Circular Motion Calculator

Circular Motion Calculator finds speed, period, frequency, RPM, angular velocity, radius and centripetal acceleration from any two known values, with arc distance and angle turned.

Circular Motion Calculator: Enter Any Two Values

Uniform circular motion has four linked quantities. They are the radius, the rotation rate, the tangential speed and the centripetal acceleration. Any two of them fix the other two, so the Circular Motion Calculator asks which pair you know and solves the rest.

The I Know list holds all six possible pairs. The default is Radius + Rotation Rate, with a 2 m radius and a 4 s period. That gives a speed of 3.142 m/s, an angular velocity of 1.571 rad/s, a frequency of 0.25 Hz and a centripetal acceleration of 4.935 m/s².

The rotation rate box accepts a period in milliseconds, seconds, minutes, hours or days. It also takes a frequency in hertz or RPM, and an angular velocity in rad/s or deg/s. Whatever is entered, the four result cards of the Circular Motion Calculator restate the motion in the other forms.

Circular Motion Formulas

To calculate circular motion by hand, convert the rotation rate to angular velocity first. Every other quantity then follows from the angular velocity and the radius. The table lists all the formulas for circular motion that the Circular Motion Calculator applies, as given in OpenStax University Physics.

The step-by-step panel under the results prints the exact line used for your entries. For the defaults it reads ω = 2π ÷ 4 = 1.5708 rad/s, then a = 1.5708² × 2 = 4.9348 m/s², then v = 1.5708 × 2 = 3.142 m/s.

Period, Frequency and RPM

The period is the time for one full lap, which answers how to calculate time in circular motion. The period of circular motion formula divides the circumference by the speed.

$$T = \frac{2 \pi r}{v}$$

Choose Radius + Speed for this. On a 2 m circle at 3.5 m/s, the lap takes 3.59 s. Frequency is the number of laps per second, so it is 1 divided by the period. A 4 s period is 0.25 Hz.

Long periods are shown in minutes, hours or days, with the seconds on the line underneath. Take the Moon, with an orbit radius of 384,400 km and a mean orbital velocity of 1.022 km/s on NASA’s fact sheet. Radius + Speed returns 27.35 days per lap, which is 2,363,300 s.

To find RPM in circular motion, multiply the frequency in hertz by 60, or divide 60 by the period in seconds. The 4 s default is 15 RPM. Going the other way, one RPM is 0.10472 rad/s, and one rad/s is 9.549 RPM. That lets the Circular Motion Calculator double as a rotation speed calculator.

$$\omega = \frac{2 \pi}{T} = 2 \pi f$$

The symbol ω is the angular velocity, also called angular speed, angular frequency or circular frequency. It is the angle turned each second, in radians. The first card shows it with the frequency and the same rate in degrees per second, which is 90 deg/s for the default.

The Angular Velocity Calculator and the Angular Frequency Calculator go deeper on ω alone. They cover an angle turned over a time, and springs and pendulums where no circle is involved.

Velocity in Circular Motion

Linear speed, or tangential speed, is the distance the object covers along the circle each second. The velocity in circular motion formula is the angular velocity times the radius.

$$v = \omega r = \frac{2 \pi r}{T}$$

The same rotation rate gives a faster speed farther out. At 1.571 rad/s, a point 1 m from the center moves at 1.571 m/s and a point 4 m out moves at 6.283 m/s. The Speed at Other Radii table inside the Circular Motion Calculator lists six such radii around your own.

The second card gives the speed in other units. The 3.142 m/s default is 11.31 km/h and 7.028 mph. If the acceleration is known instead of the rotation rate, choose Radius + Acceleration, which uses v = √(a r). A 2 m circle with 6 m/s² gives 3.464 m/s.

Velocity is never zero in circular motion while the object keeps moving. The speed stays constant, but the velocity is a vector and its direction turns all the time. Only the average velocity drops to zero, and only over a whole number of laps, because the object ends where it began.

Problems use four kinds of velocity here. Tangential velocity points along the path in m/s. Angular velocity is the turn rate in rad/s. Instantaneous velocity is the value at one moment. Average velocity is the displacement divided by the time taken.

Centripetal Acceleration

An object on a circle always accelerates toward the center, even at constant speed. With any pair entered, the third card works as a centripetal acceleration calculator and shows the result in m/s², in ft/s² and as a g-force.

$$a = \frac{v^2}{r} = \omega^2 r$$

The default comes to 4.935 m/s², or 0.5032 g. The acceleration box also accepts g directly. OpenStax works a jet at 134.1 m/s that must turn at 1 g, and gives a radius of 1.835 km. Speed + Acceleration with 134.1 m/s and 1 g returns 1,834 m.

The same card adds the ideal bank angle on Earth, which is the angle whose tangent is a ÷ g. A curve banked at 26.71° would hold the default motion with no friction. The Angle of Banking Calculator covers banked roads in full.

Acceleration here has no mass in it. Multiply by the mass to get the net inward force, which the Centripetal Force Calculator does. The outward push a rider feels is covered by the Centrifugal Force Calculator.

How to Calculate the Radius

Three of the six pairs in the Circular Motion Calculator solve for the radius of the path. Each uses a different rearrangement of the same formulas, so pick the one that matches the two values the problem gives.

$$r = \frac{v}{\omega} = \frac{v^2}{a} = \frac{a}{\omega^2}$$

Speed + Rotation Rate uses r = v ÷ ω. A speed of 3.5 m/s with a 4 s period gives 2.228 m. Speed + Acceleration uses r = v² ÷ a, and 3.5 m/s with 6 m/s² gives 2.042 m. Accel. + Rotation Rate uses r = a ÷ ω², and 6 m/s² with a 4 s period gives 2.432 m.

