of inertia about the pivot and h is the distance from the pivot to the center of mass.
Vertical axis: Horizontal axis:
On the grid provided in Figure 3, create a graph of the quantities indicated in part C(i).
Use Table 2 to record the measured or calculated quantities that you will plot.
Clearly label the axes, including units as appropriate.
Plot the points you recorded in Table 2.
Draw a best-fit line to the data graphed in part C(ii).
Using the best-fit line that you drew in part C(iii), calculate an experimental value for I_cm in kg·m².
2. A uniform rigid rod of mass M = 0.80 kg and length L = 1.2 m is suspended from a frictionless pivot at one end, forming a physical pendulum. The rod is displaced by a small angle θ₀ = 8.0° from the vertical and released from rest at time t = 0, as shown in Figure 1. The acceleration due to gravity is g = 10 m/s².
Figure 1. Physical pendulum with uniform rod
i. Calculate the moment of inertia of the rod about the pivot point.
ii. Derive an expression for the angular frequency ω of small oscillations for this physical pendulum in terms of M, g, L, and fundamental constants. Then calculate the numerical value of the period T of oscillation.
Figure 2. Angular displacement of physical pendulum
On Figure 2 provided, sketch a graph of the angular displacement θ (in degrees) as a function of time t for the interval from t = 0 to t = 3.0 s. Clearly indicate the amplitude and period on your graph.
At the instant when the rod passes through the equilibrium position (θ = 0):
i. Calculate the angular velocity of the rod.
ii. Calculate the linear speed of the bottom end of the rod.
iii. Calculate the total mechanical energy of the system.
When the rod is at an angular displacement of θ = 4.0° from vertical:
i. Determine the angular acceleration of the rod. Specify both magnitude and direction.
ii. Calculate the tangential acceleration of the bottom end of the rod.
A student claims that if the rod were instead suspended from a pivot point located at distance d from one end (where d < L), there exists a specific value of d that would minimize the period of oscillation. Is the student's claim correct? Justify your answer with appropriate physics reasoning or mathematical analysis.
4. A uniform rigid rod of mass M = 0.80 kg and length L = 1.2 m is suspended from a frictionless pivot at a point P that is a distance d = 0.30 m from one end of the rod. The rod is displaced by a small angle from its equilibrium position and released, causing it to oscillate as a physical pendulum. The moment of inertia of a uniform rod about its center of mass is I_cm = (1/12)ML². The acceleration due to gravity is g = 10 m/s².
Figure 1. Uniform rod physical pendulum
![Create a clean, black-on-white mechanics diagram with an explicit 2D coordinate system used only for placement (axes themselves are NOT drawn).
CANVAS / COORDINATE SYSTEM (for precise placement):
- Use a rectangular canvas with coordinates in centimeters.
- Canvas width: 18.0 cm, height: 14.0 cm.
- Define the origin (0,0) at the bottom-left corner of the canvas.
GLOBAL STYLE:
- All object outlines: solid black, 2 pt stroke.
- Dashed lines: black, 2 pt stroke, dash pattern 6 mm dash / 4 mm gap.
- Labels: sans-serif font, 14 pt, black.
- Angle arc: solid black, 2 pt.
- No grid lines.
ROD GEOMETRY (DRAW TWO RODS: equilibrium reference and displaced rod):
Physical scale mapping (must be used consistently): 1.0 m in the physics problem corresponds to 6.0 cm in the drawing.
- Therefore rod length L = 1.2 m corresponds to 7.2 cm.
- Pivot offset d = 0.30 m corresponds to 1.8 cm.
1) EQUILIBRIUM REFERENCE LINE (vertical dashed line):
- Draw a vertical dashed line representing the equilibrium orientation of the rod.
- Place the dashed line at x = 9.0 cm.
- The dashed line runs from y = 2.0 cm to y = 12.5 cm.
