Angular Displacement Calculator
Find angular displacement in radians and degrees from velocities or acceleration using rotational kinematics.
Choose a calculation method, enter the known values, and instantly compute the angle rotated in both radians and degrees.
Angular Displacement Calculator
Find angular displacement in radians and degrees from velocities or acceleration using rotational kinematics.
About the Angular Displacement Calculator
Angular displacement is the total angle through which an object rotates about a fixed axis during a given time interval. It is the rotational counterpart of linear displacement and is measured in radians (rad), though degrees are also commonly used. One full revolution equals 2π radians or 360 degrees.
This calculator provides two kinematic methods for computing angular displacement. The first uses the average velocity formula: θ = (ω₀ + ω) / 2 × t, where ω₀ is the initial angular velocity, ω is the final angular velocity, and t is the time. This formula assumes constant angular acceleration and is analogous to the linear kinematic formula for displacement using average velocity.
The second method applies the standard kinematic equation: θ = ω₀ × t + ½ × α × t², where α is the angular acceleration. This is particularly useful when you know the initial spin rate and how fast it is changing, but not the final angular velocity. Again, this assumes constant angular acceleration.
Angular displacement is a signed quantity: a positive value means rotation in the conventionally positive direction (usually counterclockwise in a 2D plane), while negative means clockwise rotation. This is important when combining multiple rotational motions or when setting up sign conventions for multi-body dynamics.
Practical applications of angular displacement calculations include determining how far a motor shaft has turned during start-up, predicting the position of a robotic arm after a known angular acceleration profile, analysing the rotation of a planet or moon over a time interval, computing the angle swept by a clock hand, or describing the spin-down of a flywheel under friction.
The result is displayed in both radians and degrees. Radians are the SI-coherent unit and are used directly in all physics and engineering formulae involving rotational quantities. Degrees are the more intuitive unit for everyday interpretation. The conversion is simply: degrees = radians × (180 / π). For context, one complete revolution is 6.2832 radians or 360 degrees.
Angular Displacement Examples
Three worked examples demonstrating the two kinematic methods.
| Input | Result | Notes |
|---|---|---|
| Ferris wheel: ω₀ = 0, ω = 0.5 rad/s, t = 10 s | θ = 2.5 rad ≈ 143.24° | Method: From velocities. θ = (0 + 0.5)/2 × 10 = 2.5 rad. |
| Spinning top: ω₀ = 10 rad/s, α = −0.5 rad/s², t = 4 s | θ = 36 rad ≈ 2062.65° | Method: From acceleration. θ = 10×4 + 0.5×(−0.5)×16 = 40 − 4 = 36 rad. |
| Turbine: ω₀ = 0, α = 2 rad/s², t = 5 s | θ = 25 rad ≈ 1432.39° | Method: From acceleration. θ = 0 + 0.5×2×25 = 25 rad. |
How to use the Angular Displacement Calculator
- Select the calculation method: 'From Initial & Final Velocities' if you know both angular velocities and time, or 'From Initial Velocity & Acceleration' if you know the starting speed, acceleration, and time.
- For the velocity method, enter the initial angular velocity ω₀ (rad/s), the final angular velocity ω (rad/s), and the time t (s).
- For the acceleration method, enter the initial angular velocity ω₀ (rad/s), the angular acceleration α (rad/s²), and the time t (s). α can be negative for deceleration.
- Click Calculate to see the angular displacement in radians and degrees. Click Reset to clear all fields and start over.
Angular Displacement FAQ
What is angular displacement?
Angular displacement is the angle through which a rotating object moves around its axis, measured in radians or degrees. It is a vector quantity in 3D but treated as a scalar in simple 2D rotation problems.
How does angular displacement differ from angle?
Angular displacement specifically refers to a change in angle from an initial position to a final position, including the accumulated rotation through multiple revolutions. For example, spinning 3 full circles gives an angular displacement of 6π rad ≈ 18.85 rad.
What is the formula θ = ω₀t + ½αt² derived from?
It comes from integrating the equation of motion for constant angular acceleration. Starting from α = dω/dt, integrating gives ω = ω₀ + αt, and integrating again gives θ = ω₀t + ½αt². This is directly analogous to the linear kinematic equation x = v₀t + ½at².
Can angular displacement be negative?
Yes. A negative angular displacement means rotation in the opposite direction to the defined positive sense. In most conventions, counterclockwise is positive and clockwise is negative when viewed from the standard perspective.
How do I convert radians to degrees?
Multiply radians by 180/π ≈ 57.296. Equivalently, divide the number of radians by π and multiply by 180. This calculator shows both units automatically.
What is the difference between angular displacement and arc length?
Arc length s is the actual distance traveled by a point at radius r from the axis: s = r × θ (with θ in radians). Angular displacement θ describes the rotation angle independent of radius. For the same θ, points farther from the axis travel greater arc lengths.