Pendulum Harmonic Motion Oscillator: Simulate Period, Velocity, and Energy

Adjust string length, bob mass, release angle, and gravity to see live pendulum animation and instant physics calculations. No data is sent anywhere - everything runs in your browser.

Live Animation
Period (T)
2.006
seconds per cycle
Max Velocity (v max)
1.304
m/s at bottom
Max Energy (E max)
0.851
Joules
String Length (L) 1.00 m
Bob Mass (m) 1.00 kg
Release Angle (theta) 20 deg
Gravity (g) 9.810 m/s2
Conservation of Energy: Live Breakdown
Potential Energy
0.851 J
Kinetic Energy
0.000 J

At release (top of arc), all energy is Potential Energy (PE = mgh). As the bob swings down, PE converts into Kinetic Energy (KE = 1/2 mv2). At the very bottom, all PE has become KE and velocity is at its peak. The total energy stays constant (Emax) throughout the swing - it just shifts form continuously.

Key Terms Explained
Simple Harmonic Motion
Oscillatory motion where the restoring force is proportional to displacement and directed toward the equilibrium position. A pendulum approximates SHM at small angles.
Period (T)
The time for one complete swing and return (one full cycle), measured in seconds. For a simple pendulum: T = 2 pi times sqrt(L / g).
Frequency (f)
The number of complete oscillations per second, in Hertz (Hz). Frequency is the reciprocal of the period: f = 1 / T.
Amplitude
The maximum displacement from the equilibrium (rest) position. For a pendulum this is the initial release angle. Under small-angle SHM, the period is independent of amplitude.
Pendulum Bob
The mass at the end of the string. In a simple pendulum model, all mass is assumed to be concentrated at this single point.
Small-Angle Approximation
The substitution sin(theta) = theta (in radians), valid to within 1% for angles below roughly 15 degrees. This makes the pendulum equation solvable in closed form.
Kinetic Energy (KE)
Energy of motion, equal to 1/2 mv2. Maximized at the bottom of the swing when the bob moves fastest.
Potential Energy (PE)
Stored gravitational energy, equal to mgh above the lowest point. Maximized at the top of each arc where the bob momentarily stops.
Restoring Force
The component of gravity that pulls the bob back toward the equilibrium position. Equal to -mg times sin(theta) and always directed toward center.
Isochronism
The property of a pendulum by which its period remains (nearly) constant regardless of amplitude, as long as the angle stays small. Discovered by Galileo around 1602.

The Complete Guide to Pendulum Harmonic Motion

A pendulum is one of the most elegant demonstrations in all of classical physics. From Galileo's legendary observations in Pisa Cathedral to the precision timekeeping of mechanical clocks, pendulums have shaped our understanding of oscillatory motion, gravity, and energy conservation. This simulator lets you explore all of it interactively in real time.

How to Use This Simulator

Start by setting the String Length and watch how the Period changes immediately - longer strings swing more slowly. Try the Gravity Preset dropdown to move your pendulum to the Moon or Jupiter. Notice that changing the Bob Mass has zero effect on the Period (T) but directly scales the Maximum Energy (Emax). Set a large Release Angle and watch the animation swing wider while the energy bars shift more dramatically at each extreme.

The Physics Behind the Period Formula

The period of a simple pendulum is given by:

T = 2 * pi * sqrt(L / g)

This formula tells you three important things at a glance. First, period scales with the square root of length - double the string length and the period increases by a factor of sqrt(2), roughly 1.41. Second, period is inversely proportional to the square root of gravity - move to a weaker gravitational field and the pendulum slows down. Third, mass is entirely absent from the formula, confirming that the pendulum's swing time is independent of the weight of its bob.

Velocity, Energy, and the Role of Mass

While mass does not affect the period, it absolutely affects the energy stored in the pendulum system. The maximum energy (all potential at the top of the arc, all kinetic at the bottom) is:

E_max = m * g * L * (1 - cos(theta))

And the maximum speed the bob reaches at the bottom of its arc is:

v_max = sqrt(2 * g * L * (1 - cos(theta)))

Note that v_max does not contain mass either - just like in free fall, all objects (light or heavy) reach the same speed at the bottom of an identical swing. But a heavier bob carries far more kinetic energy at that speed, because KE = 1/2 mv2.

