---
title: "Pendulum — AP Physics C: Mechanics Definition & Exam Guide"
description: "A pendulum is a mass swinging from a pivot whose small-angle motion is simple harmonic, with period T = 2π√(L/g). Core to Unit 6 SHM and physical pendulum FRQs."
canonical: "https://fiveable.me/ap-physics-c-mechanics/key-terms/pendulum"
type: "key-term"
subject: "AP Physics C: Mechanics"
unit: "Unit 6"
---

# Pendulum — AP Physics C: Mechanics Definition & Exam Guide

## Definition

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.

## What It Is

A pendulum is a [mass](/ap-physics-c-mechanics/key-terms/mass "fv-autolink") hanging from a pivot that swings back and forth because gravity creates a [restoring torque](/ap-physics-c-mechanics/unit-7/5-simple-and-physical-pendulums/study-guide/m0lcXe33VLYhg8EA "fv-autolink") 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](/ap-physics-c-mechanics "fv-autolink") 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."

## Why It Matters

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](/ap-physics-c-mechanics/unit-6 "fv-autolink") 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](/ap-physics-c-mechanics/key-terms/collision "fv-autolink"). 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.

## Connections

### Restoring Force and SHM (Unit 6)

A pendulum is SHM only because gravity provides a restoring torque proportional to [displacement](/ap-physics-c-mechanics/unit-1/2-displacement-velocity-and-acceleration/study-guide/robnlCwaanT6NImP "fv-autolink") 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](/ap-physics-c-mechanics/unit-5 "fv-autolink") 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](/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV "fv-autolink") 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.

## On the AP Exam

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.

## Pendulum vs Spring-mass oscillator

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.

## Key Takeaways

- 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.

## FAQs

### What is a pendulum in AP Physics C: Mechanics?

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.

### Does the mass of a pendulum affect its period?

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.

### Is a pendulum always simple harmonic motion?

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.

### What's the difference between a simple pendulum and a physical pendulum?

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.

### How is a pendulum different from a mass on a spring?

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.

## Related Study Guides

- [6.1 Simple Harmonic Motion, Springs, and Pendulums ](/ap-physics-c-mechanics/unit-6/simple-harmonic-motion-springs-pendulums/study-guide/V3L3GpIEmeGgpS239Xdd)

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