Hong Life CalculatorsUnderstanding the angular displacement calculator — concept and principle
Angular displacement is the angle through which a rotating object turns. This calculator finds it for rotation with constant angular acceleration, and also gives the final angular speed, the number of revolutions and the arc length travelled by a point at a given radius.
How it is calculated
- θ = ω₀t + ½αt² (radians).
- Final angular speed ω = ω₀ + αt; average speed = (ω₀ + ω) ÷ 2.
- Revolutions = θ ÷ 2π; degrees = θ × 180 ÷ π.
- Arc length at radius r: s = rθ.
- rpm = rad/s × 60 ÷ 2π.
The principle
These are the rotational versions of the familiar equations of motion with constant acceleration (s = ut + ½at², v = u + at). With constant α, the angular speed changes linearly, so the angle turned equals the average speed × time. Using radians makes the arc-length relation simple: one radian is the angle at which the arc equals the radius. If α is negative and the body reverses during the interval, θ is the net angle, not the total turned.
Worked example
Default inputs: ω₀ = 2 rad/s, α = 0.5 rad/s², t = 10 s, radius 0.3 m.
- θ = 2 × 10 + ½ × 0.5 × 10² = 20 + 25 = 45 rad.
- In degrees = 45 × 180 ÷ π ≈ 2,578.3°; revolutions = 45 ÷ 2π ≈ 7.162.
- Final speed = 2 + 0.5 × 10 = 7 rad/s ≈ 66.85 rpm.
- Average speed = (2 + 7) ÷ 2 = 4.5 rad/s.
- Arc length at 0.3 m = 0.3 × 45 = 13.5 m.
| Time t (s) | Angular displacement θ (rad) |
|---|---|
| 2.5 | 6.563 |
| 5 | 16.25 |
| 7.5 | 29.06 |
| 10 | 45 |
| 12.5 | 64.06 |
| 15 | 86.25 |
| 20 | 140 |
How to read the result
Revolutions are handy for counting turns of wheels, drums or reels; arc length tells you how far a point on the rim travelled, for example the length of cable wound on a drum of that radius. Check that the time does not include a reversal if you need total distance.
Common mistakes
- Using degrees in formulas that require radians.
- Forgetting the ½ in ½αt².
- Treating net angle as total angle when the rotation reverses.
- Mixing rpm and rad/s for ω₀.
Frequently asked questions
What is the formula for angular displacement?
θ = ω₀t + ½αt² for constant angular acceleration.
How many radians are in one revolution?
2π, about 6.283 radians.
How do I get arc length from angular displacement?
Multiply the angle in radians by the radius: s = rθ.
Sources: Halliday, Resnick & Walker, Fundamentals of Physics (rotational kinematics) · OpenStax, University Physics Volume 1, Chapter 10 Fixed-Axis Rotation