Pendulums in Oscillations

students, imagine a playground swing or a grandfather clock ticking away ⏰. Both are examples of a pendulum-like system that moves back and forth in a repeating pattern. In AP Physics C: Mechanics, pendulums are important because they show how oscillations work in a real system where gravity provides the restoring force. In this lesson, you will learn the key ideas, the equations that describe pendulums, and how to reason about them on the AP exam.

What is a Pendulum?

A pendulum is any object that can swing back and forth about a stable equilibrium position. The simplest version is a simple pendulum, which is modeled as a point mass attached to a massless string of length $L$. The string is fixed at one end, and the mass swings under the influence of gravity.

The main idea is that when the pendulum is pulled away from equilibrium and released, gravity pulls it back toward the lowest point. That pull acts as a restoring force because it tries to restore the system to equilibrium. If the displacement is small, the motion is approximately periodic, meaning it repeats at regular time intervals.

A few important terms:

Equilibrium position: the lowest point of the swing.
Displacement: the angular position $\theta$ measured from equilibrium.
Amplitude: the maximum angular displacement.
Period: the time for one full back-and-forth cycle, written as $T$.
Frequency: the number of cycles per second, written as $f$.

The relationship between period and frequency is $f=\frac{1}{T}$.

A pendulum is a classic example of oscillatory motion because the motion repeats and can be described with many of the same ideas used for springs and simple harmonic motion.

Forces Acting on a Pendulum

To analyze a pendulum, students, the first step is to draw a free-body diagram. The bob has two main forces acting on it:

the weight $mg$ downward
the tension $T$ in the string along the string direction

The weight can be split into two components:

a radial component along the string
a tangential component along the arc of motion

The tangential component is what matters for the restoring motion. It has magnitude $mg\sin\theta$ and points toward the equilibrium position.

Using Newton’s second law in the tangential direction gives

$$\sum F_t = -mg\sin\theta = ma_t$$

where $a_t$ is the tangential acceleration. The negative sign shows that the force acts opposite the displacement when $\theta$ is measured from equilibrium.

For small angles, the pendulum becomes easier to analyze. When $\theta$ is small in radians, we can use the approximation

$$\sin\theta \approx \theta$$

This turns the restoring force into a form proportional to displacement, which is the key feature of simple harmonic motion. That is why a pendulum with a small amplitude behaves approximately like an oscillator with sinusoidal motion.

Small-Angle Approximation and Simple Harmonic Motion

The motion of a pendulum is not exactly simple harmonic for all angles, but it is very close for small angles. This is one of the most important AP Physics C ideas related to pendulums.

When $\sin\theta \approx \theta$, the tangential force becomes

$$\sum F_t \approx -mg\theta$$

To connect angular motion to linear motion along the arc, use the arc length relation

$$s=L\theta$$

where $s$ is the distance along the arc. Since tangential acceleration is related to angular acceleration $\alpha$ by

$$a_t=L\alpha$$

Newton’s second law becomes

$$-mg\theta = mL\alpha$$

$$\alpha = -\frac{g}{L}\theta$$

This is the standard angular equation for simple harmonic motion. It shows that angular acceleration is proportional to displacement and directed toward equilibrium. Comparing this to the standard SHM form,

$$\alpha = -\omega^2\theta$$

we identify

$$\omega = \sqrt{\frac{g}{L}}$$

This leads directly to the period of a simple pendulum:

$$T = 2\pi\sqrt{\frac{L}{g}}$$

This is one of the most important formulas in the topic. Notice something powerful: for small oscillations, the period does not depend on the mass of the bob. It depends only on $L$ and $g$.

Period, Frequency, and What Affects Them

The formula

$$T = 2\pi\sqrt{\frac{L}{g}}$$

shows how the pendulum behaves.

Length matters

If $L$ increases, the period increases. A longer pendulum swings more slowly. This makes sense in real life: a long playground swing takes more time to complete one cycle than a short one.

Gravity matters

If $g$ increases, the period decreases. On the Moon, where $g$ is smaller, a pendulum swings more slowly than on Earth.

Mass does not matter

The mass $m$ cancels out in the derivation, so the period is independent of mass for an ideal simple pendulum. A heavy bob and a light bob with the same string length have the same period, assuming small amplitude and negligible air resistance.

The frequency is

$$f=\frac{1}{T}=\frac{1}{2\pi}\sqrt{\frac{g}{L}}$$

This can help you solve for the motion quickly if you know the pendulum’s length.

Example: If a pendulum has $L=1.00\,\text{m}$ on Earth, then

$$T=2\pi\sqrt{\frac{1.00}{9.8}}\approx 2.0\,\text{s}$$

So the pendulum completes one full cycle in about 2 seconds. That means the frequency is about

$$f\approx 0.50\,\text{Hz}$$

Energy in a Pendulum

Pendulums are also a great place to apply energy conservation. If air resistance is ignored, the total mechanical energy remains constant.

