Rotational Motion – v, ω, f, rpm, Centripetal Acceleration – Tutorial
On this page, you can find the logic, usage, and important details of the Rotational Motion – v, ω, f, rpm, Centripetal Acceleration calculator.
Rotational Motion (Uniform Circular Motion)
For an object rotating at constant angular velocity, the key quantities are: v (linear velocity), ω (angular velocity), f (frequency), T (period), rpm (revolutions per minute), and ac (centripetal acceleration).
The key insight: even though the magnitude of velocity stays constant, its direction continuously changes — so there is always acceleration. This acceleration points toward the center and is called centripetal acceleration.
1) Core Concepts and Units
1.1 Radius (r)
- r: radius of rotation, unit: metres (m)
- As r increases, linear velocity v increases for the same ω.
1.2 Linear Velocity (v)
- v: m/s — the speed of the object along its circular path
- Always directed tangentially to the circle (not toward the center).
1.3 Angular Velocity (ω)
- ω: rad/s — rate of change of angle with time (ω = dθ/dt)
- How many radians the object sweeps per second.
1.4 Frequency (f) and Period (T)
- f: Hz (1/s) — revolutions per second
- T: s — time for one complete revolution
- Relationship: f = 1/T and T = 1/f
1.5 rpm (Revolutions per Minute)
- rpm: number of revolutions in one minute
- rpm = 60·f and f = rpm / 60
2) Key Conversion Formulas
2.1 Linear and Angular Velocity (v = ω·r)
v = ω · r
A larger radius means greater linear speed for the same ω. Faster spinning (larger ω) also increases v.
Rearranged: ω = v / r | r = v / ω
2.2 Angular Velocity and Frequency (ω = 2πf)
ω = 2π · f
One revolution = 2π radians, so f revolutions per second = 2πf radians per second.
Rearranged: f = ω / (2π)
2.3 RPM Conversion
rpm = 60 · f → f = rpm / 60
ω = 2π · (rpm / 60)
3) Centripetal Acceleration (ac)
In circular motion, the velocity vector constantly changes direction. A change in direction means acceleration exists; since it points toward the center, it is called centripetal (center-seeking) acceleration.
ac = v² / r = ω² · r
- If v is known: use ac = v² / r
- If ω is known: use ac = ω² · r
- Doubling v quadruples ac (v² effect).
- Smaller radius → larger centripetal acceleration (tighter turn feels stronger).
To find centripetal force: F = m · ac. This calculator provides the acceleration; multiply by mass to get force.
4) What Gets Calculated from Each Input?
| Known | Calculated |
|---|---|
| v | ω = v/r, f, T, rpm, ac |
| ω | f, T, rpm, v = ω·r, ac |
| f | T, ω, rpm, v, ac |
| rpm | f, T, ω, v, ac |
4.1 What Happens When r = 0?
r = 0 means rotating at the center point. In that case v = ω·r = 0, and ac = v²/r involves division by zero, making it undefined. Always use a positive radius for meaningful results.
5) Full Revolution Table (t, θ, x, y)
When r > 0 and ω or f is known, the calculator finds period T and samples from t = 0 to T at regular intervals, producing four columns:
- t (s): time
- θ (rad): angular position — θ(t) = ω · t (initial angle: 0)
- θ (°): angle in degrees
- x, y (m): x = r·cos θ, y = r·sin θ (parametric circle equations)
5.1 Choosing the Time Step (Δt)
- Smaller Δt → more detailed table with more rows.
- Larger Δt → shorter table, coarser sampling of the motion.
- The table has a row limit for performance; very small Δt values may skip some steps.
6) Velocity vs. Acceleration Direction (Commonly Confused)
- v (linear velocity): always tangent to the circle
- ac (centripetal acceleration): always toward the center
In uniform circular motion, these two vectors are always perpendicular.
7) Ideal Model Assumptions
- Angular velocity is constant (no speeding up or slowing down).
- Radius is constant.
- Slip, friction, and elasticity are neglected.
In real systems, motors accelerate and decelerate, wheels may slip, and changing loads affect acceleration. When ω is not constant, angular acceleration (α) and tangential acceleration become relevant.
Note: This explanation covers uniform circular motion (constant ω) only.