Angular Displacement Calculator
Calculate how far an object turns, from its starting spin, angular acceleration and time, then link the angle to arc length.
🔄 What is Angular Displacement?
Angular displacement is the angle through which an object turns about an axis, measured from its starting orientation to its final orientation. It is the rotational counterpart of linear displacement. The symbol is theta, the SI unit is the radian, and it carries a sign: counterclockwise is normally positive and clockwise negative. Unlike a position on a circle, angular displacement can be larger than one full turn, so a wheel that spins 2.5 times has turned through 5 pi radians, 900 degrees or 2.5 revolutions. This angular displacement calculator finds that angle from the motion that produced it, and it also converts it into arc length and between angle units.
The main mode uses the rotational kinematics equation theta = omega0 t + 1/2 alpha t squared, together with omega = omega0 + alpha t. These apply when the angular acceleration is constant, in the same way that s = ut + 1/2 at squared applies to straight-line motion. You can solve for the displacement, for the starting angular velocity, for the angular acceleration, or for the time, and the page shows the final angular velocity in rad/s and rpm so you can check it against a motor or turntable rating. A chart plots the angle against time and marks your result. When you solve for time and the motion turns around, there can be two positive answers, and the calculator tells you about both. For the speed side of the problem see the angular velocity calculator, and for torque and changes in speed see the angular acceleration calculator.
The arc length mode applies s = r theta, which links rotation to the distance travelled by a point on a circle. It answers questions such as how far a bicycle moves for three turns of a wheel, what radius gives a particular arc, or what angle a given arc subtends. A diagram shows the swept sector. The conversion mode turns radians, degrees, revolutions, gradians, arcminutes, arcseconds and milliradians into each other and also gives the equivalent angle between 0 and 360 degrees, which is where a pointer sits after any number of turns.
Two points avoid common mistakes. First, s = r theta only holds when theta is in radians, which is why this tool converts degrees and revolutions before applying it. Second, the constant-acceleration equations do not apply if the acceleration changes during the motion; in that case split the motion into stages. Angular displacement is also not the same as total angle travelled: a turn forward followed by a turn back gives zero displacement.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Speeding-Up Turntable
A turntable starts at 2 rad/s and speeds up at 0.5 rad/s² for 10 seconds
Example 2 — Finding the Angular Acceleration
A wheel turns 50 revolutions in 10 s, starting at 2 rev/s
Example 3 — Time to Reach an Angle
A telescope dome turns at 90°/s initially, accelerating at 30°/s². How long to turn 720°?
Example 4 — Distance Rolled by a Wheel
A bicycle wheel of radius 0.35 m turns 3 revolutions
Example 5 — Angle from an Arc
A point on a 0.8 m radius disc moves along an arc of 2.5 m
Example 6 — Converting 750 Degrees
A valve handle is turned through 750 degrees
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for angular displacement?
With constant angular acceleration, theta = omega0 t + 1/2 alpha t squared, where omega0 is the starting angular velocity, alpha the angular acceleration and t the time. At constant speed it reduces to theta = omega t.
What is the unit of angular displacement?
The SI unit is the radian, but degrees and revolutions are common. One revolution is 360 degrees or 2 pi radians (6.28319 rad), and one radian is 57.2958 degrees.
How do you find angular displacement from angular velocity?
Multiply the angular velocity by the time if it is constant, theta = omega t. If it changes steadily, multiply the average angular velocity (omega0 + omega) / 2 by the time.
How do you convert angular displacement to arc length?
Use s = r theta with theta in radians. A wheel of radius 0.35 m that turns 3 revolutions (18.85 rad) rolls 0.35 x 18.85 = 6.597 m along its rim.
How do you convert revolutions to radians?
Multiply by 2 pi. Half a revolution is pi rad (3.14159), one revolution is 6.28319 rad and 2.0833 revolutions is 13.09 rad.
Can angular displacement be negative?
Yes. The sign shows the direction. Counterclockwise is usually positive and clockwise negative. A negative displacement means the object ended up turned in the opposite direction from the reference.
What is the difference between angular displacement and angular distance?
Angular displacement is the net change in angle, with a sign. Angular distance adds up all the turning regardless of direction. A wheel that goes forward 3 turns and back 1 has a displacement of 2 turns and an angular distance of 4.
How do you find the time to turn a given angle?
Solve 1/2 alpha t squared + omega0 t - theta = 0. With omega0 = 90 deg/s, alpha = 30 deg/s squared and theta = 720 degrees the positive root is t = 4.55 s.
What is the difference between angular displacement and linear displacement?
Angular displacement describes rotation about an axis in angle units. Linear displacement is a straight line length in metres. They are linked by s = r theta for a point at radius r.
How many degrees is 13.09 radians?
13.09 rad is 750 degrees. That is 2 full turns plus 30 degrees, so a pointer is at the same place as at 30 degrees.