---
title: "Conical Pendulum Interactive Lab | Sandbox Physics"
description: "Why does a wider cone complete a turn sooner? A wider steady cone needs a higher rotation rate at the same cord length and gravity. The cord must still support the weight, and its growing horizontal component turns the bob more strongly. The increase in speed more than offsets the longer circular path, so one turn takes less time."
canonical_url: "https://sandboxphysics.com/labs/conical-pendulum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/conical-pendulum/"
last_modified: "2026-10-24"
---

Circular motion · A steady cone

# Conical Pendulum

**Why does a wider cone complete a turn sooner?** A wider steady cone needs a higher rotation rate at the same cord length and gravity. The cord must still support the weight, and its growing horizontal component turns the bob more strongly. The increase in speed more than offsets the longer circular path, so one turn takes less time.

Grab the blue bob and widen its horizontal orbit. Watch the cord trace a cone, then compare its radius, period and tension.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The bob follows an exact steady horizontal circle on a massless, inextensible cord in uniform gravity. Changing a control prepares a new steady orbit. Air drag, transients, bob size and cord elasticity are omitted; hardware thickness is schematic.

Physics tutorial

## Why a wider cone turns faster

**Background** A pendulum can move in a horizontal circle instead of swinging through a vertical arc. Its cord then sweeps out a cone.

**Why it matters** The vertical and horizontal parts of one real force explain both support and circular motion.

Start with the essentials

- **Focus question**: At fixed cord length, does a wider orbit take longer to complete?
- **One-sentence intuition**: A wider steady cone needs more inward force per unit weight. The required increase in speed more than offsets the longer circular path, so its period is shorter.

### Core mathematical model

#### The two components of tension

$$
T\cos\theta=mg,\qquad T\sin\theta=m\omega^2r
$$

There is no vertical acceleration. The horizontal component supplies the inward acceleration.

#### Radius, angular speed and period

$$
r=L\sin\theta,\quad \omega=\sqrt{\frac{g}{L\cos\theta}},\quad P=2\pi\sqrt{\frac{L\cos\theta}{g}}
$$

At the same length and gravity, the cosine decreases as the cone opens. The period therefore decreases.

### Common difficulties

#### The bob has no outward real force

**Typical misconception** An outward centrifugal force balances the inward tension in the laboratory frame.

**Better mental model** In the laboratory frame the real horizontal force points inward and continuously turns the velocity. Centrifugal force belongs to a rotating-frame description.

#### New settings prepare a new steady motion

**Typical misconception** The instantaneous change of angle is a prediction of how the real apparatus accelerates.

**Better mental model** The model compares exact steady configurations. It does not integrate the forces during preparation, so no settling time or transient power is reported.

### Run the experiment

1. 01

   #### Grab the ball

   Pause, then drag a blue ball outward or upward in the main scene.

   **What to observe:** The cone opens, the radius grows and the calculated period falls.

2. 02

   #### Compare narrow and wide cones

   Select Narrow cone, then Wide cone, and compare a few revolutions.

   **What to observe:** The wide cone turns faster even though its circular path is longer.

3. 03

   #### Change gravity and mass

   Try Moon gravity, then change Mass in the collapsed group.

   **What to observe:** Weaker gravity slows a given cone. More mass increases tension but leaves its period unchanged.

Physical conclusion

Centripetal force is the inward part of the cord's tension, not an extra force added to it.

Scientific sources

[Richard Fitzpatrick · The conical pendulum](https://farside.ph.utexas.edu/teaching/301/lectures/node88.html) · [J. C. Maxwell · On Governors (1868)](https://www.maths.ed.ac.uk/~v1ranick/papers/maxwell1.pdf) · [TecQuipment · Flyballs and sleeve](https://www.tecquipment.com/governors)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)