At the highest point of the swing, the bob has maximum gravitational potential energy and zero speed. At the lowest point, the bob has minimum gravitational potential energy and maximum kinetic energy.

The total mechanical energy is

$$E=K+U$$

where

$$K=\frac{1}{2}mv^2$$

and gravitational potential energy can be written relative to a chosen reference level.

As the pendulum moves downward, potential energy decreases while kinetic energy increases. As it moves upward, kinetic energy decreases while potential energy increases. This exchange repeats every cycle.

A useful example: if a pendulum is released from rest at a large height, it speeds up as it approaches the bottom because gravitational potential energy is converting into kinetic energy. If the pendulum is ideal, it will return to the same height on the other side.

This energy viewpoint is useful on AP questions because it can help you find speed without doing full force analysis. For example, if the bob drops by a vertical height $h$, then

$$mgh=\frac{1}{2}mv^2$$

which gives

$$v=\sqrt{2gh}$$

That speed is greatest at the bottom of the swing.

Real-World Limits and AP Exam Reasoning

Real pendulums are not perfectly ideal. students, understanding the limits of the model is important for free-response and multiple-choice questions.

Small-angle limit

The formula

$$T = 2\pi\sqrt{\frac{L}{g}}$$

works best for small amplitudes, usually when $\theta$ is less than about $15^\circ$. For larger angles, the approximation $\sin\theta \approx \theta$ becomes less accurate, and the period becomes slightly longer than the ideal formula predicts.

String and bob assumptions

In the simple pendulum model, the string is assumed to be massless and inextensible, and the bob is treated as a point mass. If the string has mass or the object has significant size, the system may behave more like a physical pendulum, which is a different model.

Damping

Air resistance and friction at the pivot remove mechanical energy over time. This causes the amplitude to decrease gradually. The period usually changes very little for light damping, but the motion is no longer perfectly periodic forever.

What to do on the AP exam

If you see a pendulum problem, ask:

Is the angle small enough to use the simple pendulum formula?
Do I need energy, Newton’s laws, or both?
Am I solving for $T$, $f$, $v$, $a$, or a force?
Is the pendulum ideal, or are there real-world effects like damping?

For example, if the question asks how the period changes when the length is quadrupled, use

$$T\propto \sqrt{L}$$

So the new period becomes twice as large, because

$$\sqrt{4}=2$$

That kind of proportional reasoning is very common in AP Physics C.

Pendulums and the Bigger Picture of Oscillations

Pendulums connect directly to the broader topic of oscillations because they show the same core ideas as springs:

a restoring force
motion around equilibrium
periodic behavior
energy exchange between kinetic and potential forms

For a spring, the restoring force is proportional to displacement in a linear way. For a pendulum, the restoring force is proportional to $\sin\theta$, which becomes approximately proportional to $\theta$ for small angles. That is why both systems can be described with simple harmonic motion under the right conditions.

This connection is important because AP Physics C often asks students to compare systems rather than memorize isolated formulas. If you understand why a pendulum oscillates, you can transfer that reasoning to many other oscillating systems.

Conclusion

Pendulums are a foundational oscillation model in AP Physics C: Mechanics. students, the key idea is that gravity provides a restoring force that makes the bob swing back and forth around equilibrium. For small angles, a simple pendulum behaves like simple harmonic motion, with period

$$T = 2\pi\sqrt{\frac{L}{g}}$$

This means the period depends on length and gravity, but not mass. Energy shifts back and forth between kinetic and potential forms, and real pendulums show limitations such as damping and failure of the small-angle approximation at large amplitudes. Mastering pendulums gives you a strong foundation for the entire oscillations unit and helps you reason clearly on AP exam problems 🎯

Study Notes

A simple pendulum is a point mass on a massless string of length $L$.
The restoring force comes from gravity and has tangential magnitude $mg\sin\theta$.
For small angles, $\sin\theta \approx \theta$, so the motion approximates simple harmonic motion.
The angular acceleration satisfies $\alpha=-\frac{g}{L}\theta$.
The angular frequency is $\omega=\sqrt{\frac{g}{L}}$.
The period of a simple pendulum is $T=2\pi\sqrt{\frac{L}{g}}$.
The frequency is $f=\frac{1}{T}=\frac{1}{2\pi}\sqrt{\frac{g}{L}}$.
The period does not depend on mass for an ideal simple pendulum.
Mechanical energy is conserved in an ideal pendulum: $E=K+U$.
At the bottom of the swing, kinetic energy is greatest and gravitational potential energy is smallest.
Real pendulums can be affected by damping, friction, and large amplitudes.
Pendulums are a major example of oscillatory motion and connect directly to simple harmonic motion.