Conservation of Energy in the Pendulum

The live energy bars above illustrate one of the most fundamental principles in physics: conservation of mechanical energy. At the moment of release (top of the arc), the bob has maximum height h = L(1 - cos(theta)) above its rest position and is momentarily stationary. All energy is potential (PE = mgh). As it swings downward, gravity does positive work, converting PE into KE. At the very bottom, height is zero, PE is zero, and all energy has become kinetic. On the upswing, the process reverses. Ignoring air resistance and string mass, this exchange continues indefinitely with the total energy Emax remaining constant.

Limits of the Simple Pendulum Model

The period formula is exact only in two limiting cases: zero amplitude (a pendulum at rest) and truly massless, inextensible strings. In practice, the formula is accurate to better than 1% for angles below about 15 degrees, to within 2% at 23 degrees, and to within 5% at 33 degrees. At 90 degrees the true period is about 18% longer than the formula predicts. For large-angle pendulums, the exact period requires an elliptic integral and is commonly approximated using additional correction terms.

Frequently Asked Questions

Why doesn't changing the mass of the pendulum affect its swing time (period)? ▼
The period formula T = 2 pi times sqrt(L / g) contains no mass term. This is because gravity accelerates every object at the same rate regardless of mass. The gravitational restoring force on the bob is proportional to its mass, but so is its inertia, so the two effects cancel exactly. A 1 kg and a 100 kg bob on identical strings will complete each swing in exactly the same time. This mass-independence was observed by Galileo and formally derived by Newton.
What is the small-angle approximation and why is it used? ▼
The true equation of motion for a pendulum involves sin(theta), making it nonlinear and impossible to solve in a clean closed form. The small-angle approximation replaces sin(theta) with theta (in radians), which is accurate to within 1% for angles below about 15 degrees. This simplification makes the equation identical in form to that of a mass-on-a-spring, yielding the elegant period formula T = 2 pi times sqrt(L / g). For larger angles, the real period is longer than this formula predicts.
At what point in the swing is the pendulum's velocity the highest? ▼
The pendulum reaches maximum velocity at the very bottom of the swing, the equilibrium position. At this point all gravitational potential energy has been fully converted to kinetic energy. At the top of each arc, the bob momentarily stops and reverses direction, so velocity is zero at the extremes. The maximum speed is v_max = sqrt(2 times g times L times (1 - cos(theta))), where theta is the initial release angle.
How does moving a pendulum to the Moon affect its period? ▼
On the Moon, gravitational acceleration is about 1.62 m/s2, compared to 9.81 m/s2 on Earth. Since period is proportional to 1 / sqrt(g), a pendulum on the Moon swings about 2.46 times more slowly for the same string length. A clock calibrated with a 1-meter pendulum on Earth (period about 2 seconds) would lose roughly 59 minutes every hour on the Moon. Try switching to the Moon gravity preset in the simulator to see the period change instantly.
What is the difference between a simple pendulum and a physical pendulum? ▼
A simple pendulum (the model used here) assumes all mass is concentrated at a single point at the end of a massless, inextensible string. A physical (compound) pendulum is any real rigid body that swings about a fixed pivot, like a rod, a clock arm, or a human leg. Its period is T = 2 pi times sqrt(I / (m times g times d)), where I is the moment of inertia about the pivot and d is the distance from the pivot to the center of mass. Physical pendulums generally have longer periods than simple pendulums of the same length because mass is distributed away from the tip.
Educational Use Only: This simulator uses the small-angle period formula (T = 2 pi sqrt(L/g)) and energy-based velocity/energy equations. Results are ideal approximations that ignore air resistance, string mass, elasticity, and large-angle corrections. For angles above 30 degrees, actual period will be measurably longer than displayed. Not intended as a substitute for formal physics analysis.