Angular Displacement Calculator

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.

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 $\theta$.

Remember not to confuse it with linear displacement, the distance covered by an object. But just in case you want to more, then check out our 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.

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 $r$ and the distance covered along the circular path $s$. As a result, you get angular displacement. The formula looks like this:

   $\theta= s / r$

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.

   $\theta = \omega \times t$

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.

   $\theta = (\omega \times t) + (1 / 2 \times \alpha \times t^2)$

where:

• $\theta$ – Angular displacement;
• $s$ – Distance;
• $r$ – Radius of the circular path;
• $\omega$ – Angular velocity;
• $t$ – Time; and
• $\alpha$ – Angular acceleration.

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 $9 \text{ meters}$ and you run twice around the track and cover a distance of $185 \text{ meters}$, measured by your smartwatch.

Your angular displacement is: