# Angular Acceleration Calculator

Created by Krishna Nelaturu
Last updated: Aug 31, 2022

Our angular acceleration calculator is here to help you compute the angular acceleration of a body in a circular motion. Notice that this is different from linear acceleration. We shall briefly discuss the fundamentals of angular acceleration here, so join us below if you're curious about:

• What is angular acceleration?
• Angular acceleration formula.
• Units of angular acceleration.
• How to convert angular acceleration to linear acceleration and vice versa.

## What is angular acceleration? Angular acceleration formula

Like linear or translational acceleration, angular acceleration is a measure of the rate of change of angular velocity. It is a vector quantity with both magnitude and direction. The formula for angular acceleration is:

$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega_2 - \omega_1}{\Delta t}$

Where:

• $\alpha$ - Angular acceleration;
• $\omega_1,\omega_2$ - Initial and final angular velocities;
• $\Delta t$ - Time needed for the change in angular velocity.

This change in angular velocity could be due to a torque acting on the body. The SI unit of angular acceleration is radians per second squared $(\text{rad/s}^2)$. Sometimes, people omit the radians part and simply use per second squared $(\text{1/s}^2)$.

## Angular acceleration to linear acceleration

Let's tie a rope to a stone. The stone orbits around you when you swing the rope over your head. The stone undergoes translational and angular displacement at any instant in its orbit. The relationship between the rotational and translational velocities is given by:

$v = \omega \cdot r$

Where:

• $v$ - Translational or linear velocity;
• $\omega$ - Rotational or angular velocity; and
• $r$ - Radius of rotation or the distance from the axis of rotation to the object.

The same relationship exists between linear and angular acceleration:

$a = \alpha \cdot r$

Where:

• $a$ - Translational or linear acceleration; and
• $\alpha$ - Rotational or angular acceleration.

Therefore, to convert angular acceleration to linear acceleration, we need to multiply the angular acceleration with $r$, the distance between the object and its axis of rotation.

This relationship gives yet another equation for angular acceleration:

$\alpha = \frac{a}{r}$

This relationship helps determine the angular acceleration from the tangential acceleration $a$ of a body moving in a circular orbit.

## How to find angular acceleration | Angular acceleration examples

Let's go through a couple of practice problems to understand angular acceleration better:

1. You're riding a giant Ferris wheel spinning at 4 rotations per minute $(4 \text{ rpm})$. After a fun ride, it comes to a halt in 3 minutes. What angular acceleration (or rather angular deceleration) do you experience?

Given data:

• Initial angular velocity ω1: 4 rpm = 4 × 2 π/60 rad/s.
• Final angular velocity ω2: 0.
• Time t: 3 min = 180 s.

To find:

• Angular acceleration α

Using the angular acceleration formula:

\qquad \begin{align*} \alpha &= \frac{\omega_2 - \omega_1}{t}\\ &=\frac{0- \frac{8\pi}{60}}{180}\\ \alpha &= -0.002327 \text{ rad/s}^2 \end{align*}

The negative symbol indicates that the Ferris wheel is decelerating.

1. When kicking a football, your foot rotates about your hip. Let's say you moved your foot from a 30° angle to the vertical in half a second. What is the tangential acceleration at the point of impact with the ball? Assume your foot moves from rest and the distance from your foot to your hip is 85 cm.

Given data:

• Angular displacement θ: 30° = 0.524 rad.
• Initial angular velocity ω1: 0.
• Time t: 0.5 s.
• Distance from foot to the axis of rotation r: 85cm =0.85m.

To find:

• Final angular velocity ω2.
• Angular acceleration α.
• Tangential acceleration a.

Using angular velocity formula:

\qquad \begin{align*} \omega_2 &= \frac{\theta}{t} \\ &= \frac{0.524}{0.5}\\ \omega_2 &= 1.048 \text{ rad/s} \end{align*}

Let's use the angular acceleration equation now:

\qquad \begin{align*} \alpha &= \frac{\omega_2 - \omega_1}{t}\\ &= \frac{1.048 - 0}{0.5}\\ \alpha &= 2.096 \text{ rad/s}^2 \end{align*}

Finally, using the equation for angular acceleration to tangential acceleration:

\qquad \begin{align*} a &= \alpha \cdot r\\ &= 2.096 \cdot 0.85\\ a & = 1.7816 \text{ m/s}^2 \end{align*}

These problems are just to help you understand how to calculate angular acceleration. Work out more problems, and you'll soon become a master!

## How to find angular acceleration using this angular acceleration calculator

This angular acceleration calculator is a powerful tool by design:

• You can choose which method you would like to use to determine angular acceleration.
• Angular velocity difference uses the angular acceleration formula to calculate the angular velocity change rate.
• Tangential acceleration and radius uses the relationship between angular and linear acceleration.

In angular velocity difference method:

• Enter the appropriate values for initial and final velocities and time to calculate your angular acceleration instantly.
• Bonus: If you know the change in angular velocity instead of its initial and final values, you can click on the advanced mode button to directly use this value for calculation.

In tangential acceleration and radius method:

• Provide the values for the tangential acceleration and radius to calculate the angular acceleration.
Krishna Nelaturu
How would you like to calculate angular acceleration?
Method
angular velocity difference
Angular velocity difference
Initial angular velocity
Final angular velocity
Time
sec
Angular acceleration