For a formula with the period in it, the first form becomes r = v T ÷ 2π. The steps panel switches to that line when the rotation rate is entered as a period, and to r = v ÷ (2π f) when it is a frequency.

Distance, Angle and Displacement

Open Travel and Units in the Circular Motion Calculator and enter a time to see how far the object goes. Distance in circular motion is measured along the arc, and it is the speed times the time. In 3 s the default object covers 9.425 m of its 12.57 m circumference.

$$s = v t = r \theta$$

The fourth card gives the angle swept, which is ω times t. The default sweeps 4.712 rad in 3 s, which is 0.75 of a revolution or 270°. The Angular Displacement Calculator handles angles when the spin rate is changing.

Displacement is different from distance. It is the straight line from the start point to the end point, a chord of the circle. For the default it is 2.828 m, far less than the 9.425 m traveled. After each whole lap the displacement is zero.

The travel box also works from an angle. Switch its unit to degrees, radians or revolutions, and the card gives the time needed to sweep that angle. The default object takes 1 s to sweep 90°, covering 3.142 m of arc with a 2.828 m displacement. A quarter of the Moon’s orbit takes 6.838 days.

$$d = 2 r \sin \left( \frac{\theta}{2} \right)$$

Circular motion is two-dimensional, because the whole path lies in one plane. The position at time t is x = r cos(ωt) and y = r sin(ωt). Those two lines can be typed into a graphing tool such as the Desmos calculator to draw the path, with r and ω taken from the results here.

Why Work Is Zero in Circular Motion

Work is force times the distance moved in the direction of the force. In uniform circular motion the net force points to the center and the motion is along the tangent. The two are at 90°, so the centripetal force does no work.

$$W = F d \cos 90^\circ = 0$$

That is why the speed and the kinetic energy stay constant. The note under the results says the same thing for your numbers. To calculate work done in circular motion when the speed changes, use only the tangential force. The work is that force times the arc distance.

Two Types of Circular Motion

The two types are uniform and non-uniform circular motion. In uniform motion the speed is constant and the only acceleration is centripetal. The Circular Motion Calculator covers that case. In non-uniform motion the speed also changes, which adds a tangential acceleration along the path.

$$a_{total} = \sqrt{a_c^2 + a_t^2}$$

OpenStax gives a particle on a 2.0 m circle moving at 2.5 m/s while speeding up at 1.5 m/s². Its centripetal acceleration is 3.1 m/s², and the total is 3.44 m/s² at 64° from the tangent. Enter Radius + Speed to get the centripetal part, then combine it with the tangential part by hand.

The three equations of motion, or five in the SUVAT form, have matching versions for rotation. They are ω = ω₀ + αt, θ = ω₀t + ½αt² and ω² = ω₀² + 2αθ. The Angular Acceleration Calculator works with these, and the SUVAT Calculator handles the straight-line set.

A question that comes up with these is what the acceleration is when the velocity is zero. It is not always zero. A ball at the top of a throw has zero velocity for an instant and still accelerates downward at 9.81 m/s². In uniform circular motion the speed never reaches zero at all.

Examples of Circular Motion

Circular motion shows up wherever something keeps a steady distance from a fixed center. The table gives 20 real-life examples of circular motion, from the classroom to orbit. A stone whirled on a string and the hands of a clock are the usual first examples in school science.

Rotary motion is the close relative. A body in rotary motion spins about an axis through itself, like a fan, a wheel, a drill bit, a spinning top, a turbine, a potter’s wheel, a record, a drum, a gear or the Earth on its axis. Every point of such a body is in circular motion around that axis.

The next table puts numbers on some of these. Each row was worked with the formulas above, and the tool returns the same figures from the radius and the period. Orbit data are from NASA’s Earth and Moon fact sheets.

The two carousel rows show the effect of radius at one rotation rate. Three times the radius gives three times the speed and three times the acceleration, which rises from 0.5 g to 1.5 g. The Earth orbit row matches NASA’s mean orbital velocity of 29.78 km/s. For spinning lab rotors rated in g, the Centrifuge Calculator is the better fit.

Symbols and Exam Notes

Lowercase ω is the angular velocity, equal to 2π ÷ T, 2π f or v ÷ r. Capital Ω is a different symbol. In physics it is the ohm, the unit of electrical resistance, and some texts also use it for the spin rate of a rotating frame.

The Greek letter λ belongs to waves and not to circles. It is the wavelength, measured in meters, and it equals the wave speed divided by the frequency. The seven kinds of electromagnetic wave sorted by wavelength are radio, microwave, infrared, visible light, ultraviolet, X-ray and gamma ray.

Circular motion is on the NEET physics syllabus. Uniform circular motion sits in the Kinematics unit. The Laws of Motion unit adds the dynamics of uniform circular motion, with centripetal force, a vehicle on a level circular road and a vehicle on a banked road. The formulas in the table above cover the kinematics part.

An object in circular motion also carries angular momentum, which is its mass times its speed times the radius. The Angular Momentum Calculator works that out, and the other spinning-system tools are grouped on the Rotational Motion Calculators hub.