Circular Motion Calculator
Enter the radius and period (or any two known values) of a circular path to instantly find linear speed, angular velocity, centripetal acceleration, centripetal force, arc length, and more. Switch between metric and imperial units; all results update as you type.
What is uniform circular motion?
Uniform circular motion (UCM) describes an object moving at constant speed along a circular path. Although the speed is constant, the direction of the velocity vector changes continuously, so the object is always accelerating. This centripetal (centre-seeking) acceleration is directed toward the centre of the circle and is what keeps the object on its curved path. Any object in UCM - a car on a bend, a satellite in orbit, a particle in a cyclotron - must have a net inward force supplying that acceleration.
Key formulas for circular motion
All circular motion quantities flow from the radius r and the period T. Frequency f = 1/T gives revolutions per second. Angular velocity (omega) = 2pi/T gives the rate of angular change in radians per second. Linear (tangential) speed v = omega * r, which is the actual speed of the object along the path. Centripetal acceleration a_c = v^2/r = omega^2 * r, directed inward. When mass m is known, centripetal force F_c = m * a_c. Arc length for an angular displacement theta (in radians) is s = r * theta.
Centripetal force and what provides it
Centripetal force is not a new fundamental force - it is whatever real force (or combination of forces) acts inward to keep the object on its circular path. For a satellite, gravity provides it. For a car on a bend, friction between tyres and road provides it. For a stone on a string, tension in the string provides it. If that inward force disappears, the object continues in a straight line tangent to the circle - this is why a car skids outward when it takes a corner too fast: friction can no longer supply the required centripetal force.
How to use this calculator
Enter the radius of the circular path and the period of revolution. The calculator instantly returns frequency, angular velocity, linear speed, centripetal acceleration, centripetal force (for the given mass), and arc length for the given angular displacement. Switch between metric (metres, kilograms) and imperial (feet, pounds) using the unit selector at the top. The mass field is optional - if you leave it at 1 kg (or 1 lb), the force output equals centripetal acceleration per unit mass, which is useful for comparing scenarios at different masses. The arc angle field lets you find the chord distance for any portion of the circle, defaulting to a quarter-circle (90 degrees).
Common circular motion examples
| Example | Radius | Period | Speed (approx.) | Centripetal accel. |
|---|---|---|---|---|
| Car on a roundabout | 20 m | 6 s | 21 m/s (75 km/h) | 22 m/s² (2.2 g) |
| Earth orbiting the Sun | 1.496 × 10¹¹ m | 3.156 × 10⁷ s | 29,780 m/s | 0.0059 m/s² |
| Satellite in low Earth orbit | 6,771 km | 5,580 s | 7,660 m/s | 8.68 m/s² (0.88 g) |
| Washing machine spin (1,200 rpm) | 0.25 m | 0.05 s | 31.4 m/s | 3,948 m/s² (402 g) |
| Amusement park loop-the-loop | 10 m | 3.6 s | 17.5 m/s | 30.5 m/s² (3.1 g) |
| Bicycle wheel (30 km/h) | 0.35 m | 0.264 s | 8.33 m/s | 198 m/s² (20 g) |
Approximate values for everyday and engineering circular motion scenarios.
Frequently asked questions
What is the difference between angular velocity and linear speed?
Angular velocity (omega, in rad/s) measures how fast the angle changes - how many radians the object sweeps out per second. Linear (or tangential) speed (v, in m/s or ft/s) measures how fast the object actually moves along the circular path. They are related by v = omega * r: two objects on the same rotating disk have the same angular velocity but different linear speeds depending on how far from the centre they are. The object at the outer rim travels faster even though both complete one revolution in the same time.
What is centripetal acceleration and why does it always point inward?
In circular motion the velocity vector is always tangent to the circle, and it constantly changes direction. The rate of change of the velocity vector is the centripetal acceleration, which must point toward the centre because that is the direction the velocity is "turning." The magnitude is a_c = v^2/r. There is no outward centrifugal force in an inertial reference frame - in a rotating frame it appears as a fictitious force that exactly cancels centripetal acceleration.
How do I calculate the period from RPM?
RPM (revolutions per minute) is simply frequency in rev/min. Convert to rev/s by dividing by 60, then take the reciprocal: T = 60 / RPM. For example, 1,200 RPM gives T = 60 / 1200 = 0.05 s. Enter that period in this calculator along with the radius to find speed, acceleration, and force.
What happens to centripetal force if I double the speed?
Centripetal force scales with the square of speed: F_c = mv^2/r. Doubling the speed quadruples the required centripetal force. This is why safe cornering speeds are much lower than they might intuitively seem - a small speed increase demands a much larger inward force from friction or banking.
Does this calculator work for vertical circles?
The formulas are correct for the speed and geometry of vertical circular motion, but the required centripetal force varies around the loop because gravity acts downward throughout. At the bottom of a loop, the normal force must supply both centripetal force and support the weight (N = m*a_c + mg). At the top, gravity helps (N = m*a_c - mg). This calculator gives the centripetal acceleration and required net inward force at any point, which you can then split into components based on position.
What is arc length and how is it calculated?
Arc length is the distance traveled along the curved path between two points. For a circle of radius r and an angle theta in radians, s = r * theta. Converting degrees to radians: theta_rad = theta_deg * pi / 180. For a full circle (360 degrees = 2*pi radians), arc length equals the full circumference 2*pi*r.