{"id":665,"date":"2026-07-27T23:09:12","date_gmt":"2026-07-27T23:09:12","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=665"},"modified":"2026-08-24T13:03:57","modified_gmt":"2026-08-24T13:03:57","slug":"angular-velocity-formula","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/angular-velocity-formula\/","title":{"rendered":"Angular Velocity: Formula, Units and Examples"},"content":{"rendered":"\n<div class=\"pf-citation\"><div class=\"eyebrow\">Definition<\/div><p>\nAngular velocity is the rate at which an object rotates through an angle, equal to the angular displacement divided by the time taken. Its SI unit is the radian per second (rad\/s) and its symbol is omega (\u03c9). For steady rotation, angular velocity equals 2\u03c0 divided by the period, T.\n<\/p><\/div>\n\n<p>Stand near the middle of a playground roundabout and you can walk it off without much trouble. Sit on the outer rim of that same roundabout, spinning at exactly the same rate, and you will be flung sideways the instant your feet leave the ground.<\/p>\n\n<p>Nothing about the spin changed \u2014 only your distance from the centre. That is why physics needs a quantity describing the <em>whole<\/em> turning object at once, instead of the speed of one point on it.<\/p>\n\n<h2>What Is Angular Velocity?<\/h2>\n\n<p>Angular velocity is how fast an object turns, measured as the angle it sweeps out per unit of time. Ordinary velocity asks how many metres pass per second; angular velocity asks how many radians are swept per second.<\/p>\n\n<p>Take the second hand of a clock. It covers a full turn \u2014 2\u03c0 radians \u2014 every 60 seconds, so its angular velocity is 2\u03c0 \u00f7 60 \u2248 0.105 rad\/s. The tip of that hand travels far faster than its middle, yet both share the same 0.105 rad\/s.<\/p>\n\n<p>That is the whole idea. A rigid rotating body has <strong>one<\/strong> angular velocity, whatever its size and whichever point you choose to look at.<\/p>\n\n<h3>Angular Velocity vs Angular Speed<\/h3>\n\n<p>Angular speed is just the size of the rotation rate \u2014 a plain number with units of rad\/s. Angular velocity adds a direction, and that direction is a genuine surprise to most students.<\/p>\n\n<p>It does not point around the circle. It points <em>along the rotation axis<\/em>, found with the right-hand rule: curl the fingers of your right hand the way the object spins, and your thumb points along \u03c9.<\/p>\n\n<p>Why the axis? Because it is the only direction in a spinning body that stays put. Every other direction is being swung around, so none of them could label the motion consistently.<\/p>\n\n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/angular-velocity-formula-angle-theta-radius-r-arc.webp\" width=\"1440\" height=\"846\" alt=\"Angular velocity diagram showing angle theta, radius r, arc length and tangential velocity on a rotating circle, with the omega vector along the rotation axis\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:720px;display:block;margin:0 auto;\" \/><\/figure>\n<p style=\"text-align:center;font-size:14px;font-style:italic;color:#1F2E47;\">Angular velocity relates the swept angle \u03b8 to time. The vector \u03c9 lies along the rotation axis, set by the right-hand rule.<\/p>\n\n<h2>The Angular Velocity Formula<\/h2>\n\n<p>The angular velocity formula is \u03c9 = \u03b8 \/ t: divide the angle turned through by the time taken. It is the rotational twin of the familiar v = d \/ t.<\/p>\n\n<div class=\"pf-formula\">\u03c9 = \u03b8 \/ t<\/div>\n\n<ul>\n<li><strong>\u03c9<\/strong> \u2014 angular velocity, in radians per second (rad\/s)<\/li>\n<li><strong>\u03b8<\/strong> \u2014 angular displacement, the angle swept, in radians (rad)<\/li>\n<li><strong>t<\/strong> \u2014 time interval, in seconds (s)<\/li>\n<\/ul>\n\n<p>When the rotation is steady, there is a shortcut. One complete turn is always 2\u03c0 radians, so if you know how long a turn takes \u2014 the period, T \u2014 you already know \u03c9.<\/p>\n\n<div class=\"pf-formula\">\u03c9 = 2\u03c0 \/ T = 2\u03c0f<\/div>\n\n<ul>\n<li><strong>T<\/strong> \u2014 period, the time for one full revolution, in seconds (s)<\/li>\n<li><strong>f<\/strong> \u2014 rotation frequency, in hertz (Hz), where f = 1 \/ T<\/li>\n<\/ul>\n\n<p>That factor of 2\u03c0 is where marks are lost. Angular velocity and <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/waves\/frequency-formula\/\">rotation frequency<\/a> describe the same spinning, but their numbers differ by 2\u03c0 \u2014 a point the SI Brochure flags explicitly as a source of error in published work.<\/p>\n\n<p>The third formula is the bridge to everyday speed, and the one that finally explains the roundabout. Multiply angular velocity by the distance from the axis and you get the tangential speed at that point.<\/p>\n\n<div class=\"pf-formula\">v = \u03c9 r<\/div>\n\n<ul>\n<li><strong>v<\/strong> \u2014 tangential (linear) speed, in metres per second (m\/s)<\/li>\n<li><strong>r<\/strong> \u2014 perpendicular distance from the rotation axis, in metres (m)<\/li>\n<\/ul>\n\n<p>One condition matters here: \u03c9 must be in rad\/s. Feed degrees per second or rpm into v = \u03c9r and the answer is simply wrong. If you would rather check a value than grind through the algebra, our <a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/angular-velocity\">Angular Velocity Calculator<\/a> solves \u03c9 = \u03b8 \/ t for any of the three variables and handles the units for you.<\/p>\n\n<h3>Why the SI Unit Is rad\/s<\/h3>\n\n<p>A radian is a ratio: arc length divided by radius, metres over metres. It cancels, which makes the radian dimensionless \u2014 so rad\/s is really just &#8220;per second&#8221; in disguise.<\/p>\n\n<p>That is exactly why the unit is still written out in full. Frequency, angular velocity and radioactive activity all reduce to s<sup>-1<\/sup>, and <a href=\"https:\/\/www.nist.gov\/pml\/special-publication-330\/sp-330-section-2\" target=\"_blank\" rel=\"noopener\">NIST&#8217;s edition of the SI Brochure<\/a> recommends always writing Hz or rad\/s rather than s<sup>-1<\/sup> so the quantities cannot be confused.<\/p>\n\n<h2>How Do You Convert rpm to rad\/s?<\/h2>\n\n<p>To convert rpm to rad\/s, multiply by 2\u03c0 and divide by 60 \u2014 that is, multiply by roughly 0.10472. To go the other way, multiply rad\/s by 60 \/ 2\u03c0, or about 9.5493.<\/p>\n\n<p>Machines are labelled in revolutions per minute; physics equations demand radians per second. The conversion is a two-part job, and skipping half of it is the single most common slip in rotational problems.<\/p>\n\n<ol>\n<li><strong>Revolutions to radians:<\/strong> multiply by 2\u03c0, because one revolution is 2\u03c0 radians.<\/li>\n<li><strong>Minutes to seconds:<\/strong> divide by 60.<\/li>\n<li><strong>Combine:<\/strong> \u03c9 (rad\/s) = rpm \u00d7 2\u03c0 \/ 60 = rpm \u00d7 0.10472.<\/li>\n<\/ol>\n\n<p>A worked instance: a drill quoted at 1,200 rpm turns at 1,200 \u00d7 0.10472 = 125.7 rad\/s. Divide by 60 alone and you would get 20 \u2014 wrong by a factor of 2\u03c0, roughly six times too small.<\/p>\n\n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Revolutions per minute (rpm)<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Angular velocity \u03c9 (rad\/s)<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Period T (s)<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Typical example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">1<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">0.105<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">60.0<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Clock second hand<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">15<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">1.571<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">4.00<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Large wind-turbine rotor<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">33 1\/3<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">3.491<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">1.80<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Vinyl LP turntable<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">60<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">6.283<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">1.00<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">One revolution per second<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">1,200<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">125.7<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">0.0500<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Cordless drill, high gear<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">3,000<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">314.2<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">0.0200<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Car engine at motorway revs<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">7,200<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">754.0<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">0.00833<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Hard-disk drive platter<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">12,000<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">1,257<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">0.00500<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Laboratory centrifuge<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n<p>Degrees per second needs converting too. Multiply deg\/s by \u03c0 \/ 180 (about 0.01745) to reach rad\/s; one radian per second is about 57.3 deg\/s.<\/p>\n\n<h2>How Angular Velocity Works<\/h2>\n\n<p>Angular velocity works because arc length, radius and angle are locked together by s = r\u03b8, so a single rotation rate fixes the speed of every point on the body. Differentiate that relationship and v = \u03c9r falls straight out.<\/p>\n\n<p>Follow it step by step. A point at distance r sweeping an angle \u03b8 travels an arc of length s = r\u03b8, provided \u03b8 is in radians.<\/p>\n\n<p>Divide both sides by the time taken. The left-hand side becomes arc length per second \u2014 the tangential speed v. The right-hand side becomes r \u00d7 (\u03b8 \/ t), which is r \u00d7 \u03c9.<\/p>\n\n<p>So v = \u03c9r, and the radius is the only thing separating one point from another. Double the distance from the axis and you double the speed, while \u03c9 sits there unchanged.<\/p>\n\n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/angular-velocity-formula-same-every-radius-tangential-speed.webp\" width=\"1440\" height=\"804\" alt=\"Diagram showing that angular velocity is the same at every radius while tangential speed grows in proportion to distance from the axis\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:720px;display:block;margin:0 auto;\" \/><\/figure>\n<p style=\"text-align:center;font-size:14px;font-style:italic;color:#1F2E47;\">Every point shares one angular velocity, but tangential speed v = \u03c9r rises in direct proportion to the radius.<\/p>\n\n<p>This is the mechanism behind almost every rotating machine. Gears, belts and pulleys are all devices for trading angular velocity against radius, and it is the same relationship at the heart of <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/circular-motion-physics\/\">circular motion<\/a>.<\/p>\n\n<p>Drag the sliders below to watch it happen: change the radius or the spin rate and see how the tangential speed responds while \u03c9 stays put.<\/p>\n\n<div class=\"pf-sim-slot\"><div class=\"pf-sim-slot-header\"><span class=\"icon-dot\"><\/span><span class=\"label\">Angular Velocity Lab<\/span><\/div><div class=\"pf-sim-slot-body\"><style>.pf-sim-frame{width:100%;border:none;height:600px}@media(max-width:760px){.pf-sim-frame{height:1000px}}<\/style><iframe src=\"\/labs\/angular-velocity.html?embed=1\" class=\"pf-sim-frame\" loading=\"lazy\"><\/iframe><\/div><\/div>\n\n<h2>Real-World Examples of Angular Velocity<\/h2>\n\n<p>Angular velocity spans an enormous range in ordinary life, from a planet creeping round once a day to a centrifuge rotor screaming at thousands of radians per second. Five cases show the spread.<\/p>\n\n<h3>1. The Earth Spinning on Its Axis<\/h3>\n\n<p>Earth completes one rotation relative to the stars in 23 h 56 min 4 s \u2014 not 24 hours. NASA notes that this sidereal day runs <a href=\"https:\/\/science.nasa.gov\/learn\/basics-of-space-flight\/chapter2-1\/\" target=\"_blank\" rel=\"noopener\">3 minutes 56.55 seconds short<\/a> of the mean solar day, because Earth also moves along its orbit.<\/p>\n\n<p>That gives T \u2248 86,164 s, so \u03c9 = 2\u03c0 \/ 86,164 \u2248 7.29 \u00d7 10<sup>-5<\/sup> rad\/s. Tiny \u2014 yet at the equator, where r \u2248 6,378 km, it still works out to v = \u03c9r \u2248 465 m\/s.<\/p>\n\n<h3>2. A Vinyl Record<\/h3>\n\n<p>An LP turns at 33 1\/3 rpm, which is 3.49 rad\/s. The needle near the outer edge (r \u2248 0.15 m) moves at 0.52 m\/s, but by the final track (r \u2248 0.06 m) that has dropped to 0.21 m\/s.<\/p>\n\n<p>Same \u03c9 throughout, less groove passing the stylus every second \u2014 which is precisely why the inner tracks of an LP sound worse.<\/p>\n\n<h3>3. A Wind Turbine<\/h3>\n\n<p>A large turbine rotor turns slowly, around 15 rpm, giving \u03c9 \u2248 1.57 rad\/s. On a 40 m blade, though, the tip is doing v = 1.57 \u00d7 40 \u2248 63 m\/s \u2014 over 220 km\/h.<\/p>\n\n<h3>4. A Hard-Disk Drive<\/h3>\n\n<p>A 7,200 rpm platter spins at 754 rad\/s, completing a revolution in 8.3 milliseconds. That period sets the drive&#8217;s rotational latency: on average the read head waits half a turn, roughly 4 ms, for the right sector to arrive.<\/p>\n\n<h3>5. A Laboratory Centrifuge<\/h3>\n\n<p>At 12,000 rpm a rotor reaches 1,257 rad\/s. A sample sitting 8.0 cm from the axis then experiences a centripetal acceleration of \u03c9<sup>2<\/sup>r \u2248 1.3 \u00d7 10<sup>5<\/sup> m\/s<sup>2<\/sup> \u2014 about 13,000 g, which is what drives the separation.<\/p>\n\n<figure style=\"margin:32px auto;max-width:640px;text-align:center;\">\n  <img decoding=\"async\" src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/07\/Turntable-Category-Header-Mobile-2.webp\"\n       alt=\"Vinyl record turntable illustrating constant angular velocity with different tangential speeds across the disc\"\n       loading=\"lazy\"\n       style=\"width:100%;height:auto;border-radius:4px;\" width=\"1080\" height=\"810\">\n  <figcaption style=\"font-size:13px;color:#1F2E47;font-style:italic;margin-top:8px;\">A turntable holds one angular velocity, yet the stylus meets slower-moving grooves as it tracks inwards.<\/figcaption>\n<\/figure>\n\n<h2>Common Misconceptions About Angular Velocity<\/h2>\n\n<h3>Misconception 1: Angular Velocity and Linear Velocity Are the Same Thing<\/h3>\n\n<p>They are not, and conflating them wrecks otherwise correct working. Every point on a rigid body shares one angular velocity; almost none of them share a linear velocity.<\/p>\n\n<p>On a spinning bicycle wheel, the valve near the hub and the tread at the rim both complete a turn in the same time. The tread simply has further to travel, so it moves faster.<\/p>\n\n<h3>Misconception 2: Converting rpm Just Means Dividing by 60<\/h3>\n\n<p>Dividing by 60 converts minutes to seconds and stops there \u2014 it leaves the answer in revolutions per second, not radians per second. You still owe the factor of 2\u03c0.<\/p>\n\n<p>In practice, a missing 2\u03c0 shows up as an answer that is about six times too small. If a result looks suspiciously modest for something visibly spinning fast, check this first.<\/p>\n\n<h3>Misconception 3: Constant Angular Velocity Means No Acceleration<\/h3>\n\n<p>A body turning at constant \u03c9 is accelerating the entire time. Its speed is steady, but its <em>direction<\/em> changes continuously, and velocity is a vector.<\/p>\n\n<p>That change of direction is a real acceleration pointing at the axis, of size a = \u03c9<sup>2<\/sup>r. It is supplied by a genuine <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/centripetal-force\/\">centripetal force<\/a> \u2014 string tension, friction, gravity \u2014 and if that force disappears, the object leaves along the tangent.<\/p>\n\n<h3>Misconception 4: Angular Velocity Points Around the Circle<\/h3>\n\n<p>Angular velocity is an axial vector: it lies along the rotation axis, not along the circular path. Only its magnitude, the angular speed, behaves like the scalar most people expect.<\/p>\n\n<p>This matters as soon as rotations start interacting. Gyroscopes, precessing tops and the stability of a moving bicycle all depend on \u03c9 having a fixed direction in space that torques can push against.<\/p>\n\n<h2>How Angular Velocity Relates to Torque, Momentum and Oscillation<\/h2>\n\n<p>Angular velocity sits at the centre of rotational dynamics: change it and you need torque, multiply it by inertia and you get angular momentum, square it and you get rotational energy. Each linear quantity has a rotational partner.