Conical pendulum calculator
A 1 m string held at 30° from the vertical traces a circle of 0.5 m radius. Enter length, angle and gravity to get the period and speed of each orbit, plus cone height, angular speed and frequency. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Length
- 1
- Half angle deg
- 30
- Gravity
- 9.80665
- Length m
- 1
- Gravity m s2
- 9.80665
- Radius m
- 0.5
- Height m
- 0.8660254038
A conical pendulum is a mass on a string that moves in a steady horizontal circle so the string sweeps out a right circular cone. Unlike a simple pendulum that swings back and forth, the bob never stops: tension, weight, and centripetal demand stay in continuous balance at a fixed half-angle from the vertical. Use it for classroom checks, lab design, game physics, and any pipeline that needs a deterministic conical-pendulum baseline without spreadsheets or ad-hoc scripts.
What a conical pendulum is and which formulas apply
In the ideal model a point mass hangs from a massless inextensible string of length L fixed at a pivot. The bob travels at constant speed on a horizontal circle of radius r equals L times sin of theta, where theta is the constant angle between the string and the vertical. Vertical force balance sets tension times cos theta equal to weight; horizontal balance sets tension times sin theta equal to centripetal force.
Eliminating tension yields the orbital speed v equals the square root of g L sin theta tan theta, and the period of one full revolution is T equals two pi times the square root of L cos theta over g. Angular speed is two pi over T. These formulas assume steady circular motion, no air drag, and a rigid length.
They are not the small-angle simple-pendulum period, which describes planar oscillation rather than conical orbit. This tool returns the ideal conical values so you can compare experiment against theory, size a demonstration, or seed a simulation with a transparent physical baseline. Mass does not appear in T or v for the ideal string model, just as it cancels in the simple-pendulum derivation.
Where teams use conical-pendulum numbers in practice
Physics teachers and lab assistants compute expected period and speed before students time revolutions with a stopwatch or photogate. Exhibit and museum designers size string length and target angle so a hanging demonstration orbits at a comfortable visual rate. Game and animation engineers need a transparent period and speed when a wrecking ball, hanging lamp, or carnival ride should feel physically plausible at a given rope length and cone angle.
Instrumentation groups compare measured orbital times against the ideal T to flag damping, stretch, or incorrect effective length.
What you can do with it
Classroom and lab expected values
Generate ideal period and orbital speed for a given string length and cone half-angle before students measure revolution times and compute percent error.
Demo and exhibit design
Choose length and half-angle so a hanging conical demonstration completes each orbit at a target period under local gravity.
Game and animation physics
Set rope length and cone angle so hanging props and circular rides move at a predictable speed without hand-tuning timers.
FAQ
What formulas does the conical pendulum calculator use?
Radius r = L sin θ is also returned.
How is this different from a simple pendulum?
A simple pendulum oscillates in a plane and, for small angles, has period 2π √(L/g) independent of amplitude. A conical pendulum moves in a continuous horizontal circle at fixed θ; both period and speed depend on that angle.
Why is a zero or 90° half-angle rejected?
At zero degrees there is no cone—only a bob at rest under the pivot. At ninety degrees or more the geometry is not a hanging conical pendulum with positive vertical support.
Does the mass of the bob matter?
Not in the ideal string model implemented here. Mass cancels in the force balance, so only length, half-angle, and gravity enter the period and speed formulas.
Which units should I enter?
Length in metres, the half-angle in degrees measured from the vertical, and gravity in metres per second squared. Convert centimetres or inches to metres first, since the tool does not convert units for you.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/mech/conical-pendulum \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"length":1,"half_angle_deg":30,"gravity":9.80665}'const res = await fetch("https://api.kit.forhosting.com/mech/conical-pendulum", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"length": 1,
"half_angle_deg": 30,
"gravity": 9.80665
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/conical-pendulum",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"length": 1,
"half_angle_deg": 30,
"gravity": 9.80665
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/conical-pendulum", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"length":1,"half_angle_deg":30,"gravity":9.80665}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"length":1,"half_angle_deg":30,"gravity":9.80665}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/conical-pendulum", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"length": 1,
"half_angle_deg": 30,
"gravity": 9.80665
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.conical_pendulum",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This capability applies the classic closed-form results for period and tangential speed from length, half-angle, and gravity, and returns radius, height, angular speed, and frequency as clean JSON.
Inputs, valid ranges, outputs, and units
Length and gravity must be finite and strictly greater than zero. The half-angle must be strictly greater than zero and strictly less than ninety degrees: zero degrees is a bob hanging at rest under the pivot with no cone, and ninety degrees or more is not a hanging conical geometry with positive vertical support. A one-meter string at thirty degrees under standard Earth gravity is a useful sanity check when wiring clients. Convert centimeters or inches to meters before calling. Send three numbers: length in meters, half_angle_deg in degrees, and gravity in meters per second squared. The response echoes length_m, half_angle_deg, and gravity_m_s2, then radius_m, height_m (vertical pivot-to-plane distance L cos theta), period_s, speed_m_s, angular_speed_rad_s, frequency_hz, and the two formula strings for audit logs. Rounding is fixed to twelve decimal places so browser and API results match byte-for-byte on the same inputs. Invalid or missing fields fail fast with invalid_input and are not charged.
Where teams use conical-pendulum numbers in practice
Documentation writers embed the API in interactive guides so readers can try their own L, theta, and g without leaving the page. Because each successful call is a single analytic item priced at $0.002, batch scripts can recompute tables across lengths, angles, or planetary gravities without a subscription or model cost. Prefer this endpoint when you want the steady conical closed form—not a simple pendulum swing, not a spherical pendulum trajectory, and not a numerical ODE solver for large transient motion.
Technical questions
How much does each API call cost?
$0.002 per successful request. The browser tool above runs free locally; the API draws from your prepaid KIT balance only when the task succeeds.