Angular Displacement Calculator

Created by Komal Rafay
Reviewed by Steven Wooding
Last updated: Apr 21, 2022

Our angular displacement calculator has your back if you are a physics enthusiast or a student looking to complete an assignment. Even if you are neither, we are still here for you. And would you say, learning about various methods to calculate the angular displacement is on your bucket list?

You'd be glad to know the range of topics we have in store, from the angular displacement formula, its unit, and equations, to understanding angular displacement from angular velocity and angular acceleration.

What is angular displacement?

Imagine an object moving along a circular path, an angle forms along the radius, and this angle is the angular displacement of the object. It means that the body or object is in rotational motion. It is a vector quantity as it has a magnitude and a direction.

To denote angular displacement, the symbol we use is θ\theta.

Remember not to confuse it with linear displacement, the distance covered by an object.

Visual representation of angular displacement θ.
Angular displacement is the angle between the point of rest or starting point and radius.

Angular displacement calculator

The angular displacement calculator is based on multiple formulae to cater to all the various ways it can be determined.
You have three options to choose from:

  1. Radius of the circular path is known;
  2. Angular velocity is known; or
  3. Angular acceleration is known.

When selecting the first option, you know the radius of the circular path. Input the radius and the distance traveled. As a result, you have angular displacement. The default angular displacement unit is radians, but you have a list of units to choose from.

The second option is when you have the angular velocity. This calculation is based on the simple fact that angular velocity is the rate of change of angular displacement. To use this method, input the angular velocity and the time taken for the object to cover the distance, and the result is angular displacement.

The third option is when you may want to determine the angular displacement with respect to angular acceleration. This method then does not need the radius or distance covered. Here you can input the angular velocity, the time taken for the object to cover the particular distance, and the angular acceleration. This is one of the most common formulas used to calculate angular displacement because it considers all the aspects of that object in a circular motion.

One thing to remember is that angular acceleration is in radians per second squared. Be mindful when adjusting the units of other variables.

✅ Since we are all about rotational motions today, why not take a look at our angular momentum calculator.

Angular displacement formula

We have used various angular displacement equations to formulate our calculator. Let's take a look at all three of them one by one.

  1. Angular displacement from the radius of the circular path
    This method uses the radius of the circular path rr and the distance covered along the circular path ss. As a result, you get angular displacement. The formula looks like this:

    θ=s/r\theta= s / r

  2. Angular displacement from angular velocity
    Angular velocity is the rate of change of angular displacement. If we shuffle this formula, we can quickly determine the angular displacement from the angular velocity.

    θ=ω×t\theta = \omega \times t

  3. Angular displacement from angular acceleration
    The most common method used to determine angular displacement is through angular acceleration. This formula uses angular velocity, angular acceleration, and time to estimate the angular displacement of the object.

    θ=(ω×t)+(1/2×α×t2) \theta = (\omega \times t) + (1 / 2 \times \alpha \times t^2)

where:

  • θ\theta – Angular displacement;
  • ss – Distance;
  • rr – Radius of the circular path;
  • ω\omega – Angular velocity;
  • tt – Time; and
  • α \alpha – Angular acceleration.

How to find angular displacement

Let us consider an example to understand how to find angular displacement!

Imagine you are out for your morning jog on your favorite track in the neighborhood park. You get curious about what is your angular displacement? You are determined to find out the answer.

You estimate the radius by measuring the distance from the fountain in the middle of the park to the edge. And the radius is 9 meters9 \text{ meters} and you run twice around the track and cover a distance of 185 meters185 \text{ meters}, measured by your smartwatch.

Your angular displacement is:

θ=s/r=185/9=20.556 radians\small \begin{align*} \theta &= s / r \\ &= 185 / 9\\ &= 20.556 \text { radians} \end{align*}

FAQ

How can I calculate angular displacement from angular acceleration?

The formula for angular displacement given angular acceleration is:

θ = (ω × t) + (1 / 2 × ɑ × t²)

where:

  • θ – Angular displacement;
  • ω – Angular velocity;
  • t – Time; and
  • ɑ – Angular acceleration.

If you observe, this formula uses Newton's second equation of motion, which determines the distance covered by an object moving with uniform acceleration.

How do I calculate angular displacement from angular velocity?

The formula for angular displacement given angular velocity is:

θ = ω × t

where:

  • θ – Angular displacement;
  • ω – Angular velocity; and
  • t – Time.

So this means all you are required to do is multiply the value of angular velocity by time, and your result is angular displacement.

A wheel rotates 3 times, what is its angular displacement?

A wheel rotating three times has an angular displacement of 1080°. One complete rotation of a circle is equal to 360°. To estimate the value of θ (angular displacement) from it:

  1. Note the number of rotations;
  2. Multiply the number of rotations by 360.
  3. The result is angular displacement.

Remember, this is a general estimate as you might get different results using the angular displacement equations.

What is the difference between angular and linear displacement?

Angular displacement is the angle between the distance covered by an object on a circular path and the radius of the said circular path. In contrast, linear displacement is the shortest distance covered by an object from one point to another.

The angular displacement unit is radians or degrees as it measures the angle, whereas linear displacement is in meters.

Both angular and linear displacement are vector quantities, which means they have a magnitude and direction.

Komal Rafay
Formula:
Radius of circular path is known
Distance travelled (s)
m
Radius of the circular path (r)
m
Angular displacement (θ)
rad
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