Angular Displacement Calculator

Angular Displacement Calculator finds theta in radians, degrees or turns from arc length and radius or angular velocity and time. It also takes acceleration, two angles or a clock.

What the Angular Displacement Calculator Finds

The Angular Displacement Calculator gives the angle a body turns through about an axis. With the loaded values, a point that travels 6 m along a circle of radius 4 m has an angular displacement of 1.5 rad. The cards show the same angle as 85.94°, 0.2387 revolutions and 0.4775 π rad.

The Calculate From list has seven starting points. They are arc length and radius, angular velocity and time, velocity with acceleration and time, two velocities and time, two velocities and acceleration, a start and end angle, and a clock hand. An Angle Diagram under the results draws the turn on a circle.

Displacement counts every full turn, while position on the circle repeats every 360°. The tool reports both. A turn of 15 rad shows as 859.4°, and also as 2 full turns plus 139.4°.

Angular Displacement Formula in Radians

Angular displacement is the change in the angular position of a rotating body. Its symbol is the Greek letter theta, θ, often written Δθ to mark it as a change. The SI unit of angular displacement is the radian, and the basic formula defines the radian itself.

$$\theta = \frac{s}{r}$$

Here s is the arc length the point travels and r is the radius of its circle, in the same length unit. OpenStax University Physics, section 10.1 gives this as equation 10.1. This is also the angular displacement formula used in A Level Physics, where the answer must stay in radians.

To calculate angular displacement, divide the distance along the circle by the radius. A runner who covers 185 m on a circular track of radius 9 m turns through 20.56 rad. The Angular Displacement Calculator adds that this is 1,178°, or 3 full laps plus 97.75°.

What unit is used for angular displacement?

Radians are the SI unit, and degrees and revolutions are also common. A full turn is 2π rad, 360° or 1 revolution, so 1 rad is about 57.3°. A wheel that rotates 3 times has an angular displacement of 1,080°, which is 6π or 18.85 rad. The Result Unit bar switches the main answer between the three.

Angular displacement to linear displacement

Turn the formula around and the arc length is the radius times the angle in radians. An angle of 2 rad on a 4 m radius is 8 m of arc. The same option solves for the radius, so 6 m of arc over 2 rad means a 3 m radius.

$$s = r \times \theta$$

The arc is the distance travelled, not the straight-line displacement. That is the chord, which the Angular Displacement Calculator also gives. For the loaded 1.5 rad on a 4 m radius the arc is 6 m and the chord is 5.453 m. Past one full turn, the sector area on the same card covers only the leftover angle.

This is the difference between linear and angular displacement. Linear displacement is a change in position in meters, a vector written Δx or s, and it is negative when it points against the chosen positive direction. Angular displacement is a change in angle. Straight-line cases belong in the Displacement Calculator.

Formula with Angular Velocity

At a steady spin, angular displacement is the angular velocity times the time. This is the formula for calculating angular rotation whenever the speed does not change, and the Angular Displacement Calculator loads it with 3 rad/s and 5 s.

$$\theta = \omega \times t$$

Those values give 15 rad. OpenStax works a flywheel at 45.0 rad/s for 30 s and gets 1,350 rad, about 215 revolutions. The tool returns the same 1,350 rad and shows 214.9 revolutions.

The same option finds the other two values. The formula to calculate angular speed is the angle divided by the time, so 12 rad in 5 s is 2.4 rad/s, or 22.92 rpm. At 3 rad/s the same 12 rad takes 4 s. Velocities can be entered in rad/s, rpm, degrees per second or revolutions per second.

From Angular Acceleration

To find angular displacement from angular acceleration, add two parts. The first is what the starting velocity alone would cover. The second is what the acceleration adds, which grows with the square of the time. OpenStax lists this velocity and acceleration formula as equation 10.12 in section 10.2.

$$\theta = \omega_0 \times t + \frac{1}{2} \times \alpha \times t^2$$

ω₀ is the initial angular velocity and α is the angular acceleration. Starting at 2 rad/s with 0.5 rad/s² for 10 s, the steady part is 20 rad and the acceleration adds 25 rad. The total is 45 rad, which is 7 full turns plus 58.31°, and the final velocity is 7 rad/s.

The Solve For list in this option works the formula backwards when the angle is known. A turn of 45 rad from 2 rad/s at 0.5 rad/s² takes 10 s. Over 10 s from 2 rad/s, the same 45 rad needs 0.5 rad/s². With 0.5 rad/s² for 10 s, the body must start at 2 rad/s.

