The circular motion calculator above takes a radius and any single rate and returns the whole family: period, frequency, rotation speed in RPM, angular velocity in radians per second, tangential speed in metres per second, and the centripetal acceleration that keeps the object on its path. Uniform circular motion is fully determined by two numbers, so once you supply them there is nothing left to guess.
Arb Digital builds free calculators that show a relationship rather than a single output, because in rotational problems the useful insight is almost always the link between the angular description and the linear one. This page is deliberately the overview tool. If you only need the force, the centripetal force calculator is narrower and quicker; if you are converting an angular rate between unit conventions, the angular velocity calculator does that one job.
What This Circular Motion Calculator Does
It closes the loop between five equivalent descriptions of the same motion. Period and frequency are reciprocals. RPM is frequency scaled by sixty. Angular velocity is frequency multiplied by two pi, because one revolution is 2π radians. Tangential speed is angular velocity multiplied by the radius. Give the tool any one of them plus a radius and it fills in the rest, then computes the centripetal acceleration that the motion implies.
The selector matters because these quantities are routinely quoted in different conventions in the same problem. A motor is specified in RPM, a physics equation wants radians per second, a belt speed is in metres per second, and an oscillation is described by its period. Converting between them by hand is where sign-free but silent errors creep in, usually a missing factor of 2π or 60.
The optional mass field adds the force that would be required to sustain the motion, so you can sanity-check whether the string, bearing or gravitational attraction in your problem could actually supply it. That figure is reported as supporting information; the force-first version of this question lives on its own page.
How to Use It
- Enter the radius in metres. This is what links the angular quantities to the linear ones, so it is required in every mode.
- Choose which rate you know. The other four rate fields are ignored, and the tool recalculates them from the one you selected.
- Enter that rate. Period in seconds, frequency in hertz, speed in RPM, tangential speed in metres per second, or angular velocity in radians per second.
- Add a mass if you want the force. Leave it at one kilogram and the force figure doubles as the acceleration in newtons per kilogram.
- Read the note. It reports the acceleration as a multiple of standard gravity and the distance covered per revolution, which are the two figures people usually want next.
The Formula: How Circular Motion Is Calculated
Start from the definition of the period as the time for one full revolution. In that time the object covers a circumference of 2πr, so the tangential speed is v = 2πr/T. Angular velocity is the angle swept per unit time, ω = 2π/T, and combining the two gives the relation v = ωr. OpenStax University Physics Volume 1, section 4.4 on uniform and nonuniform circular motion, states both ω = 2π/T and v = 2πr/T directly, and gives the centripetal acceleration as ac = v2/r, directed toward the centre of the circle.
Work the defaults. A radius of 0.5 m with a period of 2 s gives a frequency of 0.5 Hz, which is 30 RPM. Angular velocity is 2π ÷ 2 = 3.1416 rad/s. Tangential speed is 3.1416 × 0.5 = 1.5708 m/s. Centripetal acceleration is 1.57082 ÷ 0.5 = 4.9348 m/s2, which is just over half of standard gravity. A one-kilogram object would need 4.93 N of inward force to hold that path.
Because ac can also be written as ω2r, the acceleration is proportional to the radius at fixed angular velocity but inversely proportional to it at fixed tangential speed. Those two statements sound contradictory and are not: they answer different questions. The first is what happens further out on a spinning disc; the second is what happens when a vehicle at constant speed takes a tighter corner. Section 10.1 on rotational variables in the same text develops the angular description in full.
Angular Quantities Belong to the Body, Linear Ones to the Point
This is the distinction that makes rotational problems click. Every point on a rigid rotating object shares the same period, the same frequency and the same angular velocity, because they all complete a revolution together. None of them share a tangential speed, because the outer points travel a longer circumference in that same time.
A bicycle wheel makes the point concretely. The valve near the hub and the tread at the rim have identical angular velocity, but the tread may be moving several times faster in metres per second. That is why rim brakes work on a fast-moving surface, why a longer propeller blade approaches the speed of sound at its tip while the hub barely moves, and why a centrifuge's force depends on how far the sample sits from the axis — the case handled by our centrifuge RCF calculator.
Practically, it means you must always ask which radius a quoted speed refers to. A stated surface speed without a stated radius is incomplete information, whereas an angular velocity stands on its own.
Uniform Motion Still Has an Acceleration
Objects in uniform circular motion move at constant speed and are nonetheless accelerating the whole time, which is the single most common conceptual objection to this topic. The resolution is that velocity is a vector. Speed is unchanged but direction changes continuously, and any change of direction is a change of velocity.
That acceleration points toward the centre at every instant, perpendicular to the motion. Being perpendicular, it does no work and cannot change the speed — it only bends the path. The moment you remove whatever supplies it, the object continues in a straight line along the tangent, which is exactly what happens when a string snaps or a car loses grip mid-corner. Nothing flings the object outward; it simply stops being pulled inward.
If the speed does change as well, the motion is non-uniform and there is a tangential acceleration on top of the centripetal one. This tool covers the uniform case only. For the rate at which angular velocity itself changes, use the angular acceleration calculator.
Radians, Degrees and the Factor of 2π
Angular velocity in physics is expressed in radians per second because the radian makes arc length equal to angle times radius with no conversion factor attached. That is the entire reason the unit exists and the reason v = ωr is so clean. Express the angle in degrees and every formula acquires a stray π/180.
The traps follow from the different conventions. Hertz counts complete revolutions per second, so multiplying by 2π converts to radians per second. RPM counts revolutions per minute, so dividing by 60 gives hertz first. Missing either step produces an answer wrong by a factor of about 6.28 or exactly 60, both large enough to notice but not always large enough to look obviously absurd. The frequency converter and the angle converter handle those rescalings when you need them separately.
Reading Acceleration as a Multiple of g
Raw acceleration figures in metres per second squared are hard to feel. Dividing by standard gravity turns them into something intuitive: a fairground ride at 3 g presses you into the seat with three times your weight, a centrifuge at 1,000 g is doing something no everyday experience prepares you for, and 0.5 g means the inward pull is half what gravity supplies.
The note under the results reports that multiple automatically. It is worth checking against the mechanism that has to supply the force. A string, a bearing, a rail or friction between tyre and road each has a limit, and the calculation tells you the demand rather than the capacity. Our centripetal force calculator converts that demand into newtons for a given mass, and the friction force calculator estimates what a contact surface can actually provide.
How This Differs From the Neighbouring Tools
The boundary in one sentence each. This page relates the five rate descriptions of circular motion to one another and reports the acceleration. The centripetal force calculator answers the force question for a known mass and is the right tool when force is the output you actually want. The angular velocity calculator converts an angular rate between conventions without needing a radius at all.
Use this one when you have a mixed problem — a period here, a surface speed there — and need everything in a single consistent set. Use the others when the question is narrow. All three describe the same physics; they differ in what they take in and what they hand back.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Dropping the factor of 2π — hertz counts revolutions and radians per second counts angle, and the two differ by 2π in every direction.
- Quoting a tangential speed without its radius — angular velocity stands alone, but a surface speed is meaningless unless you say where on the body it was measured.
- Assuming constant speed means no acceleration — velocity is a vector, and continuous change of direction is acceleration even when the speed never moves.
- Treating centrifugal force as a real inward-balancing force — in an inertial frame there is only the inward force; the outward sensation belongs to the rotating frame.
- Using degrees in v = ωr — the relation only holds with the angle in radians, which is why the radian exists in the first place.
Related Free Tools From Arb Digital
For force from a known mass use the centripetal force calculator, and for a changing rotation rate the angular acceleration calculator. Convert rates with the angular velocity calculator, the frequency converter and the speed converter, and angles with the angle converter. For laboratory rotation see the centrifuge RCF calculator, and for the distance travelled along part of a circle the arc length calculator. The full free online tools hub lists everything else.
Frequently Asked Questions
Motion in a circle at constant speed. The velocity still changes because its direction changes continuously, so there is a constant acceleration directed toward the centre even though the speed never varies.
Period and frequency are reciprocals of each other. Angular velocity is 2π times the frequency, because one complete revolution corresponds to 2π radians of angle. RPM is simply frequency counted per minute rather than per second.
Because every point on a rigid rotating body sweeps the same angle in the same time, while points further from the axis travel a longer path to do it. Angular velocity describes the body; tangential speed describes a particular point on it.
As the square of the tangential speed divided by the radius, or equivalently as the square of the angular velocity multiplied by the radius. Both expressions give the same value and point toward the centre of the circle.
Not in an inertial frame. There the only force is the inward one that bends the path. The outward push you feel is what appears when the description is written from inside the rotating frame, and it disappears when the frame does.
The object travels in a straight line along the tangent at the speed it already had. It is not thrown outward; it simply stops being pulled inward and continues as Newton's first law requires.
This page relates the rate descriptions of circular motion — period, frequency, RPM, angular velocity and tangential speed — and reports the acceleration. The centripetal force calculator takes a mass and returns the force in newtons, which is a narrower question.
This tool is provided for educational and estimating use. It assumes uniform circular motion at constant speed on a fixed radius, and does not model tangential acceleration, non-inertial reference frames, drag or the strength limits of whatever supplies the force.
