String linear density from frequency calculator
A string vibrating at 110 Hz over 0.65 m under 80 N has a linear density of 0.003912171744 kg/m. Enter the measured frequency, length and tension to find how heavy each metre of string is, with every step shown. Free in your browser.
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Run — free
Worked example
- Frequency
- 110
- Length
- 0.65
- Tension
- 80
- Linear density
- 0.0039121717
- Linear density unit
- kg/m
- Model
- ideal_taut_string_fundamental
This string linear density calculator finds mass per unit length from a measured fundamental frequency, vibrating length, and tension. Enter frequency in hertz, length in metres, and tension in newtons to obtain linear density in kilograms per metre. The calculation rearranges the standard ideal-string frequency relation rather than estimating density from a material name or string diameter. It also returns the substituted formula and intermediate value, making the result useful for laboratory work, musical-string checks, physics exercises, and reproducible software tests.
How frequency reveals linear mass density
A stretched string supports transverse waves whose speed depends on tension T and linear mass density μ. For an ideal string fixed at both ends, the fundamental mode has wavelength twice the vibrating length L. Combining wave speed v = √(T/μ) with f = v/(2L) gives f = (1/2L)√(T/μ).
When frequency, length, and tension are known, squaring and rearranging produces μ = T/(2Lf)². This calculator performs that inverse calculation directly. Linear density means mass per unit length, not ordinary volumetric density: a result of 0.004 kg/m says each metre of the string has a mass of four grams.
A greater tension raises frequency if length and density remain fixed, while a heavier string lowers frequency because its inertia is larger. In the inverse formula, raising the specified tension raises the inferred density, because a heavier string is required to retain the same frequency under a stronger pull. Raising frequency or length lowers the inferred density through the squared denominator.
These relationships are useful sanity checks before accepting any numerical answer. The returned formula, intermediate 2Lf value, and final SI result make every stage visible instead of presenting an unexplained number.
Entering measurements and reading the result
Provide three positive finite values. Use the fundamental, not an overtone: if a measured peak is actually the second harmonic and is entered as the fundamental, the inferred density will be one quarter of the correct value because frequency is squared. Likewise, measure only the portion free to vibrate; including string wrapped around a tuning post makes the length too large and drives the calculated density too low.
To express the answer in grams per metre, multiply kg/m by 1000. All three inputs must be greater than zero because a positive fundamental mode requires nonzero length, tension, and frequency in this model.
Assumptions, measurement quality, and practical checks
The equation describes a uniform, perfectly flexible string under constant tension, fixed at both ends, vibrating with small transverse amplitude. Real strings can depart from that model. Stiffness shifts partials away from exact integer multiples, tension can change slightly as a string moves, supports may not be perfectly rigid, and a thick or wound string may have nonuniform construction near its ends.
Air loading and damping usually affect decay more than the basic frequency, but precise metrology may require a richer model. Treat the output as an effective average linear density over the vibrating span when those effects matter. Measurement uncertainty also deserves attention: because μ is proportional to tension and inversely proportional to both length squared and frequency squared, percentage errors in length and frequency contribute approximately twice their percentage size to the density result.
Measure length carefully, use calibrated tension where possible, and average repeated frequency readings. A direct cross-check is simple: weigh a known length of the same string and divide mass by length, then compare that value with the frequency-derived result. You can also insert the returned μ into v = √(T/μ), divide the resulting wave speed by 2L, and confirm that it reproduces the entered fundamental frequency.
Large disagreement often indicates unit conversion trouble, an overtone mistaken for the fundamental, an inaccurate tension estimate, or a string that violates the uniform ideal assumptions.
What you can do with it
Identify an unknown laboratory string
Infer mass per unit length from a measured resonant frequency, known span, and calibrated hanging tension.
Check musical-string specifications
Compare frequency-derived linear density with a manufacturer value when scale length and playing tension are known.
FAQ
What formula calculates string linear density from frequency?
For the ideal fundamental mode, μ = T/(2Lf)², where T is tension, L is vibrating length, and f is fundamental frequency.
What units should I enter?
Enter frequency in hertz, vibrating length in metres, and tension in newtons. The result is linear mass density in kilograms per metre.
Can I use a harmonic frequency?
Not directly. Enter the fundamental frequency. If you measured harmonic n, divide that frequency by n before using this calculator under the ideal-string model.
How do I convert the result from kg/m to g/m?
Multiply it by 1000. A result of 0.004 kg/m, for example, means that each metre of the string has a mass of four grams.
Why does the calculated density come out too low?
Usually because the length is too long or the frequency is an overtone. Measure only the part that actually vibrates, not string wrapped around a tuning post, and enter the fundamental: a second harmonic entered by mistake gives one quarter of the true value.
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/optics/string-linear-density-from-frequency \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"frequency":110,"length":0.65,"tension":80}'const res = await fetch("https://api.kit.forhosting.com/optics/string-linear-density-from-frequency", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"frequency": 110,
"length": 0.65,
"tension": 80
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/string-linear-density-from-frequency",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"frequency": 110,
"length": 0.65,
"tension": 80
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/string-linear-density-from-frequency", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"frequency":110,"length":0.65,"tension":80}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"frequency":110,"length":0.65,"tension":80}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/string-linear-density-from-frequency", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"frequency": 110,
"length": 0.65,
"tension": 80
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.string_linear_density_from_frequency",
"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.
Limits
max_abs | 1000000000000 |
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
Entering measurements and reading the result
Frequency is the measured or specified fundamental frequency f in hertz, length is the active vibrating length L in metres between fixed endpoints, and tension is T in newtons. The response echoes the normalized inputs, reports linear_density in kg/m, identifies the ideal_taut_string_fundamental model, and supplies ordered substitution steps. Missing values, zero, negative values, non-finite numbers, or extreme combinations that overflow the calculation return invalid_input rather than a misleading result. Successful API requests cost $0.002; the browser calculation runs locally with the same deterministic solver. Numeric strings are accepted for convenient form and API use, but values should not contain unit symbols or thousands separators.
Integration use cases
Verify physics calculations and code
Generate deterministic SI results and visible substitution steps for coursework, simulations, or unit-test fixtures.
Technical questions
How do I convert kg/m to g/m?
Multiply the returned linear_density by 1000. For example, 0.005 kg/m equals 5 g/m.
What does the API request cost?
A successful API request costs $0.002. Invalid input is rejected, and the browser calculator runs locally for free.