In AP Physics C: Mechanics, a pendulum is a mass suspended from a pivot that oscillates under gravity's restoring torque; for small angles its motion is simple harmonic with period T = 2π√(L/g) for a simple pendulum, independent of mass and amplitude.
A pendulum is a mass hanging from a pivot that swings back and forth because gravity creates a restoring torque pulling it toward the lowest point. Displace it by an angle θ and the torque is proportional to sin θ. For small angles, sin θ ≈ θ, so the restoring torque becomes proportional to the displacement itself. That proportionality is the defining condition for simple harmonic motion, which is why the small-angle approximation matters so much. Swing it too far and the motion is still periodic, but it's no longer SHM and the standard period formulas stop being exact.
AP Physics C cares about three flavors. A simple pendulum is a point mass on a massless string, with T = 2π√(L/g). A physical pendulum is any rigid body swinging about a pivot, with T = 2π√(I/mgd), where I is the rotational inertia about the pivot and d is the distance from the pivot to the center of mass. A torsional pendulum twists instead of swings, with a wire supplying restoring torque τ = -kθ and period T = 2π√(I/k). Notice the pattern. Every version is the same math, just with different things playing the roles of "inertia" and "restoring stiffness."
The pendulum lives in Topic 6.1 (Simple Harmonic Motion, Springs, and Pendulums) and is one of the two canonical SHM systems on the exam, alongside the spring-mass oscillator. It's also where Unit 6 reaches backward into earlier units. Deriving the physical pendulum's period requires rotational inertia and torque from rotation, and classic FRQs bolt a pendulum onto energy conservation or a collision. The 2019 FRQ released by College Board did exactly that, swinging a block on a string into a collision at the bottom of its arc. If you can write Newton's second law for rotation about the pivot, apply the small-angle approximation, and recognize the resulting equation as SHM, you've demonstrated the central skill Topic 6.1 is built around.
Keep studying AP Physics C: Mechanics Unit 6
Restoring Force and SHM (Unit 6)
A pendulum is SHM only because gravity provides a restoring torque proportional to displacement when angles are small. Spot the form α = -ω²θ in your equation and you can read the period straight off it, no memorized formula needed.
Rotational Inertia and Torque (Unit 5)
The physical pendulum is really a Unit 5 problem in disguise. You write τ = Iα about the pivot, often using the parallel axis theorem to find I, and the SHM falls out. The 2023 FRQ's torsional pendulum (a disk on a twisting wire) tested exactly this crossover.
Conservation of Energy (Unit 3)
Speed at the bottom of a swing comes from energy conservation (mgh converts to ½mv²), not from SHM equations. Large-amplitude swings, like a string released from horizontal, are energy problems first.
Momentum and Collisions (Unit 4)
A favorite FRQ mashup drops a swinging pendulum into a collision at the lowest point, ballistic-pendulum style. The 2019 FRQ did this. The trap is using energy through the collision; you must use momentum conservation there instead.
Pendulums show up in both MCQs and FRQs. Multiple-choice stems love proportional reasoning, asking what happens to the period when you double L (it increases by √2), change the mass (nothing, for a simple pendulum), or move the setup to the Moon (T increases because g decreases). FRQs go deeper. The 2023 FRQ asked about a torsional pendulum made from a uniform disk, which means deriving the period from τ = Iα with the disk's rotational inertia. The 2019 FRQ used a pendulum released from horizontal as the setup for an energy-then-collision problem. Expect to derive, not just plug in. The highest-value skill is starting from Newton's second law for rotation, applying sin θ ≈ θ, and showing the equation matches the SHM form so you can identify ω and T. Also be ready to justify in words why the small-angle approximation is required for SHM.
Both are SHM systems in Topic 6.1, but their periods depend on different things. A spring's period T = 2π√(m/k) depends on mass and not gravity, while a simple pendulum's T = 2π√(L/g) depends on gravity and not mass. Take both to the Moon and the spring's period stays the same while the pendulum's period grows. That contrast is a classic MCQ.
A simple pendulum's period is T = 2π√(L/g), which depends on length and gravitational field strength but not on the mass of the bob or (for small angles) the amplitude.
Pendulum motion is only simple harmonic when the small-angle approximation sin θ ≈ θ holds, because that's what makes the restoring torque proportional to displacement.
A physical pendulum (any rigid swinging body) has period T = 2π√(I/mgd), so you need rotational inertia about the pivot, often via the parallel axis theorem.
A torsional pendulum twists with restoring torque τ = -kθ and period T = 2π√(I/k), and the 2023 FRQ built an entire problem around one made from a uniform disk.
For large swings, like a release from horizontal, use energy conservation to find speed at the bottom, and use momentum conservation if a collision happens there, as in the 2019 FRQ.
Doubling the length multiplies the period by √2, and moving to a weaker gravitational field makes the period longer.
It's a mass suspended from a pivot that oscillates under gravity's restoring torque. For small angles the motion is simple harmonic, with T = 2π√(L/g) for a simple pendulum and T = 2π√(I/mgd) for a physical pendulum. It's covered in Topic 6.1.
No, for a simple pendulum. Mass cancels out of the equation of motion, so T = 2π√(L/g) has no m in it. For a physical pendulum the mass distribution matters through I, but a uniform scaling of mass still cancels.
No. It's only approximately SHM when the amplitude is small enough that sin θ ≈ θ. At large angles the motion is still periodic, but the restoring torque is no longer proportional to displacement, so T = 2π√(L/g) is no longer exact.
A simple pendulum is an idealized point mass on a massless string with T = 2π√(L/g). A physical pendulum is a real rigid body swinging about a pivot, so its period T = 2π√(I/mgd) depends on its rotational inertia about that pivot. The simple pendulum is just the special case where I = mL² and d = L.
Both are SHM, but a spring's period T = 2π√(m/k) depends on mass and ignores gravity, while a simple pendulum's T = 2π√(L/g) depends on gravity and ignores mass. On the Moon, the spring oscillates with the same period and the pendulum slows down.