<\/p>\n\n<p>When \u03c9 itself changes, the rate of change is the angular acceleration, \u03b1 = \u0394\u03c9 \/ \u0394t, measured in rad\/s<sup>2<\/sup>. Producing it takes <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/torque-physics\/\">torque<\/a>, through \u03c4 = I\u03b1 \u2014 the rotational form of Newton&#8217;s second law.<\/p>\n\n<p>Multiply angular velocity by the moment of inertia and you get angular momentum, L = I\u03c9. Like linear <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/momentum-and-impulse\/\">momentum<\/a>, it is conserved when no external torque acts.<\/p>\n\n<p>That single fact explains the spinning skater. Pulling her arms in cuts I, so \u03c9 must rise to keep L constant \u2014 the same physics that makes a collapsing star spin up into a pulsar.<\/p>\n\n<p>Rotational kinetic energy follows the same pattern: KE = \u00bdI\u03c9<sup>2<\/sup>, the exact echo of \u00bdmv<sup>2<\/sup>. The \u03c9 in a flywheel is squared, which is why doubling the spin rate quadruples the energy stored.<\/p>\n\n<p>One caution on vocabulary. In <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/simple-harmonic-motion\/\">simple harmonic motion<\/a> the symbol \u03c9 means angular <em>frequency<\/em>, and nothing is physically rotating \u2014 it is a bookkeeping device, since \u03c9 = 2\u03c0f describes how fast the phase advances.<\/p>\n\n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Linear quantity<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Symbol and unit<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Rotational analogue<\/th>\n<th style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;background:#0A1628;color:#FAF6EE;\">Symbol and unit<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Displacement<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">s (m)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Angular displacement<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">\u03b8 (rad)<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Velocity<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">v (m\/s)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Angular velocity<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">\u03c9 (rad\/s)<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Acceleration<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">a (m\/s<sup>2<\/sup>)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Angular acceleration<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">\u03b1 (rad\/s<sup>2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Mass<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">m (kg)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Moment of inertia<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">I (kg m<sup>2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Force<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">F (N)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Torque<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">\u03c4 (N m)<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Momentum<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">p = mv (kg m\/s)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Angular momentum<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">L = I\u03c9 (kg m<sup>2<\/sup>\/s)<\/td>\n<\/tr>\n<tr>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Kinetic energy<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">\u00bdmv<sup>2<\/sup> (J)<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">Rotational kinetic energy<\/td>\n<td style=\"border:1px solid #D9CFB8;padding:9px 11px;text-align:left;\">\u00bdI\u03c9<sup>2<\/sup> (J)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n<h2>Worked Problems<\/h2>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 1<\/div><div class=\"pf-problem-question\">The second hand of a clock completes one full revolution every 60 s. Find its angular velocity in rad\/s.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: For steady rotation, one full turn is 2\u03c0 radians, so \u03c9 = 2\u03c0 \/ T.<\/p>\n<p>Step 2: Substitute T = 60 s. \u03c9 = 2\u03c0 \/ 60 s = 6.2832 \/ 60 rad\/s.