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Circular Motion Calculator

This calculator solves all key circular motion quantities — centripetal acceleration, centripetal force, period, frequency, angular velocity, and linear speed — from any combination of inputs you provide.

Enter radius and at least one of speed, period, frequency, or angular velocity to see results.

Quick answer

Circular motion occurs when an object moves at constant speed along a circular path, always accelerating toward the center. Centripetal acceleration equals v²/r, where v is linear speed and r is radius. Centripetal force is F = mv²/r, directed inward toward the center of the circle. Period T = 2πr/v and frequency f = 1/T describe how long one revolution takes and how many occur per second. Angular velocity ω = 2πf = v/r connects linear speed to the rate of rotation in radians per second.

Formula & method

a_c = v² / r
  • a_c — Centripetal acceleration (m/s²)
  • v — Linear (tangential) speed (m/s)
  • r — Radius of circular path (m)

Centripetal acceleration — the inward acceleration required to maintain circular motion.

F_c = m·v² / r = m·a_c
  • F_c — Centripetal force (N)
  • m — Mass of the object (kg)
  • v — Linear speed (m/s)
  • r — Radius (m)

Centripetal force — the net inward force required to keep mass m moving in a circle.

T = 2πr / v = 2π / ω
  • T — Period (s)
  • r — Radius (m)
  • v — Linear speed (m/s)
  • ω — Angular velocity (rad/s)

Period — time for one complete revolution.

ω = v / r = 2π / T = 2πf
  • ω — Angular velocity (rad/s)
  • v — Linear speed (m/s)
  • r — Radius (m)
  • T — Period (s)
  • f — Frequency (Hz)

Angular velocity — rate of rotation in radians per second.

Examples

Example 1: Centripetal acceleration of a car on a curve
Input
Speed v = 10 m/s, Radius r = 5 m
Result
a_c = 20 m/s²
Why
Using a_c = v²/r: a_c = (10)²/5 = 100/5 = 20 m/s². A car travelling at 10 m/s around a 5 m radius bend requires 20 m/s² of centripetal acceleration — about twice the acceleration due to gravity.
Example 2: Centripetal force on a ball swung in a horizontal circle
Input
Mass m = 2 kg, Speed v = 8 m/s, Radius r = 4 m
Result
F_c = 32 N
Why
F_c = mv²/r = 2 × (8)²/4 = 2 × 64/4 = 128/4 = 32 N. The string must exert 32 N of inward tension to keep the 2 kg ball moving at 8 m/s in a 4 m circle.
Example 3: Speed and angular velocity of a wheel
Input
Radius r = 0.5 m, Period T = 0.25 s
Result
v ≈ 12.566 m/s, ω ≈ 25.133 rad/s
Why
v = 2πr/T = 2π × 0.5 / 0.25 = π / 0.25 ≈ 12.566 m/s. ω = 2π/T = 2π/0.25 ≈ 25.133 rad/s. A point on the rim of a 0.5 m radius wheel completing one revolution every 0.25 s moves at about 12.57 m/s.
Example 4: Finding speed from centripetal force
Input
Force F = 50 N, Mass m = 5 kg, Radius r = 2 m
Result
v ≈ 4.472 m/s, a_c = 10 m/s²
Why
Rearranging F = mv²/r gives v = √(Fr/m) = √(50 × 2 / 5) = √(100/5) = √20 ≈ 4.472 m/s. Centripetal acceleration a_c = F/m = 50/5 = 10 m/s².

Frequently asked questions

What is the difference between centripetal and centrifugal force?

Centripetal force is the real inward force that causes circular motion — it is supplied by tension, gravity, friction, or a normal force. Centrifugal force is a fictitious (pseudo) force felt in a rotating reference frame that appears to push outward. Only centripetal force appears in Newton's laws in an inertial frame.

Does the direction of velocity change in uniform circular motion?

Yes. Even though the speed (magnitude) stays constant in uniform circular motion, the direction of the velocity vector continuously changes — it always points tangent to the circle. This change in direction is what produces the centripetal acceleration pointing toward the center.

How is angular velocity different from linear velocity?

Linear velocity v is the distance covered per unit time along the circular path (m/s). Angular velocity ω is the angle swept per unit time (rad/s). They are related by v = ωr, so a larger radius means a higher linear speed for the same angular velocity.

What happens to centripetal acceleration if the radius doubles?

If the radius doubles while speed stays the same, centripetal acceleration halves, since a_c = v²/r. Conversely, if speed doubles with constant radius, acceleration quadruples because it depends on v squared.

How do I calculate the period from frequency?

Period T and frequency f are reciprocals: T = 1/f and f = 1/T. If an object completes 4 revolutions per second (f = 4 Hz), its period is T = 1/4 = 0.25 s. Angular velocity is then ω = 2πf = 8π ≈ 25.13 rad/s.

Is circular motion the same as simple harmonic motion?

No, but they are related. Circular motion involves motion along a circular path. Simple harmonic motion (SHM) is the one-dimensional projection of uniform circular motion — for example, the x-coordinate of a point on a circle traces out a sine wave over time.

Sources & references

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