{"id":100052,"date":"2026-02-15T20:09:26","date_gmt":"2026-02-15T20:09:26","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100052"},"modified":"2026-07-11T22:47:14","modified_gmt":"2026-07-11T22:47:14","slug":"center-of-gravity-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/pl\/calculators\/center-of-gravity-calculator\/","title":{"rendered":"3D Center of Mass Arithmetic | Discrete Components"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Discrete 3D Center-of-Mass Arithmetic\",\"description\":\"Calculate the three-dimensional mass-weighted mean position of discrete components in a common coordinate system.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/center-of-gravity-calculator\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-11\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"When is center of mass the same as center of gravity?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"For engineering arithmetic in an approximately uniform gravitational field, the mass-weighted center of mass is used as the center of gravity.\"}},{\"@type\":\"Question\",\"name\":\"May coordinates be negative?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Yes. All x, y and z values must use one declared coordinate frame and one consistent length unit.\"}},{\"@type\":\"Question\",\"name\":\"Does this result prove that a lift or machine is stable?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"No. Stability and rigging require geometry, support or suspension conditions, load path, dynamics, equipment limits and a competent plan beyond a point-coordinate calculation.\"}}]}<\/script>\r\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"3D Center of Mass\",\"item\":\"https:\/\/vibromera.eu\/calculators\/center-of-gravity-calculator\/\"}]}<\/script>\r\n<link rel=\"preconnect\" 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.15s}.vc-related-link:hover{border-color:var(--vc-accent);color:var(--vc-accent);background:var(--vc-accent-light)}.vc-promo{margin-top:24px;padding:20px;background:linear-gradient(135deg,var(--vc-accent-light),var(--vc-surface));border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);display:flex;align-items:center;gap:16px}.vc-promo-text{flex:1;font-size:14px;color:var(--vc-ink-secondary)}.vc-promo-text strong{color:var(--vc-ink)}.vc-promo-link{padding:8px 20px;font-size:13px;font-weight:700;color:#fff;background:var(--vc-accent);border-radius:var(--vc-radius);text-decoration:none;white-space:nowrap;transition:background .15s}.vc-promo-link:hover{background:var(--vc-accent-hover)}.vc-footer{text-align:center;padding:32px 16px;font-size:13px;color:var(--vc-ink-muted)}.vc-footer a{color:var(--vc-accent);text-decoration:none}.vc-footer a:hover{text-decoration:underline}.vc-footer-links{margin-top:8px;font-size:12px}.vc-footer-links a{margin:0 8px}.vc-viz{margin:20px 0;background:var(--vc-surface);border:1px solid var(--vc-border-light);border-radius:var(--vc-radius);padding:16px;overflow:hidden}.vc-viz svg{width:100%;height:auto;display:block}.vc-viz-title{font-family:var(--vc-mono);font-size:11px;font-weight:500;letter-spacing:.1em;text-transform:uppercase;color:var(--vc-ink-muted);margin-bottom:12px}@media print{.vc-section-body{max-height:none!important}.vc-faq-a{max-height:none!important}.vc-results{max-height:none!important;padding:24px!important}.vc-copy-btn,.vc-section-chevron,.vc-faq-chevron,.vc-presets,.vc-promo{display:none!important}}<\/style>\r\n<div class=\"vc-calculator\" id=\"vc-center-mass\"><header class=\"vc-header\"><p class=\"vc-header-eyebrow\">Strict discrete-component arithmetic<\/p><h1 class=\"vc-header-title\">3D Center of Mass (Center of Gravity in Uniform Gravity)<\/h1><p class=\"vc-header-subtitle\">Compute the mass-weighted mean position of any number of discrete components. Every row must supply a positive mass and x, y, z coordinates in one common frame.<\/p><div class=\"vc-badges\"><span class=\"vc-badge\">3D coordinates<\/span><span class=\"vc-badge\">Strict rows<\/span><span class=\"vc-badge\">No stability verdict<\/span><\/div><\/header>\r\n<div class=\"vc-card\"><form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\"><div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-components\">Components: mass; x; y; z <span class=\"vc-label-hint\">(one component per line)<\/span><\/label><textarea class=\"vc-input\" id=\"vc-components\" rows=\"8\" spellcheck=\"false\" placeholder=\"2; 0; 0; 0&#10;3; 10; -2; 4\"><\/textarea><p style=\"margin:.5rem 0 0;color:var(--vc-ink-secondary)\">Use semicolons between columns. Decimal point or decimal comma is accepted. Masses must share one unit; x, y and z must share one length unit.<\/p><p id=\"vc-error\" role=\"alert\" style=\"margin:.5rem 0 0;color:#b42318\"><\/p><\/div><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>Model boundary:<\/strong> each row is treated as a point mass at the stated component center. Use a fixed coordinate frame and the same physical configuration throughout. Fluids, moving payloads, fuel burn, deformation or reconfiguration can move the result.<\/p><\/div><\/form>\r\n<div class=\"vc-results\" id=\"vc-results\"><div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Mass-weighted position<\/h2><button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\">Copy<\/button><\/div><div class=\"vc-result-grid\"><div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\">Center of mass (x\u0304, \u0233, z\u0304)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-cg\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Total mass \u03a3m<\/div><div class=\"vc-rcard-value\" id=\"vc-r-mass\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Mass moments (\u03a3mx, \u03a3my, \u03a3mz)<\/div><div class=\"vc-rcard-value\" id=\"vc-r-moments\">\u2014<\/div><\/div><div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Valid components<\/div><div class=\"vc-rcard-value\" id=\"vc-r-count\">\u2014<\/div><\/div><\/div><p style=\"margin:0;color:var(--vc-ink-secondary)\">Coordinate units are the length unit entered; mass moments are mass-unit \u00d7 length-unit.<\/p><\/div><\/div>\r\n<div class=\"vc-section vc-open\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udcd8<\/span><span class=\"vc-section-title\">Equation, assumptions and safety boundary<\/span><\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner vc-theory\"><h3>Discrete component equation<\/h3><div class=\"vc-formula-box\">M = \u03a3m\u1d62; &nbsp; r\u0304 = \u03a3(m\u1d62r\u1d62)\/M; &nbsp; x\u0304 = \u03a3(m\u1d62x\u1d62)\/M, \u0233 = \u03a3(m\u1d62y\u1d62)\/M, z\u0304 = \u03a3(m\u1d62z\u1d62)\/M<\/div><p>The <a href=\"https:\/\/www1.grc.nasa.gov\/beginners-guide-to-aeronautics\/center-of-gravity\/\" target=\"_blank\" rel=\"noopener\">NASA Glenn center-of-gravity explanation<\/a> defines the location through the weighted distribution of weight. In an approximately uniform gravitational field, weight is proportional to mass, so this discrete mass-weighted equation gives the center of gravity used in ordinary engineering arithmetic.<\/p><h3>What this coordinate does not establish<\/h3><p>A center coordinate alone is not a stability, lifting, rigging or machine-safety approval. A real assessment also needs the support polygon or suspension geometry, force direction, attachment points, load path, acceleration and dynamic effects, structural and equipment capacities, uncertainty, and the applicable procedure. <a href=\"https:\/\/www.osha.gov\/laws-regs\/regulations\/standardnumber\/1926\/1926.1404\" target=\"_blank\" rel=\"noopener\">OSHA 29 CFR 1926.1404<\/a>, for example, requires the center of gravity to be identified when needed for stability and requires measures against dangerous unintended movement when information is insufficient.<\/p><div class=\"vc-warning-box\"><p style=\"margin:0\"><strong>Do not infer \u201csafe to lift\u201d from this output.<\/strong> Lifting and rigging require a qualified plan and the governing equipment, site and legal requirements.<\/p><\/div><\/div><\/div><\/div>\r\n<div class=\"vc-section\"><button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\u2753<\/span><span class=\"vc-section-title\">Frequently Asked Questions<\/span><\/span><\/button><div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-faq-list\" id=\"vc-faq-list\"><\/div><\/div><\/div><\/div><footer class=\"vc-footer\"><p>\u00a9 2024\u20132026 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a><\/p><p>Discrete static arithmetic only; not a lifting, stability or safety approval. Scientific review: July 2026.