<\/p>\n<p>Step 3: Divide. \u03c9 = 0.10472 rad\/s.<\/p>\n<p><strong>Answer: \u03c9 \u2248 0.105 rad\/s<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 2<\/div><div class=\"pf-problem-question\">A car engine turns at 3,000 rpm at motorway speed. Convert this to an angular velocity in rad\/s.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: One revolution is 2\u03c0 radians and one minute is 60 s, so \u03c9 = rpm \u00d7 2\u03c0 \/ 60.<\/p>\n<p>Step 2: The combined factor is 2\u03c0 \/ 60 = 0.10472 rad\/s per rpm.<\/p>\n<p>Step 3: \u03c9 = 3,000 \u00d7 0.10472 = 314.16 rad\/s.<\/p>\n<p><strong>Answer: \u03c9 \u2248 314 rad\/s<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 3<\/div><div class=\"pf-problem-question\">A vinyl LP turns at 33 1\/3 rpm. Find (a) its angular velocity in rad\/s and (b) the tangential speed of a groove 0.15 m from the centre.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: (a) Convert with \u03c9 = rpm \u00d7 0.10472.<\/p>\n<p>Step 2: \u03c9 = 33.333 \u00d7 0.10472 = 3.4907 rad\/s.<\/p>\n<p>Step 3: (b) Apply v = \u03c9r with r = 0.15 m. v = 3.4907 \u00d7 0.15 = 0.5236 m\/s.<\/p>\n<p><strong>Answer: \u03c9 \u2248 3.49 rad\/s and v \u2248 0.524 m\/s<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 4<\/div><div class=\"pf-problem-question\">A car travels at 25 m\/s on wheels of radius 0.35 m, rolling without slipping. Find the angular velocity of a wheel in rad\/s and in rpm.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Rolling without slipping means the rim speed equals the road speed, so v = \u03c9r and \u03c9 = v \/ r.<\/p>\n<p>Step 2: \u03c9 = 25 m\/s \u00f7 0.35 m = 71.43 rad\/s.<\/p>\n<p>Step 3: Convert back with rpm = \u03c9 \u00d7 60 \/ 2\u03c0 = 71.43 \u00d7 9.5493 = 682.1 rpm.<\/p>\n<p><strong>Answer: \u03c9 \u2248 71.4 rad\/s, which is about 682 rpm<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 5<\/div><div class=\"pf-problem-question\">Earth completes one rotation relative to the fixed stars in 86,164 s. Find its angular velocity, and the speed of a point on the equator where r = 6.378 x 10^6 m.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Use \u03c9 = 2\u03c0 \/ T for one complete rotation.<\/p>\n<p>Step 2: \u03c9 = 6.2832 \u00f7 86,164 s = 7.292 \u00d7 10<sup>-5<\/sup> rad\/s.<\/p>\n<p>Step 3: Apply v = \u03c9r. v = (7.292 \u00d7 10<sup>-5<\/sup>) \u00d7 (6.378 \u00d7 10<sup>6<\/sup>) = 465.1 m\/s.<\/p>\n<p><strong>Answer: \u03c9 \u2248 7.29 \u00d7 10<sup>-5<\/sup> rad\/s and v \u2248 465 m\/s<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 6<\/div><div class=\"pf-problem-question\">A fan accelerates uniformly from rest to 1,200 rpm in 8.0 s. Find its angular acceleration, and how many revolutions it completes while speeding up.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Convert the final rate. \u03c9 = 1,200 \u00d7 0.10472 = 125.66 rad\/s.<\/p>\n<p>Step 2: Angular acceleration is \u03b1 = (\u03c9 \u2212 \u03c9<sub>0<\/sub>) \/ t = (125.66 \u2212 0) \u00f7 8.0 s = 15.71 rad\/s<sup>2<\/sup>.<\/p>\n<p>Step 3: From rest, \u03b8 = \u00bd\u03b1t<sup>2<\/sup> = 0.5 \u00d7 15.71 \u00d7 8.0<sup>2<\/sup> = 502.7 rad. Revolutions = 502.7 \u00f7 2\u03c0 = 80.0.<\/p>\n<p><strong>Answer: \u03b1 \u2248 15.7 rad\/s<sup>2<\/sup>, completing 80 revolutions<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 7<\/div><div class=\"pf-problem-question\">A centrifuge rotor spins at 12,000 rpm with a sample 8.0 cm from the axis. Find the angular velocity and the centripetal acceleration, expressed as a multiple of g.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Convert, then use a = \u03c9<sup>2<\/sup>r for centripetal acceleration.<\/p>\n<p>Step 2: \u03c9 = 12,000 \u00d7 0.10472 = 1,256.6 rad\/s, and r = 0.080 m.<\/p>\n<p>Step 3: a = (1,256.6)<sup>2<\/sup> \u00d7 0.080 = 1.579 \u00d7 10<sup>6<\/sup> \u00d7 0.080 = 1.263 \u00d7 10<sup>5<\/sup> m\/s<sup>2<\/sup>. Dividing by g = 9.81 m\/s<sup>2<\/sup> gives 1.29 \u00d7 10<sup>4<\/sup>.<\/p>\n<p><strong>Answer: \u03c9 \u2248 1.26 \u00d7 10<sup>3<\/sup> rad\/s and a \u2248 1.26 \u00d7 10<sup>5<\/sup> m\/s<sup>2<\/sup>, roughly 13,000 g<\/strong><\/p>\n<\/div><\/details><\/div>\n\n<h2>Frequently Asked Questions<\/h2>\n\n<details class=\"pf-faq-item\"><summary>What is angular velocity in simple terms?