Circular Motion Calculator

Calculate period, frequency, angular velocity, linear speed, and centripetal quantities for circular motion.

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Period, Frequency, and Angular Speed Describe the Same Repetition

Uniform circular motion repeats after each revolution, so period, frequency, and angular velocity are different ways to describe the same timing. The period T is seconds per revolution, frequency f is revolutions per second, and angular speed ω is radians per second. They satisfy f=1/T and ω=2πf. Linear speed follows from the distance around the circle, v=ωr=2πr/T.

Even when speed is constant, velocity changes direction continuously. That requires inward centripetal acceleration ac=v2/r=ω2r. The word centripetal describes the net inward acceleration or force requirement; it is not a new extra force added to gravity, tension, friction, or normal force.

f=1/T,   ω=2πf,   v=ωr,   ac=ω2r
SymbolMeaningWhy it appears / units
TPeriods per revolution.
fFrequencyHz or s−1; revolutions per second.
ωAngular speedrad/s; 2π radians per revolution.
rRadiusm; distance from the rotation axis.

Increasing frequency raises both linear speed and centripetal acceleration. At fixed radius, ac grows with f², so doubling rotation rate produces four times the required inward acceleration.

One revolution provides a useful circular-motion check. If T doubles, f and ω must both be halved. If frequency is given in hertz, multiplying by 2π should produce rad/s; if speed is given in rpm, convert revolutions per minute to revolutions per second before applying 2π.

Worked Examples

Example 1: Earth orbit: r=6.371e6+400000m, T=5520s
ω=2π/5520, v=ωr
Result: 7660 m/s = 7.66 km/s ISS speed
Low Earth orbit speed
Example 2: Car wheel: r=0.3m, 1500 RPM
f=25Hz, ω=157 rad/s
Result: v=47.1 m/s at rim
Car at ~170 km/h tire speed
Example 3: 120 rpm wheel
120rpm=2Hz → ω=4π
Result: ω≈12.57rad/s, T=0.50s
Converting rpm to hertz first prevents a common factor-of-60 error.
Example 4: Rim acceleration
r=0.25m, ω=12.57rad/s
Result: ac≈39.5m/s2
A modest radius can still produce large inward acceleration at high rotation rate.

Common Mistakes

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Treating rpm as hertz

Divide revolutions per minute by 60 to obtain revolutions per second before using SI formulas.

⚠️
Calling centripetal force a separate physical force

Centripetal force is the net inward component supplied by real forces such as tension, gravity, friction, or normal force.

⚠️
Using diameter where radius is required

Circular-motion formulas use distance from the axis, which is half the diameter.

Frequently Asked Questions

Why objects 'fly outward' in circular motion?
They don't — no outward force exists. Objects naturally travel in straight lines (Newton 1st law). Circular motion requires inward centripetal force. Remove the force → object goes straight (tangentially), which appears as 'flying outward'.
Banking angle for a turn?
tan θ = v²/rg for no-friction ideal bank. Racing tracks bank corners to reduce tire friction requirements. At the ideal speed for a given bank, friction force is zero.
Why is there acceleration at constant speed?
Acceleration measures change in the velocity vector, not only change in speed. In circular motion the direction of velocity changes continuously, producing inward acceleration.
How are angular and linear speed different?
Angular speed is shared by all points on a rigid rotating body, while linear speed v=ωr increases with distance from the axis.
What direction is centripetal acceleration?
It always points radially inward toward the instantaneous center of curvature. The velocity is tangent to the circular path and therefore perpendicular to that acceleration.
Does frequency determine centripetal acceleration by itself?
Not completely. The acceleration is a=4π²f²r, so radius is also required. Two objects with the same frequency but different radii have different linear speeds and inward accelerations.
How are period, frequency, and angular speed related?
Period and frequency are reciprocals, f=1/T, while angular speed is ω=2πf=2π/T. One complete revolution is 2π radians, so a shorter period must correspond to both higher frequency and higher angular speed. Keep rpm conversions separate from rad/s until the final step.

Formula Explorer connections

Interpretation: This formula is the rotational counterpart of linear mechanics, relating angle, angular motion, torque, inertia or rotational energy. Assumption: Define the rotation axis and sign convention. Rigid-body behavior, no slipping, steady rotation or negligible bearing losses may be assumed.

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