<\/p><\/footer><\/div>\r\n<script>(function(){'use strict';function $(i){return document.getElementById(i)}function number(raw,positive){var s=String(raw).trim().replace(\/\\s+\/g,'');if(!\/^[+-]?(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)$\/.test(s))return NaN;var n=Number(s.replace(',','.'));return Number.isFinite(n)&&(!positive||n>0)?n:NaN}function fmt(n){if(Math.abs(n)<5e-13)n=0;return n.toFixed(9).replace(\/(\\.\\d*?)0+$\/,'$1').replace(\/\\.$\/,'')}function parse(){var raw=$('vc-components').value,lines=raw.split(\/\\r?\\n\/),rows=[];for(var i=0;i<lines.length;i++){if(!lines[i].trim())continue;var c=lines[i].split(';');if(c.length!==4)return{error:'Line '+(i+1)+': expected exactly 4 semicolon-separated values (mass; x; y; z).'};var m=number(c[0],true),x=number(c[1],false),y=number(c[2],false),z=number(c[3],false);if(!Number.isFinite(m))return{error:'Line '+(i+1)+': mass must be a positive finite number.'};if(![x,y,z].every(Number.isFinite))return{error:'Line '+(i+1)+': x, y and z must be finite numbers.'};rows.push([m,x,y,z])}if(!rows.length)return{empty:true};return{rows:rows}}function calc(){var p=parse();$('vc-error').textContent=p.error||'';if(!p.rows){$('vc-results').classList.remove('vc-visible');return}var M=0,mx=0,my=0,mz=0;p.rows.forEach(function(r){M+=r[0];mx+=r[0]*r[1];my+=r[0]*r[2];mz+=r[0]*r[3]});if(![M,mx,my,mz].every(Number.isFinite)||M<=0){$('vc-error').textContent='The totals exceed the finite calculation range.';$('vc-results').classList.remove('vc-visible');return}$('vc-r-cg').textContent='('+fmt(mx\/M)+', '+fmt(my\/M)+', '+fmt(mz\/M)+')';$('vc-r-mass').textContent=fmt(M);$('vc-r-moments').textContent='('+fmt(mx)+', '+fmt(my)+', '+fmt(mz)+')';$('vc-r-count').textContent=String(p.rows.length);$('vc-results').classList.add('vc-visible');var u=new URL(location);u.searchParams.set('components_mass_xyz',$('vc-components').value.trim().replace(\/\\r?\\n\/g,'|'));history.replaceState(null,'',u.toString())}$('vc-form').addEventListener('input',calc);$('vc-form').addEventListener('change',calc);$('vc-copy-btn').addEventListener('click',function(){if(navigator.clipboard)navigator.clipboard.writeText($('vc-results').innerText)});document.querySelectorAll('.vc-section-toggle').forEach(function(b){b.addEventListener('click',function(){var s=this.closest('.vc-section');s.classList.toggle('vc-open');this.setAttribute('aria-expanded',s.classList.contains('vc-open'))})});var q=[['When is this also the center of gravity?','When gravity is approximately uniform across the object, weight is proportional to mass and the mass-weighted point is used as the center of gravity.'],['Can coordinates be negative?','Yes. Choose any declared origin and axes, then use that same frame and length unit for every component.'],['Does the result prove stability or lifting safety?','No. Those decisions require support or suspension geometry, forces, dynamics, capacities, uncertainty and the applicable qualified procedure.']];var l=$('vc-faq-list');q.forEach(function(v){l.innerHTML+='<div class=\"vc-faq-item\"><button type=\"button\" class=\"vc-faq-q\">'+v[0]+'<\/button><div class=\"vc-faq-a\"><div class=\"vc-faq-a-inner\"><p>'+v[1]+'<\/p><\/div><\/div><\/div>'});l.addEventListener('click',function(e){var b=e.target.closest('.vc-faq-q');if(b)b.closest('.vc-faq-item').classList.toggle('vc-open')});var z=new URLSearchParams(location.search);if(z.has('components_mass_xyz'))$('vc-components').value=z.get('components_mass_xyz').split('|').join('\\n');calc()})();<\/script>\r\n","protected":false},"excerpt":{"rendered":"<p>Strictly calculate the three-dimensional mass-weighted mean position of discrete components in one common coordinate frame; not a stability or lifting approval.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100052","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/calculator\/100052","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/calculator\/100052\/revisions"}],"predecessor-version":[{"id":102342,"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/calculator\/100052\/revisions\/102342"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/media?parent=100052"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/categories?post=100052"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/pl\/wp-json\/wp\/v2\/tags?post=100052"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}