The time comes from a quadratic, so it can have two answers, and the first positive one is shown. Some targets are out of reach. A body that starts at 2 rad/s with −0.5 rad/s² stops after 4 rad, so it never reaches 100 rad, and the message under the results says so.

When both velocities are known, the mean angular velocity does the job. With steady acceleration it is midway between the two, and the angle is that mean times the time. Going from 10 to 40 rad/s in 6 s has a mean of 25 rad/s and a displacement of 150 rad.

$$\theta = \frac{\omega_1 + \omega_2}{2} \times t$$

That card also shows the acceleration behind the change, 5 rad/s² here. The Angular Acceleration Calculator works the same motion from the other side, including torque and moment of inertia.

Angular displacement without time

Sometimes the time is not given at all. The two velocities and the angular acceleration are then enough, since the angle follows from the change in the squares of the velocities. OpenStax lists this as equation 10.13, rearranged here for the angle.

$$\theta = \frac{\omega_2^2 – \omega_1^2}{2 \times \alpha}$$

Going from 10 to 40 rad/s at 5 rad/s², the squares differ by 1,500 and the angle is 150 rad. A card adds the time taken, 6 s. OpenStax works a fishing reel that speeds up from rest to 220 rad/s at 110 rad/s² and turns through 220 rad, or 35.0 revolutions. The tool gives the same 220 rad in 2 s.

The sign of α has to fit the two velocities. Slowing that reel from 220 rad/s to rest needs a negative value, and at −300 rad/s² it turns another 80.67 rad in 0.7333 s. Enter a positive α for a body that slows down and a Check the Entry message asks for the sign to be flipped.

Start and End Angle

To find the angle of displacement between two positions, subtract the start angle from the end angle. A body that moves from 30° to 150° has an angular displacement of 120°, or 2.094 rad. The Angular Displacement Calculator also shows the other way round, which is −240°.

The sign carries the direction. OpenStax takes counterclockwise as positive, so a negative result means a clockwise turn. Going from 350° to 10° by the formula gives −340°, while the short way round is only 20°.

Can angular displacement be zero?

Yes. Angular displacement is zero whenever the body ends at the angle it started from. Start at 10 rad/s with an α of −2 rad/s² and run for 10 s. The spin reverses after 5 s and comes back, so the main result is 0 rad. The cards still show the motion, with 50 rad, or 2,865°, swept in total and 5 s until the spin reverses.

That example shows the difference between angular distance and angular displacement. Angular distance adds up everything swept, whichever way the body turned. Angular displacement is only the net change between start and end.

The two also differ when the body does not come all the way back. Start at 2 rad/s with −0.5 rad/s² and run for 6 s. The body turns 4 rad forward, then 1 rad back. Its angular displacement is 3 rad, and the Total Angle Swept card reads 5 rad.

Why is angular displacement not a vector?

A large angular displacement has a size and an axis, but it does not add like a vector. Turn a book 90° about one axis and then 90° about another, and the result depends on the order. True vectors give the same sum in either order. Only very small angular displacements behave as vectors, which is why angular velocity is one.

A real-life example of angular displacement

A bicycle wheel gives a simple one. With a radius of 0.35 m, three full turns of the wheel are an angular displacement of 18.85 rad. If the tire does not slip, the bike moves forward by the arc length, which is 6.597 m.

Clock Hand Angles

The Clock Hand option of the Angular Displacement Calculator answers questions such as the angular displacement of a minute hand in 20 minutes. The minute hand makes one turn in 60 minutes, so it moves 6° each minute. In 20 minutes it turns 120°, which is 2.094 rad or one third of a turn.

On a clock face 1 hour is 30° for the hour hand, not 15°. The figure of 15° per hour belongs to the Earth, which turns 360° in 24 hours. As a unit of angle, a minute is something else again. One arcminute is a sixtieth of a degree, so 2 arcminutes are 0.0333°.

What is a 20 degree angle called?

A 20° angle is an acute angle, since it is less than 90°. Angles are usually sorted into seven types. A zero angle is 0°, an acute angle is under 90° and a right angle is exactly 90°. An obtuse angle lies between 90° and 180°, a straight angle is 180°, a reflex angle is over 180° and a full angle is 360°.

The other tools for spinning bodies are listed under Rotational Motion Calculators, and each one takes angles in radians, degrees or revolutions in the same way as the Angular Displacement Calculator.