<\/summary><div class=\"pf-faq-item-answer\">\nAngular velocity is how much angle an object turns through each second. Instead of measuring how far a point travels, it measures how far the whole object rotates, in radians per second. A clock second hand has an angular velocity of about 0.105 rad\/s, because it sweeps 2\u03c0 radians every 60 seconds.\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>What is the SI unit of angular velocity?<\/summary><div class=\"pf-faq-item-answer\">\nThe SI unit of angular velocity is the radian per second, written rad\/s. Because a radian is a length divided by a length it is dimensionless, so rad\/s reduces to inverse seconds. Standards bodies still recommend writing rad\/s in full, so that angular velocity is never confused with frequency in hertz.\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>How do you convert rpm to rad\/s?<\/summary><div class=\"pf-faq-item-answer\">\nMultiply the rpm value by 2\u03c0 and divide by 60, which is the same as multiplying by 0.10472. For example, 1,500 rpm becomes 1,500 \u00d7 0.10472 = 157.1 rad\/s. To reverse it, multiply rad\/s by 9.5493. Dividing by 60 alone is the classic mistake, and leaves the answer 2\u03c0 times too small.\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>Is angular velocity a vector or a scalar?<\/summary><div class=\"pf-faq-item-answer\">\nAngular velocity is a vector, and specifically an axial vector. Its magnitude is the angular speed in rad\/s, and its direction lies along the rotation axis, given by the right-hand rule: curl your right fingers the way the body spins and your thumb points along the vector. Angular speed alone is the scalar version.\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>What is the difference between angular velocity and angular frequency?<\/summary><div class=\"pf-faq-item-answer\">\nThey share the symbol \u03c9 and the unit rad\/s, but describe different things. Angular velocity describes something physically rotating through an angle. Angular frequency describes how fast the phase of an oscillation advances, as in a pendulum or a wave, where nothing spins. Both are found from 2\u03c0f, which is why the mathematics looks identical.\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>What is the angular velocity of the Earth?<\/summary><div class=\"pf-faq-item-answer\">\nEarth rotates at about 7.29 x 10^-5 rad\/s, or roughly 0.0000729 rad\/s. This comes from one rotation every 86,164 seconds, the sidereal day, which is just under four minutes shorter than the 24-hour solar day. At the equator this gives a surface speed of about 465 m\/s.\n<\/div><\/details>\n\n<details class=\"pf-faq-item\"><summary>Does angular velocity depend on the radius?<\/summary><div class=\"pf-faq-item-answer\">\nNo. Every point on a rigid rotating body has the same angular velocity, whether it sits near the axis or out at the rim. Radius affects tangential speed instead, through v = \u03c9r. That is why the rim of a wheel moves quickly while the hub barely moves, even though both share one value of \u03c9.\n<\/div><\/details>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Angular velocity measures how fast something rotates, in radians per second. This guide covers the formula, the rpm to rad\/s conversion, real examples and seven worked problems.<\/p>\n","protected":false},"author":1,"featured_media":667,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-665","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mechanics"],"_links":{"self":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/665","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/comments?post=665"}],"version-history":[{"count":5,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/665\/revisions"}],"predecessor-version":[{"id":1538,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/665\/revisions\/1538"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media\/667"}],"wp:attachment":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media?parent=665"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/categories?post=665"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/tags?post=665"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}