Torsional pendulum period
A body with a moment of inertia of 0.05 kg·m² on a wire of 0.2 N·m/rad swings with a period of 3.14159265359 s. Enter both values to get the period, frequency and angular frequency of the twist. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Moment of inertia
- 0.05
- Torsion constant
- 0.2
- Period s
- 3.1415926536
- Frequency hz
- 0.3183098862
- Angular frequency rad s
- 2
A torsional pendulum is a rigid body suspended by a wire, rod, or fiber that twists about a fixed axis and then oscillates under the restoring torque of the elastic suspension. For small twists the torque is proportional to the angular displacement, so the motion is simple harmonic with period determined only by the moment of inertia about the torsion axis and the torsion constant of the wire. Use it for undergraduate lab prep, materials torsion checks, sensor calibration, textbook verification, and any pipeline that needs a deterministic torsional period without spreadsheets or one-off scripts.
What the torsional-pendulum period formula assumes
The moment of inertia I is taken about the fixed torsion axis of the suspension, and the torsion constant κ (kappa) is the torque per unit radian of twist of the wire or fiber. Mass distribution enters only through I; amplitude drops out once the linear torque law holds. Real setups still face air drag, internal friction in the suspension, large-angle nonlinear elasticity, and imperfect axis alignment; this tool deliberately returns the ideal undamped value so you can compare experiment against theory, seed a simulation, or document the textbook baseline.
Moment of inertia must be positive and expressed in kg·m². The torsion constant must be strictly positive in N·m/rad—zero kappa is rejected because the formula would divide by zero and no restoring torque would exist.
Where teams plug this into real workflows
Physics teachers and lab assistants use the calculator to prepare expected periods before students time oscillations of a bar or disk on a torsion wire. Materials and metrology groups estimate κ from wire geometry and shear modulus, then check that the predicted T matches measured zero-crossing times. Instrument designers sizing galvanometer suspensions, torsion balances, or MEMS resonators need a transparent period when choosing I and κ targets.
What you can do with it
Classroom and lab expected values
Generate the ideal torsional period for a known disk or bar inertia and wire kappa before students measure swing times and compute percent error.
Instrument and sensor design
Size moment of inertia and suspension stiffness so a torsion balance, galvanometer, or resonant sensor lands on a target period.
FAQ
What formula does the torsional pendulum period calculator use?
Period is returned in seconds along with frequency and angular frequency.
What happens if the torsion constant is zero?
A zero (or non-positive) torsion constant means no restoring torque, so the period formula is undefined; failed tasks are not charged.
Does the period depend on the twist amplitude?
Not in the ideal linear model used here. Once torque is proportional to angle, amplitude cancels and only I and κ set the period.
What is the period for 0.05 kg·m² and 0.2 N·m/rad?
The period is 3.14159265359 s, with a frequency of 0.318309886184 Hz and an angular frequency of 2 rad/s. It comes from two pi times the square root of inertia over the torsion constant.
Why does my measured period differ from the result?
The result is the ideal value without damping. Real setups add air drag, internal friction in the wire, large-angle effects and imperfect axis alignment, so use the calculation as the textbook baseline to compare.
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/torsional-pendulum-period \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"moment_of_inertia":0.05,"torsion_constant":0.2}'const res = await fetch("https://api.kit.forhosting.com/mech/torsional-pendulum-period", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"moment_of_inertia": 0.05,
"torsion_constant": 0.2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/torsional-pendulum-period",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"moment_of_inertia": 0.05,
"torsion_constant": 0.2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/torsional-pendulum-period", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"moment_of_inertia":0.05,"torsion_constant":0.2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"moment_of_inertia":0.05,"torsion_constant":0.2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/torsional-pendulum-period", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"moment_of_inertia": 0.05,
"torsion_constant": 0.2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.torsional_pendulum_period",
"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 standard formula T equals two pi times the square root of I over kappa, returning period, frequency, and angular frequency as clean JSON.
What the torsional-pendulum period formula assumes
The closed-form period T = 2π √(I/κ) follows from Newton’s second law for rotation when the restoring torque is linear: τ = −κθ. The equation of motion becomes I θ̈ + κ θ = 0, which is simple harmonic motion with angular frequency ω = √(κ/I) and period T = 2π/ω.
Inputs, outputs, and units you should send
Both must be finite and strictly greater than zero. If you only know g·cm² or dyne·cm/rad, convert to SI before calling. Send two numbers: moment_of_inertia (kg·m²) and torsion_constant (N·m/rad). A disk of I = 0.05 kg·m² on a wire with κ = 0.2 N·m/rad returns a period near a few seconds, which is a useful sanity check when wiring clients. The response echoes the two inputs, then period_s (seconds per full cycle), frequency_hz (cycles per second), angular_frequency_rad_s (radians per second), and the formula string for audit logs. Rounding is fixed to twelve decimal places so browser and API results match byte-for-byte on the same inputs. Missing fields, non-finite values, non-positive I, or a zero or negative torsion constant fail fast with invalid_input and are not charged.
Where teams plug this into real workflows
Documentation writers embed the API in interactive guides so readers can try their own I and kappa without leaving the page. Because each successful call is a single analytic item priced at $0.002, batch scripts can recompute tables of periods across disks, rods, and wire stiffnesses without a subscription or model cost. Prefer this endpoint when you want a stable, reviewable formula—not a full viscoelastic model, multi-axis dynamics, or large-twist nonlinear torsion, which need richer constitutive laws and different inputs.
Integration use cases
Wire and fiber torsion checks
Convert geometry and shear modulus into κ, combine with measured I, and verify that T = 2π√(I/κ) matches stopwatch or encoder data.
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.
Is this the same as a simple pendulum period?
No. A simple pendulum uses T = 2π √(L/g) with length and gravity. A torsional pendulum twists about an axis and uses T = 2π √(I/κ) with inertia and torsion constant.