Unit 7 is about systems that oscillate back and forth around an equilibrium position because a restoring force proportional to displacement acts on them. You will write and solve the SHM differential equation, calculate period and frequency for springs and pendulums, track kinetic and potential energy through a cycle, and analyze physical pendulums using rotational Newton's second law.
Fiveable's AP Physics C: Mechanics Unit 7 review covers 5 topics, with guided practice questions for the unit.
SHM occurs when a restoring force is proportional to displacement: F = -kx. Newton's second law gives the differential equation d²x/dt² = -(k/m)x, whose solution is sinusoidal. The equilibrium position is where net force is zero.
Period, frequency, and angular frequency are linked by T = 2π/ω = 1/f. Spring-mass period is T = 2π√(m/k); simple pendulum period is T = 2π√(l/g). Period is independent of amplitude for ideal SHM.
Position follows x(t) = A cos(ωt + φ). Velocity and acceleration are obtained by differentiation. Velocity is maximum at equilibrium; acceleration is maximum at turning points. All three quantities are sinusoidal and phase-shifted relative to each other.
Total energy E = (1/2)kA² is constant. Energy shifts between kinetic and potential throughout each cycle. Speed at any position is v(x) = ω√(A² - x²). Doubling amplitude quadruples total energy.
A physical pendulum is a rigid body oscillating under gravity with period T = 2π√(I/mgd). The small-angle approximation linearizes the restoring torque. A simple pendulum is the point-mass special case. Torsion pendulums use a wire's restoring torque instead of gravity.
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SHM is periodic motion in which the net force on an object is always directed toward an equilibrium position and has magnitude proportional to the displacement from that position. Applying Newton's second law gives m(a_x) = -kΔx, which rearranges to the second-order differential equation d²x/dt² = -(k/m)x. The negative sign is essential: it guarantees the force always opposes displacement, producing oscillation rather than runaway motion. The equilibrium position is where net force equals zero; the restoring force is zero there and maximum at the turning points.
If a spring constant doubles while mass stays the same, what happens to ω and to the period T?
The period T, frequency f, and angular frequency ω are always related by T = 2π/ω = 1/f. For a mass-spring system, substituting ω = √(k/m) gives T_s = 2π√(m/k). For a simple pendulum at small angles, the restoring force component along the arc is -mg sinθ ≈ -mgθ, which produces ω = √(g/l) and T_p = 2π√(l/g). A critical exam point: period does not depend on amplitude for ideal SHM. Increasing mass increases T for a spring but has no effect on T for a simple pendulum.
A pendulum on the Moon (g smaller) compared to Earth: does T increase, decrease, or stay the same? Justify using T_p = 2π√(l/g).
| System | Angular frequency ω | Period T | Depends on mass? |
|---|---|---|---|
| Mass-spring | √(k/m) | 2π√(m/k) | Yes |
| Simple pendulum | √(g/l) | 2π√(l/g) | No |
| Physical pendulum | √(mgd/I) | 2π√(I/mgd) | Yes (through I and d) |
The general solution to d²x/dt² = -ω²x is x(t) = A cos(ωt + φ), where A is amplitude and φ is the phase constant set by initial conditions. Differentiating gives v(t) = -Aω sin(ωt + φ) and a(t) = -Aω² cos(ωt + φ) = -ω²x(t). Velocity and displacement are 90° out of phase: velocity is maximum when displacement is zero (at equilibrium) and zero when displacement is maximum (at turning points). Acceleration is always proportional to and opposite in sign to displacement. On graphs, x(t), v(t), and a(t) are all sinusoidal with the same period but shifted relative to each other.
At the moment an object passes through equilibrium moving in the positive direction, what are the signs of velocity and acceleration?
For an undamped spring-mass oscillator, total mechanical energy is constant: E_total = K + U = (1/2)mv² + (1/2)kx² = (1/2)kA². At the turning points (x = ±A), kinetic energy is zero and potential energy equals (1/2)kA². At equilibrium (x = 0), potential energy is zero and kinetic energy is maximum at (1/2)mv_max² = (1/2)kA². The velocity at any position can be found without tracking time: v(x) = ω√(A² - x²). Increasing amplitude increases total energy because E_total scales as A².
If the amplitude of a spring-mass oscillator is doubled, by what factor does the total energy change?
