Circular Motion: Angular Speed and Centripetal Force
An object going round at steady speed is still accelerating, because the direction of its velocity keeps changing. The resultant force points to the centre and has size mv squared over r, which is also m omega squared r.
What a learner can do afterwards
- Converts between period, frequency, linear speed and angular speed in radians per second
- Names the real force acting as the centripetal one in a given setup instead of adding a new force
- Finds the greatest speed for a whirled mass before the string reaches its breaking tension
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Rotation uses the radian: arc length over radius, with 2 pi per full turn. Angular speed is angle per second, tied to frequency by omega equals 2 pi f and to period by 2 pi over T. Every point on a disc shares one omega, but rim points move faster since v equals omega r.
Centripetal force is not a new force but a job title. On a whirled ball it is string tension, on a cornering car it is tyre friction, on the Moon it is gravity. Inertia tries to carry things straight while the real force hauls them round. There is no outward force in this picture.
Demand grows as m v squared over r. Spin a 0.5 kg ball on a 1 m string that snaps at 50 N. Setting 0.5 times v squared over 1 equal to 50 gives v equals 10 metres per second. Past that demand the string snaps or the car slides.
Link angle, speed, and time in radians, name the real inward force, and check the demand before it snaps.
More to read
Introduction to Uniform Circular Motion and Gravitation
Uniform circular motion looks constant but hides a permanent acceleration. The speed stays fixed while the velocity direction rotates, and any change of direction is an acceleration by definition. That acceleration always points to the centre, which is why it is called centripetal, centre seeking. Its size is v squared over r, or equivalently omega squared r. Tight bends and high speeds demand huge inward acceleration, which your car provides through tyre friction and your body feels as being flung outward.
6.1 Rotation Angle and Angular Velocity
Rotation has its own angle unit, the radian, defined as arc length over radius. A full circle is 2 pi radians, a neat fit with circumference 2 pi r. Angular speed omega is the angle swept per second, tied to frequency by omega = 2 pi f and to period by omega = 2 pi over T. Each point on a spinning disc shares one omega, but rim points travel faster because v = omega r. Convert with care: degrees feel natural, but every formula wants radians.
6.2 Centripetal Acceleration
Acceleration in a circle needs no speeding up or slowing down. Picture the velocity arrows at two nearby instants: same length, slightly rotated, and their difference points inward. Divide that change by the time step and you get a = v squared over r toward the centre. The derivation works for any steady circular motion, from a roundabout to an electron in a magnetic trap. Note the direction rule: changing speed needs a tangential push, but changing direction alone is pure centripetal.
6.3 Centripetal Force
Centripetal force is not a new kind of force but a job description for whichever real force points inward. On a whirled ball it is the string tension, on a cornering car it is tyre friction, on the Moon it is gravity. Multiply mass by v squared over r to find how much inward force the motion demands, then ask which agent supplies it. If the demand exceeds the supply, the circle fails: the string snaps or the car slides. There is no outward centrifugal force in an inertial frame, only inertia trying to carry the object straight while the real force hauls it round.
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2 · Watch
Take it off screen
Where it sits
Learn first
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.