Definition: Angular acceleration of an object undergoing circular motion is defined as the rate with which its angular velocity changes with time. Angular acceleration 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 Î± and is expressed in the units of rad/s2 or radians per second square.

Formula:

Angular acceleration can be expressed as given below,

\(\alpha =\frac{d\omega }{dt}\)

And also in terms of the double differentiation of the angular displacement, as given below,

\(\alpha = \frac{d^{2}\theta}{ dt^{2}}\)

Derivation:

Angular acceleration is the rate of change of angular velocity with respect to time, or we can write it as,

\(\alpha = \frac{d\omega }{dt}\)

Here, Î± is the angular acceleration that is to be calculated, in terms of rad/s2,Â Ï‰Â is the angular velocity given in terms of rad/s and t is the time taken expressed in terms of seconds.

Angular velocity as we know, can be expressed as given below.

\(\omega = \frac{v}{r}\)

Here, Ï‰ is the angular velocity in terms of rad/s, v is the linear velocity and r is the radius of the path taken.

Angular VelocityÂ can also be expressed as the change in angular displacement with respect to time, as given below.

\(\omega = \frac{\theta }{t}\)

Where,Â Î¸Â is the angular rotation of the object and t is the total time taken.

Using the above formula, we can write angular accelerationÂ Î±Â as

\(\alpha = \frac{d^{2}\theta }{dt^{2}}\)

**Real Life Example:**

**Example Â 1:Â **

An ant is sitting at the edge of a rotating circular disc. It’s angular velocity changes at the rate of 60 rad/s for 10 seconds. Calculate its angular acceleration during this time?

**Solution:**

Given: The change in angular velocity is equal to dÏ‰ = 60 rad/s. The time taken for this change to occur is equal to t = 10s.

Using the formula for angular acceleration and substituting the above values, we get,

\(\alpha = \frac{d\omega }{dt}=\frac{30}{5}=6\;rad/s^{2}\)

**Example Â 2:Â **

The rear wheel of a motorcycle has an angular acceleration of 20 rad/s2Â in a second. What can be said about its angular velocity?

**Answer:**

Given: The angular acceleration of the wheel is equal toÂ Î±Â = 10 \(rad/s^{2}\),

Â Â Â Â Â Â Â Â Â Â Â Â Time taken t = 1 s,Â

According to the formula for angular acceleration,

\(\alpha = \frac{d\omega }{dt}\)

Upon substituting the values, we get,

Angular velocity d is

dÏ‰ = Î±dt

dÏ‰ =20Ã—1 = 20 rad/s