Circular Motion Calculator

Solve centripetal acceleration a = v²/r, centripetal force F = mv²/r, period T, angular velocity ω for any circular motion. Planet orbit and car curve presets.

Presets:

Understanding Circular Motion

Uniform circular motion occurs when an object moves in a circle at constant speed. Although the speed stays the same, the direction of velocity continuously changes — meaning the object is always accelerating. This centripetal acceleration always points toward the centre of the circle.

a_c = v²/r
Centripetal acceleration
F_c = mv²/r
Centripetal force
v = ωr
Speed–angular velocity

Centripetal Force vs Centrifugal Force

Centripetal force is the net inward force required to maintain circular motion. It is a real force provided by gravity, tension, friction, or a normal force.

Centrifugal force is a fictitious (pseudo) force that appears in a rotating reference frame. An observer in the rotating frame "feels" pushed outward, but this is just inertia — there is no outward force in an inertial frame.

Property Centripetal Centrifugal
Direction Inward (toward centre) Outward (away from centre)
Reality Real force Fictitious (pseudo) force
Frame Inertial frame Rotating reference frame

Period, Frequency & Angular Velocity

Three quantities describe how fast the rotation happens. They are all linked:

T = 1/f = 2π/ω   |   ω = 2πf = 2π/T   |   v = ωr = 2πr/T

T — Period (s)

Time for one complete revolution. Earth's period around the Sun ≈ 3.156 × 10⁷ s (1 year).

f — Frequency (Hz)

Revolutions per second. A 3000 rpm motor has f = 50 Hz and T = 20 ms.

ω — Angular velocity (rad/s)

Radians swept per second. 1 revolution = 2π rad, so ω = 2πf.

Real-World Examples

Cars on curves

Friction provides centripetal force. Maximum safe speed for radius r: v_max = √(μgr). Reduce speed or increase r to prevent skidding.

Satellites in orbit

Gravity provides centripetal force. For circular orbit: v = √(GM/r). At higher altitude, orbital speed is lower but period is longer.

Centrifuges

Spin samples at high ω to create large a_c = ω²r, separating materials by density. Medical centrifuges reach 10,000–20,000 rpm.

Roller coaster loops

At the top of a loop, gravity + normal force = centripetal force. Minimum speed to maintain contact: v = √(gr).

Worked Examples

Example 1 — Earth's centripetal acceleration toward the Sun

Given: r = 1.496×10¹¹ m, T = 3.156×10⁷ s
Step 1: ω = 2π / T = 2π / 3.156×10⁷ = 1.991×10⁻⁷ rad/s
Step 2: a_c = ω²r = (1.991×10⁻⁷)² × 1.496×10¹¹
Answer: a_c ≈ 5.93×10⁻³ m/s²

Example 2 — Car on a curve (1500 kg, r = 200 m, v = 80 km/h)

Given: m = 1500 kg, r = 200 m, v = 80/3.6 = 22.22 m/s
a_c: v²/r = 22.22²/200 = 493.1/200 = 2.466 m/s²
F_c: m × a_c = 1500 × 2.466
Answer: F_c = 3,699 N ≈ 3.7 kN

Example 3 — Centrifuge (r = 0.1 m, ω = 5000 rpm)

Given: r = 0.1 m, ω = 5000 rpm = 523.6 rad/s
a_c: ω²r = 523.6² × 0.1 = 274,156 × 0.1
Answer: a_c = 27,416 m/s² ≈ 2,795 g

Example 4 — Hammer throw (m = 7.26 kg, r = 1.8 m, v = 27 m/s)

Given: m = 7.26 kg, r = 1.8 m, v = 27 m/s
F_c: mv²/r = 7.26 × 729 / 1.8 = 5,292.54 / 1.8
Answer: F_c = 2,940 N ≈ 2.94 kN

Frequently Asked Questions

Uniform circular motion is the motion of an object travelling in a circular path at constant speed. Although the speed is constant, the velocity is not — its direction continuously changes, meaning the object is always accelerating toward the centre. This inward acceleration is centripetal acceleration: a_c = v²/r = ω²r.
Centripetal force is a real force directed toward the centre of the circular path. It is provided by tension, gravity, friction, or a normal force depending on the situation. Centrifugal force is a fictitious (pseudo) force that appears to push outward in a rotating reference frame — it does not exist in an inertial frame.
Centripetal force is not a separate type of force — it is the name for whatever real force provides the inward (centripetal) acceleration. For a satellite orbiting Earth, gravity provides it. For a car on a curve, friction provides it. For a ball on a string, tension provides it. The centrifugal force (outward) is fictitious and only appears in rotating reference frames.
Period T (seconds) is the time for one complete revolution. Frequency f (Hz) is the number of revolutions per second. They are reciprocals: T = 1/f and f = 1/T. Angular velocity relates to both by ω = 2π/T = 2πf. A wheel at 60 rpm has f = 1 Hz, T = 1 s, ω = 2π rad/s.
Centripetal force is directly proportional to mass: F_c = mv²/r = mω²r. Double the mass at the same speed and radius, and the required centripetal force doubles. This is why heavier vehicles need more friction to safely navigate a curve at the same speed.
When the centripetal force is removed, the object moves in a straight line tangent to the circle (Newton's first law). If a string holding a ball breaks, the ball flies off tangentially. A car skidding off a curved road travels approximately in a straight line from the point of skidding.
Angular velocity (ω, rad/s) is the rate of rotation measured in radians per second. It relates to linear speed by v = ωr. In rpm: ω = rpm × 2π/60. For a full revolution in time T: ω = 2π/T.
Circular motion principles are applied in centrifuges (separating materials by density), turbines (converting rotational energy), flywheels (storing kinetic energy), roller coasters (loop design), vehicle dynamics (cornering limits), satellite orbits (balancing gravity with centripetal needs), and washing machines (spin-drying by centrifugal effect in the rotating frame).

Related Calculators

Newton's Second Law Calculator Gravitational PE Calculator Pendulum Period Calculator Momentum Calculator Kinetic Energy Calculator