Angular Displacement Calculator

Find the angular displacement, final angular velocity, and arc length of a rotating object from its initial angular velocity, angular acceleration, and time using θ = ω₀t + ½αt².

Quick Facts

Formula
θ = ω₀t + ½αt²
Same form as linear kinematics x = v₀t + ½at², with angle in place of distance.
Conversion
1 revolution = 2π rad = 360°
1 rad ≈ 57.2958°; positive angles are counterclockwise by convention.

Your Results

Calculated
Angular displacement (θ)
-
θ = ω₀t + ½αt², in radians
Angular displacement (degrees)
-
θ × 180/π
Final angular velocity (ω)
-
ω = ω₀ + αt
Arc length (s)
-
s = r × θ

Ready

Enter ω₀, α, t, and r, then press Calculate.

Understanding Angular Displacement

Angular displacement is the angle through which an object rotates about a fixed axis, measured from a starting orientation to an ending orientation. Unlike linear displacement, which is measured in meters, angular displacement is measured in radians (or degrees), and it can be positive or negative depending on the direction of rotation.

The formula

For an object rotating with constant angular acceleration, angular displacement follows the rotational kinematics equation:

θ = ω₀t + ½αt²

where θ is the angular displacement (rad), ω₀ is the initial angular velocity (rad/s), α is the angular acceleration (rad/s²), and t is the elapsed time (s). This equation is the rotational counterpart of the linear kinematics formula x = v₀t + ½at², with angle replacing distance, angular velocity replacing velocity, and angular acceleration replacing linear acceleration.

Related quantities

  • Final angular velocity: ω = ω₀ + αt — how fast the object is rotating at the end of the interval.
  • Arc length: s = rθ — the actual distance a point at radius r travels along its circular path (θ must be in radians).
  • Revolutions: N = θ / (2π) — how many full turns the angular displacement represents.

Units and sign convention

Radians are the natural unit for angular displacement because they relate angle directly to arc length and radius. One radian is about 57.2958°, and one full revolution equals 2π radians, or 360°. By the standard right-hand-rule convention, a positive angular displacement or angular velocity represents counterclockwise rotation (viewed from the positive axis direction); a negative value represents clockwise rotation.

When this formula does not apply

The equation θ = ω₀t + ½αt² assumes angular acceleration α is constant over the time interval. If the acceleration changes with time — as with a motor ramping up under variable load — angular displacement must instead be found by integrating the actual angular velocity function, and this constant-acceleration formula will not give an exact answer.

Frequently Asked Questions

What is angular displacement?
Angular displacement is the angle swept by a rotating object between a starting and ending position, measured in radians or degrees. For constant angular acceleration it is calculated as θ = ω₀t + ½αt², where ω₀ is the initial angular velocity, α is the angular acceleration, and t is the elapsed time.
How do I convert radians to degrees?
Multiply the angle in radians by 180/π (approximately 57.2958). One full revolution equals 2π radians, which equals 360°. This calculator shows both units so you don't need to convert by hand.
What is the difference between angular displacement and arc length?
Angular displacement (θ) is the angle swept, independent of radius. Arc length (s) is the actual distance traveled along the circular path by a point at radius r, calculated as s = rθ with θ in radians. Two points at different radii on the same rotating object share the same angular displacement but travel different arc lengths.
Can angular displacement be negative?
Yes. By the standard right-hand-rule convention, counterclockwise rotation is positive and clockwise rotation is negative. A negative result from this calculator means the net rotation over the time interval is clockwise.

Related calculators