Calculator · Engineering Mechanics
Rotational Kinematics Calculator
Planar rotation at constant angular acceleration: ω = ω₀ + α·t, Δθ = ω₀·t + ½·α·t² and ω² = ω₀² + 2·α·Δθ. Angles are in radians, exactly as ω and α require.
Known values
θ₀ only shifts the reported angle; it never changes ω or α.
Results
| Relation used | θ = θ₀ + ω₀·t + ½·α·t² and ω = ω₀ + α·t |
|---|---|
| Angle swept Δθ | 32 rad (1833.5°) |
| Angle θ = θ₀ + Δθ | 32 rad (1833.5°) |
| Initial angular velocity ω₀ | 2 rad/s |
| Angular velocity ω | 14 rad/s |
| Average angular velocity | 8 rad/s |
| Angular acceleration α | 3 rad/s² |
| Time t | 4 s |
Technical reference
ω = ω₀ + α·t
Δθ = ω₀·t + ½·α·t²
ω² = ω₀² + 2·α·Δθ
- θ₀
- reference angle at t = 0[rad]
- ω₀, ω
- initial and final angular velocity, CCW positive[rad/s]
- α
- constant angular acceleration, CCW positive[rad/s²]
- Δθ
- angle swept over the interval[rad]
- t
- elapsed time, t ≥ 0[s]
Each solve mode applies exactly one of the three relations above, matching the values the mode treats as known — nothing is inferred from a contradictory mix.
For a point on the body at radius r from the rotation axis, v = r·ω and the tangential acceleration a_t = r·α; this calculator works with ω and α directly and leaves the r-scaling to a specific point, as in Rolling Without Slipping.
Assumptions
- · Planar rotation about a fixed axis, with α constant over the interval.
- · Angles are in radians and a positive ω or α is counter-clockwise.
- · θ₀ is a reference offset only: Δθ, ω and α never depend on it.
- · ω² = ω₀² + 2·α·Δθ gives a magnitude; the reported sense follows the direction of motion, so a reversal within the interval is not resolved by that relation alone.
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The same ω₀, α and time window as a rotating body, with G held at the origin.