A physical pendulum is any rigid body that pivots about a fixed axis and oscillates under gravity. When displaced by angle θ, the gravitational force on the center of mass produces a restoring torque τ = -mgd sinθ, where d is the distance from the pivot to the center of mass. Applying Newton's second law in rotational form (τ = Iα) and using the small-angle approximation sinθ ≈ θ gives d²θ/dt² = -(mgd/I)θ, which is the SHM differential equation with ω = √(mgd/I). The period is T_phys = 2π√(I/mgd). A simple pendulum is the special case where all mass is concentrated at distance l from the pivot, so I = ml² and T reduces to 2π√(l/g). A torsion pendulum uses a twisted wire supplying restoring torque τ = -κθ, giving ω = √(κ/I).
A uniform rod of mass m and length L is pivoted at one end. Write an expression for its period using I = (1/3)mL² and d = L/2.
| Pendulum type | Restoring mechanism | Period formula | Key quantity |
|---|---|---|---|
| Simple pendulum | Gravity on point mass | 2π√(l/g) | Length l |
| Physical pendulum | Gravity on rigid body's CM | 2π√(I/mgd) | Moment of inertia I |
| Torsion pendulum | Twisted wire torque τ = -κθ | 2π√(I/κ) | Torsional constant κ |
| Term | Definition |
|---|---|
| amplitude | The maximum displacement from equilibrium in SHM; determines the maximum potential energy and total energy of the system via E_total = (1/2)kA². |
| angular displacement | The angle in radians through which a pendulum or rotating body moves from its equilibrium position; used in the rotational form of Newton's second law for physical pendulums. |
| damped oscillation | Oscillatory motion in which amplitude decreases over time because non-conservative forces such as friction dissipate energy from the system. |
| equilibrium position | The position where net force on the oscillating object is zero; the center of the oscillation where speed is maximum and potential energy is minimum. |
| oscillation | Repeated back-and-forth motion of an object about an equilibrium position; SHM is the special case where the restoring force is proportional to displacement. |
| periodic motion | Motion that repeats at regular time intervals; SHM is a special case characterized by a sinusoidal position function and a restoring force proportional to displacement. |
| small-angle approximation | The approximation sinθ ≈ θ (in radians) for small angular displacements; converts the pendulum's nonlinear restoring torque into the linear SHM form needed to derive period formulas. |
| torsional constant | The proportionality constant κ relating restoring torque to angular displacement in a torsion pendulum: τ = -κθ; determines the angular frequency ω = √(κ/I). |
| torsional pendulum | A system in which a mass suspended by a wire oscillates rotationally; the wire supplies a restoring torque τ = -κθ and the period is T = 2π√(I/κ). |
Miss rates come from Fiveable multiple-choice practice.
Angular frequency ω is in radians per second, not hertz. Always convert using T = 2π/ω before comparing to period or frequency values. Plugging ω directly into a formula that expects f will give a wrong answer by a factor of 2π.
For ideal SHM, period is independent of amplitude. A larger push does not make a spring or pendulum oscillate faster or slower. This is only true for small angles in pendulums; large-angle swings do not follow ideal SHM.
If the object does not start at maximum displacement at t = 0, φ is not zero. Use initial conditions x(0) and v(0) to solve for both A and φ before writing the full position function.
T_p = 2π√(l/g) only applies when all mass is concentrated at a single point distance l from the pivot. For any rigid body, you must use T_phys = 2π√(I/mgd) with the correct moment of inertia about the pivot.
Speed is maximum at equilibrium (x = 0), not at the turning points. At x = ±A, the object is momentarily at rest. Students who mix up where kinetic and potential energy peak will get energy and velocity questions wrong.
Free-response problems frequently ask students to start from F = ma or τ = Iα, apply the restoring force or torque, and derive the period formula rather than simply recall it. Practice writing the differential equation, identifying ω², and substituting into T = 2π/ω for both spring-mass and pendulum systems.
Multiple-choice and free-response items often present x(t), v(t), or a(t) graphs and ask students to read off amplitude, period, or phase, or to match a graph to an equation. Be ready to identify where each quantity is zero or extremal and to explain the physical meaning of those points.
When a problem asks for speed at a specific position rather than at a specific time, energy conservation via E_total = (1/2)kA² = (1/2)mv² + (1/2)kx² is faster and less error-prone than the time-based kinematic equations. Exam problems often reward students who recognize when to switch methods.
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get printablesThe 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.