- Label near the top of this dashed line: text 'equilibrium' centered at (9.0 cm, 12.9 cm).
2) PIVOT LOCATION P AND EQUILIBRIUM ROD (faint reference rod):
- Define the pivot point P at coordinate P = (9.0 cm, 10.0 cm).
- Draw a small filled black circle of diameter 3 mm centered at P.
- Label 'P' with text baseline left of the point at (9.3 cm, 10.2 cm).
- Draw the equilibrium rod as a thin solid line (1 pt) collinear with the dashed equilibrium line, centered such that the pivot is located along the rod at the correct offset distance from the upper end.
- Because d = 1.8 cm from the upper end to pivot: place the upper end of the rod at coordinate A_eq such that distance A_eq to P along the rod is 1.8 cm upward.
- Thus set the equilibrium upper end A_eq = (9.0 cm, 11.8 cm).
- The lower end is then B_eq = A_eq shifted down by L = 7.2 cm: B_eq = (9.0 cm, 4.6 cm).
- Mark the endpoints with small open circles (3 mm diameter) at A_eq and B_eq.
- Label the endpoints:
- Label 'top end' at (9.6 cm, 11.8 cm) aligned horizontally with A_eq.
- Label 'bottom end' at (9.6 cm, 4.6 cm) aligned horizontally with B_eq.
3) DISPLACED ROD (main bold rod at angle θ to the right):
- The displaced rod is the primary object: draw it as a solid black line, 2 pt stroke, rotating about the same pivot point P.
- Set the angular displacement to a specific displayed value so there is no ambiguity: θ = +15.0° (clockwise from the vertical equilibrium line, i.e., the rod leans to the right).
- The displaced rod must have the same length L = 7.2 cm and the pivot must be located d = 1.8 cm from its upper end measured along the rod.
- Compute displaced-rod direction unit vector for 15.0° clockwise from vertical:
- Relative to +x right, +y up: the rod’s axis points downward-right from the upper end toward the lower end.
- From pivot to upper end is 1.8 cm along the rod in the upward-left direction.
- Place displaced upper end A at:
- A = P + (1.8 cm) * [sin(15.0°) left negative x, cos(15.0°) up positive y] with clockwise tilt meaning upper end is left of pivot.
- Numerically: sin15 = 0.258819, cos15 = 0.965926.
- Δx = -1.8*0.258819 = -0.466 cm; Δy = +1.8*0.965926 = +1.739 cm.
- So A = (9.0 - 0.466, 10.0 + 1.739) = (8.534 cm, 11.739 cm).
- Place displaced lower end B by moving from A along the rod length 7.2 cm downward-right:
- From A to B: Δx = +7.2*0.258819 = +1.864 cm; Δy = -7.2*0.965926 = -6.954 cm.
- So B = (8.534 + 1.864, 11.739 - 6.954) = (10.398 cm, 4.785 cm).
- Draw the displaced rod as a straight line segment from A to B.
- Mark endpoints A and B with open circles (3 mm diameter).
- Do NOT redraw P as a second dot; the displaced rod must pass through P.
4) CENTER OF MASS (CM) MARKING ON DISPLACED ROD:
- The rod is uniform, so the center of mass is at its geometric midpoint (halfway along length).
- Midpoint of displaced rod: CM = ((8.534+10.398)/2, (11.739+4.785)/2) = (9.466 cm, 8.262 cm).
- Draw a small filled black circle of diameter 3 mm at CM.
- Label 'CM' with text placed at (9.9 cm, 8.4 cm).
5) DIMENSION ANNOTATIONS (must show exact given values L and d):
- Rod length L:
- Draw a dimension bracket parallel to the displaced rod, offset 0.6 cm to the right of the rod.
- The bracket endpoints project perpendicularly from A and B to the bracket line.
- Put arrowheads at both ends of the bracket.
- Center the text 'L = 1.2 m' at the midpoint of the bracket.
