## Introduction

In circular motion, angular acceleration is the rate with which the angular velocity changes with time. It is also referred to as rotational acceleration. It is a vector quantity, that is, it has both magnitude and direction.

Angular acceleration is denoted by \[\alpha \].

If \[\theta \] is the angular displacement, \[\omega \]is the angular velocity and \[\alpha \], the angular acceleration, then;

\[\alpha = \frac{{d\omega }}{{dt}} = \frac{{{d^2}\theta }}{{d{t^2}}}\] (as; \[\omega = \frac{{d\theta }}{{dt}}\])

The above formula gives the instantaneous angular acceleration.

If \[\Delta \omega \] is the change in angular velocity over a time interval \[\Delta t\], then average angular acceleration is given by:

\[\alpha = \frac{{\Delta \omega }}{{\Delta t}}\]

In the case of uniform rotation, the average and instantaneous values coincide.

It is expressed in the units of rad/s^{2} or radians per second squared.

### Angular Acceleration Formula - Equation & Solved Examples

Acceleration is caused by varying velocity. We call it angular acceleration if an object is spinning and changing its speed. Let us explore angular acceleration in detail in this article.

Example 1

If the angular velocity of a body in rotational motion changes from \[\frac{\pi }{2}\]rad/s to \[\frac{{3\pi }}{4}\] in 0.4 s. Find the angular acceleration.

Solution:

\[{\omega _1} = \frac{\pi }{2}rad/s,\,\,\,{\omega _2} = \frac{{3\pi }}{4}rad/s,\,\,\,\Delta t = 0.4\,s,\,\,\,\alpha = ?\]

\[\alpha = \frac{{\Delta \omega }}{{\Delta t}} = \frac{{{\omega _2}--{\omega _1}}}{{\Delta t}} = \frac{{\frac{{3\pi }}{4}--\frac{\pi }{2}}}{{0.4}} = \frac{{5\pi }}{8}rad/{s^2}\]

Example 2

The angular displacement of an object in rotational motion depends on time t according to the relation

\[\theta = 2\pi \,{t^3}--\pi {t^2} + 3\pi --6\], where \[\theta \]is in radians and t in seconds. Find its angular acceleration at t = 2 s.

Solution:

We have: \[\theta = 2\pi \,{t^3}--\pi \,{t^2} + 3\pi \,t--6\] rad

Angular velocity: \[\omega = \frac{{d\theta }}{{dt}} = 6\pi \,{t^2}--2\pi \,t + 3\pi \] rad/s

Angular acceleration: \[\alpha = \frac{{d\omega }}{{dt}} = 12\pi \,t--2\pi \] rad/s2

At t = 2s, \[{\alpha _{t = 2s}} = 12\pi \, \times 2--2\pi = 22\pi \,\,\,\,rad/{s^2}\]

Question: A wheel rotating at 10 rad/s2 is imparted with a constant angular acceleration of 4 rad/s^{2} for 5 seconds. The number of rotations made by the wheel in this 5 s interval is:

Options:

(a) \[\frac{{20}}{\pi }\]

(b) \[\frac{{40}}{\pi }\]

(c) \[\frac{{100}}{\pi }\]

(d) \[\frac{{50}}{\pi }\]

Answer: (d)

### What is Angular Acceleration?

Time rate of change in angular velocity is defined as angular acceleration. Radians per second per second is a common unit of measurement. Hence,

\[\alpha = \frac{d\omega}{dt} \]

Also known as rotational acceleration, angular acceleration occurs when a body rotates. An angular velocity change per unit of time can be expressed as a quantitative vector.

A pseudo scalar is an angle acceleration. An increase in angular speed counter clockwise is regarded as positive angular acceleration; an increase in speed clockwise is regarded as negative angular acceleration. A real-life example of angular acceleration is the study of objects that rotate, such as the wheel, fan, and earth.

### Angular Acceleration Unit

As the acceleration vector is perpendicular to the plane where the rotation occurs, the acceleration vector is perpendicular to that plane. An acceleration velocity that points away from the observer is created by increasing angular velocity clockwise. The vector of angular acceleration points to the viewer if the increase in angular velocity is counter clockwise.

The SI unit of angular acceleration is the radian per second squared (rad/s^{2}) and it is often denoted by the alpha (α).

### Angular Acceleration Formula

The rate at which the angular velocity changes with time is known as the angular acceleration in circular motion. It is also known as rotational acceleration. The quantity has both magnitude and direction, so it is a vector quantity.

Angular acceleration is denoted by α.

Let θ be the angular displacement, \[\omega\] be the angular velocity and α, be the angular acceleration, then;

\[\alpha = \frac{d\omega}{dt} = \frac{d^{2}\theta}{dt^{2}}\] (as; \[\omega = \frac{d\theta}{dt}\])

The above formula can be used to calculate a constant angular acceleration.

If \[ \Delta\omega\] is the change in angular velocity over a time interval \[\Delta t\], then the average angular acceleration is given by:

\[\alpha = \frac{\Delta\omega}{\Delta t}\]

The average value and the instantaneous value coincide for uniform rotation.

The units are rad/s^{2} or radians per second squared.

## FAQs on Angular Acceleration Formula

**1. Is it possible to have negative angular acceleration?**

When the angular speed increases anticlockwise, the sign of the angular acceleration is considered positive, and when it increases clockwise, it is taken to be negative.

**2. What is angular acceleration?**

In order to study rotational objects such as the wheel, fan and earth, angular acceleration is used.

**3. How do you calculate angular acceleration?**

It is defined as the rate at which angular velocity changes. An angular acceleration describes a change in angle of motion as follows: \[\alpha = \frac{\Delta \omega}{\Delta t}\] \[ \alpha = \frac{\Delta \omega}{\Delta t}\] , where Δω is the change in angular velocity and Δt is the change in time.

**4. How do we define angular acceleration?**

Average Angular Acceleration

The blade of a fan is at rest; it starts rotating after 10 seconds at a speed of 5 radians per second. We can calculate angular acceleration from the slope of any line on this graph, which is y/x. This gives us radians per second per second (rad/s^{2}).

The rotation's velocity is not the same as its speed. Rotational speed can be measured by the number of radians it covers per second using the Greek letter omega (ω). The spinning object accelerates angularly if its speed changes, which is indicated by the variable symbol alpha (α). Angular acceleration is divisible into two types: average acceleration and instantaneous acceleration. They are calculated differently, so we will start with the average angular acceleration (avg) in Equation 1.

**5. How are Angular accelerations measured?**

Angular acceleration is measured in SI units in the form of radians per second squared (rad/s^{2}). The symbol used for angular acceleration is alpha (α).

**6. What is the importance of angular acceleration?**

For angular velocity, the angular acceleration is the rate of change of the angle of approach: the magnitude of the angular acceleration tells us the rate at which the angle of approach changes, and the direction tells us the direction of the change.