Circular Motion Calculator — Period, Frequency, Speed & Centripetal Acceleration
Calcugo's circular motion calculator allows you to calculate all the parameters of uniform circular motion. Enter the period and radius of a circular path to instantly find the frequency, angular velocity, linear speed, centripetal acceleration, and circumference — in both metric (m, m/s, m/s²) and US customary (ft, ft/s, ft/s²) unit systems.
Definition of Circular Motion
When an object moves in a circular path, we call its motion circular motion. Some common examples are:
- A race car moving on a circular curve;
- An artificial satellite revolving around the Earth;
- The tip of the minute/second hand of a clock;
- Whirling a stone tied to a string.
In a circular motion the direction of motion changes continuously. At any instant, the direction of motion is along the tangent to the circle at that point — an easy way to visualize this is to imagine whirling a stone on a string: when the string breaks, the stone flies off tangentially.
What Is Uniform Circular Motion?
In uniform circular motion the object travels on a circular path at constant speed, covering equal arc lengths in equal time intervals. Although the speed is constant, the velocity changes continuously because its direction rotates — this produces an acceleration directed toward the center of the circle called the centripetal acceleration.
From Newton's First Law a body cannot change its direction on its own; the external force that keeps an object on its circular path is called the centripetal force.
Circular Motion Formulas
All formulas assume uniform circular motion (constant speed):
-
Frequency: f = 1 / T (unit: Hz)
The number of complete revolutions per second. -
Angular velocity: ω = 2π / T = 2πf (unit: rad/s)
The rate of change of the angle swept; independent of radius. -
Linear (tangential) speed: v = 2πr / T = ωr (unit: m/s or ft/s)
The distance traveled along the circumference per second. -
Centripetal acceleration: ac = v² / r = ω²r (unit: m/s² or ft/s²)
The inward acceleration that keeps the object on its circular path. -
Circumference: C = 2πr (unit: m or ft)
The total path length for one revolution.
How to Calculate Circular Motion Parameters — Example
- Select the measurement system (Metric or Imperial).
- Enter the period T = 4 s (time for one full revolution).
- Enter the radius r = 2 m (distance from center to object).
- Click Calculate. The calculator displays:
- Frequency: f = 1/4 = 0.25 Hz
- Angular velocity: ω = 2π/4 ≈ 1.5708 rad/s
- Linear speed: v = 2π·2/4 ≈ 3.1416 m/s
- Centripetal acceleration: ac = v²/r ≈ 4.9348 m/s²
- Circumference: C = 2π·2 ≈ 12.5664 m
FAQs
What is the difference between period and frequency?
The period T is the time (in seconds) for one complete revolution. The frequency f is how many revolutions occur per second (in Hz). They are reciprocals: f = 1/T.
Why is centripetal acceleration directed inward?
Even though the speed is constant, the direction of the velocity vector continuously rotates. The rate of change of velocity (acceleration) always points toward the center — this is centripetal acceleration. Without it (i.e., without centripetal force), the object would travel in a straight line.
What is angular velocity?
Angular velocity ω measures how fast the angle changes, expressed in radians per second (rad/s). One full revolution equals 2π radians, so ω = 2π/T. It is related to linear speed by v = ωr.
How do I convert between metric and imperial units?
Use the Measurement System toggle in the calculator. When you select Imperial, the radius is in feet (ft), the speed in ft/s, and the acceleration in ft/s². The calculator handles the conversion automatically (1 ft = 0.3048 m).