- Pivot offset d:
- Draw a shorter dimension bracket along the displaced rod between A (upper end) and pivot point P.
- Offset this bracket 0.6 cm to the left of the displaced rod.
- Arrowheads at both ends; centered label 'd = 0.30 m'.
6) ANGLE θ ANNOTATION:
- Draw an angle arc centered at the pivot P between the vertical dashed equilibrium line and the displaced rod.
- Use a radius of 1.4 cm.
- Arc starts on the equilibrium line at point (9.0 cm, 11.4 cm) (this is exactly 1.4 cm above P) and ends on the displaced rod at the point located 1.4 cm from P toward the lower end along the displaced rod.
- That endpoint coordinate is: P + 1.4*[+sin15, -cos15] = (9.0 + 0.362, 10.0 - 1.352) = (9.362 cm, 8.648 cm).
- Place the label 'θ = 15°' near the arc, at (9.7 cm, 10.9 cm).
7) FORCE-VECTOR DRAWING SPACE (explicit blank region):
- Leave a clear blank region around the displaced rod for student-drawn vectors by ensuring no text overlaps the rod within a 1.5 cm radius of the CM point (9.466, 8.262).
- Include a light instruction text (still black, 14 pt) in the lower-left quadrant: 'Draw forces here' with its baseline starting at (1.0 cm, 2.0 cm).
8) GIVEN VALUES TEXT BOX (to ensure all numerical values in the prompt appear on the figure):
- Add a small text box (no border) in the upper-left corner listing constants exactly:
- Line 1 at (1.0 cm, 12.6 cm): 'M = 0.80 kg'
- Line 2 at (1.0 cm, 12.0 cm): 'L = 1.2 m'
- Line 3 at (1.0 cm, 11.4 cm): 'd = 0.30 m'
- Line 4 at (1.0 cm, 10.8 cm): 'g = 10 m/s²'
- Line 5 at (1.0 cm, 10.2 cm): 'pivot: frictionless'
FINAL CHECKS FOR NUMERICAL ACCURACY:
- The displaced rod must pass exactly through P = (9.0, 10.0).
- Distance along the rod from A to P must be exactly 1.8 cm (corresponding to 0.30 m).
- Total rod length from A to B must be exactly 7.2 cm (corresponding to 1.2 m).
- The angle between the displaced rod and the vertical dashed line must be exactly 15.0° clockwise.
- CM must be exactly at the midpoint coordinate (9.466, 8.262).](/_next/image?url=https%3A%2F%2Fkcbwzropdukwhmmvdylp.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Ffrq_stimulus%2Fap-physics-c-mechanics%2F4bc05c34-36da-48a0-9796-06cfbd3d1ef1_fig1.jpg&w=1080&q=75)
On the diagram below, draw and label vectors to represent all forces acting on the rod when it is displaced at a small angle θ from the vertical equilibrium position. Draw the vectors with their tails at the point of application of each force.
i. Derive an expression for the period T of small oscillations of this physical pendulum in terms of M, L, d, and g. Begin with the rotational form of Newton's second law.
ii. Calculate the numerical value of the period T for this physical pendulum.
When the rod is at its maximum angular displacement of θ_max = 0.15 rad from equilibrium, determine:
i. The angular velocity ω of the rod
ii. The magnitude of the linear velocity v of the free end of the rod (the end farthest from the pivot)
As the physical pendulum passes through its equilibrium position, the rod has angular velocity ω_eq. Derive an expression for ω_eq in terms of θ_max, M, L, d, and g using energy conservation principles.
