Uniform Circular Motion Calculator
Kinematics of uniform circular motion (constant angular velocity ω): enter the known quantities, pick the quantity to solve for — v, ω, r, T, f, or ac — then calculate. All common units are supported.
Results
| Quantity | Value (SI) | Conversion |
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Formulas used
v = ω × r— linear velocityac = v² / r = ω² × r = v × ω— centripetal acceleration, pointing toward the circle’s centerT = 2π / ω— period (time for one revolution)f = 1 / T = ω / 2π— frequencyω (rad/s) = RPM × 2π / 60— revolutions-per-minute conversion
All formulas above are equivalent and consistent: from any sufficient pair of quantities, all six quantities can be derived.
Verified example
Radius r = 0.5 m, rotation speed 120 RPM:
- ω = 120 × 2π / 60 = 12.5664 rad/s
- v = ω × r = 6.2832 m/s (≈ 22.62 km/h)
- ac = v² / r = 78.96 m/s²
- T = 2π / ω = 0.5 s, f = 1 / T = 2 Hz
Frequently asked questions
What is the linear velocity formula for uniform circular motion?
v = ω × r. Example: a 0.5 m radius with ω = 12.5664 rad/s gives v = 6.2832 m/s (≈ 22.62 km/h).
How do I calculate centripetal acceleration?
a_c = v²/r = ω² × r, and its direction is always toward the circle’s center. Example: v = 6.2832 m/s with r = 0.5 m gives a_c = 78.96 m/s² (about 8.05 g).
How do I convert RPM to rad/s?
ω (rad/s) = RPM × 2π / 60. Example: 120 RPM = 12.5664 rad/s. Conversely, 1 RPM = 0.10472 rad/s.
What is the relationship between period (T) and frequency (f)?
T = 2π/ω is the time for one full revolution, while f = 1/T. Example: ω = 12.5664 rad/s → T = 0.5 s and f = 2 Hz.
What is the difference between uniform and non-uniform circular motion?
Uniform means ω is constant — that is what this tool computes. If ω changes over time (there is angular acceleration α), use the angular acceleration calculator.
Where do I compute the centripetal force?
This tool is kinematics only (no mass, no force). For F = m × v²/r, use the centrifugal force calculator. Example: m = 2 kg, v = 6.2832 m/s, r = 0.5 m → F = 157.91 N. Centripetal and centrifugal forces have the same magnitude; only the reference frame differs.
Which quantities can be solved for, and their units?
Six quantities: v (m/s, km/h), ω (rad/s, RPM), r (m, cm), T (s, ms), f (Hz), a_c (m/s²). Enter the known quantities, then pick the target.
Can I see a complete worked example?
An object moves in a circle with r = 2 m and period T = 4 s: ω = 2π/T = 1.5708 rad/s; v = ω × r = 3.1416 m/s; f = 1/T = 0.25 Hz; a_c = ω² × r = 4.9348 m/s².
Free tool by NRTechStudio