| # | Radius (m) | Given input | Computed a (m/s²) | Notes |
|---|---|---|---|---|
| 1 | 0.50 | v = 10 m/s | 200 | a = v²/r |
| 2 | 1.20 | ω = 6 rad/s | 43.2 | a = ω²r |
| 3 | 0.30 | T = 0.8 s | 18.50 | a = 4π²r/T² |
| 4 | 0.75 | RPM = 120 | 118.44 | ω = 2π·RPM/60 |
Uniform circular motion has inward (centripetal) acceleration pointing to the center.
- a = v² / r where v is linear speed and r is radius.
- a = ω² r where ω is angular velocity in rad/s.
- a = 4π² r / T² where T is the period.
- a = 4π² r f² where f is frequency.
- ω = 2π f and ω = 2π·RPM/60 for conversions.
- Enter the radius and choose the correct radius unit.
- Provide one additional value: speed, angular velocity, period, frequency, or RPM.
- Choose units for speed or period when you use them.
- Click Calculate to view acceleration and derived motion values.
- Use Download CSV or Download PDF after a successful calculation.
Centripetal Acceleration in Uniform Circular Motion
In uniform circular motion, speed stays constant while direction changes continuously. That change in direction creates inward (centripetal) acceleration. This calculator reports a in m/s² and compares related motion values in consistent SI units. On Earth, 1 g equals about 9.81 m/s², useful for comparisons during design.
Choosing the Best Input Pair
You must enter radius and one additional quantity. If you know linear speed, use v and r for the simplest path. If your data comes from a motor or turntable, RPM with radius is usually fastest. For timing tests, period with radius gives stable results.
Typical Radius and Speed Ranges
Small lab setups often use r = 0.10–1.00 m with v = 1–20 m/s. A roller coaster loop might use r ≈ 12 m at v ≈ 25 m/s, giving a ≈ 52 m/s², about 5.3 g. A road curve with r = 150 m at v = 25 m/s yields a ≈ 4.17 m/s², roughly 0.43 g.
Period, Frequency, and RPM Conversions
Frequency f (Hz) is cycles per second, while period T (s) is time per cycle. They are reciprocals: f = 1/T. RPM is revolutions per minute, so f = RPM/60. The calculator converts these into angular velocity ω using ω = 2πf, then computes a from ω²r.
Error Checks and Unit Consistency
Radius must be greater than zero because division by r appears in v²/r. Negative speeds or frequencies are rejected since they represent magnitudes here. If you switch units, confirm the numeric value matches the unit label, for example 36 km/h equals 10 m/s.
Interpreting the Output Values
The derived speed v = ωr helps validate your input. If you entered both v and T, compare the implied v = 2πr/T to your measured v. Large discrepancies often indicate unit mix-ups or a non-uniform motion assumption.
Real World Examples
For r = 0.50 m and v = 10 m/s, the calculator returns a = 200 m/s². For a centrifuge at r = 0.10 m and 3000 RPM, ω ≈ 314 rad/s and a ≈ 9860 m/s². For low-Earth orbit, r ≈ 6.8×10^6 m and v ≈ 7700 m/s give a ≈ 8.7 m/s².
What inputs are required?
Enter radius and any one of: speed, angular velocity, period, frequency, or RPM. The calculator selects the first complete pair in its priority order and derives the rest.
Does the result include tangential acceleration?
No. It assumes uniform speed, so tangential acceleration is zero. The reported value is centripetal (radial) acceleration only.
Why is acceleration nonzero if speed is constant?
Acceleration measures change in velocity direction, not only speed. In circular motion the velocity vector rotates each instant, producing inward acceleration toward the center.
Can I compute acceleration without radius?
Not for a numeric a. Radius sets curvature; without it you cannot convert ω to v or v to a. Measure r from the rotation center to the moving point.
Why do my v and T disagree with ω or RPM?
They should agree when data is consistent: v = 2πr/T and ω = 2π/T. Differences usually come from unit mistakes, timing errors, or motion not being perfectly uniform.
How can I reduce centripetal acceleration?
Reduce speed, increase radius, or lower angular velocity/RPM. Because a scales with v² and with ω², small reductions in speed or RPM can significantly reduce acceleration.