A student claims that if the pivot point P were moved closer to the center of the rod, the period of oscillation would decrease. Is this claim correct? Justify your answer qualitatively by describing how the relevant physical quantities would change.
| Term | Definition |
|---|---|
| simple harmonic motion (SHM) | simple harmonic motion (SHM) is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| periodic motion (special case) | periodic motion (special case) is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| restoring force proportional to displacement | restoring force proportional to displacement is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| Hooke's law F = -kx | Hooke's law F = -kx is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| mass-spring system | mass-spring system is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| m a_x = -k Δx | m a_x = -k Δx is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| linear second-order ODE d^2x/dt^2 + (k/m)x = 0 | linear second-order ODE d^2x/dt^2 + (k/m)x = 0 is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| sinusoidal solution x(t)=A cos(ωt+φ) | sinusoidal solution x(t)=A cos(ωt+φ) is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| angular frequency ω = √(k/m) | angular frequency ω = √(k/m) is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
| period T = 2π√(m/k) | period T = 2π√(m/k) is worth reviewing because it connects directly to Unit 7, Oscillations and can help you answer AP-style questions with more specific evidence. |
Definitions help, but AP questions usually ask you to apply terms to a new example, source, data set, or prompt.
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Start with the big picture, then move into the topic or skill guide that matches the exact thing you missed.
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Before the exam, use this page to decide whether Unit 7, Oscillations needs a quick skim, targeted practice, or a deeper guide review.
Open the individual guides for Unit 7 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Physics C: Mechanics Unit 7 covers 5 topics: Defining Simple Harmonic Motion (SHM), Frequency and Period of SHM, Representing and Analyzing SHM, Energy of Simple Harmonic Oscillators, and Simple and Physical Pendulums. Together these topics build from the restoring force definition of SHM through energy analysis and pendulum systems. See the full topic breakdown at /ap-physics-c-mechanics/unit-7.
Unit 7: Oscillations makes up 10-15% of the AP Physics C: Mechanics exam, making it one of the more heavily tested units. That weight covers everything from defining simple harmonic motion and analyzing displacement, velocity, and acceleration, to the energy of oscillators and the behavior of simple and physical pendulums.
The AP Physics C: Mechanics Unit 7 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all five unit topics: Defining SHM, Frequency and Period of SHM, Representing and Analyzing SHM, Energy of Simple Harmonic Oscillators, and Simple and Physical Pendulums. MCQ questions typically test conceptual understanding and equation application, while the FRQ portion asks you to derive expressions, sketch graphs of displacement or energy, and analyze pendulum systems. For matched progress check practice, visit /ap-physics-c-mechanics/unit-7.
The best way to practice Unit 7 FRQs is to focus on the three topics that generate the most free-response questions: Representing and Analyzing SHM, Energy of Simple Harmonic Oscillators, and Simple and Physical Pendulums. FRQs in this unit typically ask you to derive equations of motion using Newton's second law, sketch or interpret displacement and energy graphs over time, and compare simple versus physical pendulum periods. Start by writing out full solutions by hand, checking that your calculus steps are shown clearly, since AP Physics C FRQs award method points. Find practice problems and worked examples at /ap-physics-c-mechanics/unit-7.
You can find AP Physics C: Mechanics Unit 7 practice questions, including multiple-choice and practice test sets, at /ap-physics-c-mechanics/unit-7. That page organizes MCQ and FRQ practice by topic, covering Simple Harmonic Motion, frequency and period, energy of oscillators, and pendulums, so you can target whichever area needs the most work before your exam.
Start with Topic 7.1 and make sure you can state the SHM condition precisely: the restoring force must be proportional to displacement. From there, work through frequency and period relationships in Topic 7.2, then move to Topic 7.3 where you practice writing and solving the differential equation of motion. Topic 7.4 on energy is high-yield, so practice converting between kinetic and potential energy at different points in the oscillation cycle. Finish with Topic 7.5 by deriving the period formulas for both simple and physical pendulums and knowing when each applies. A few concrete habits that help: draw a free-body diagram before every problem, practice sketching x(t), v(t), and a(t) graphs from scratch, and do at least one full FRQ under timed conditions. Visit /ap-physics-c-mechanics/unit-7 for topic guides and practice sets organized in this order.