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# Angular Acceleration

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 Sub Topics You might have already gone through the concept of acceleration. The Angular acceleration is nothing but the acceleration in circular path. Many a times you see turns or twists in the path of rotating body or body in circular path where you can experience this concept.Lets see more about this.

## Angular Acceleration Definition

Angular acceleration is that acceleration a body experiences in a circular path. It gives the change in angular velocity rate for a given time and is a vector quantity. The acceleration is in general given in meters per second square (m/s2) and since the acceleration here is in circular path the angular acceleration unit will be the same i.e. (m/s2).It is denoted by $\alpha$ given as
$\alpha$ = $\frac{change\ in\ angular\ velocity}{time\ taken}$

## Angular Acceleration Formula

Angular acceleration is that where there is change in angular velocity to the given time. It is given by
$\alpha$ = $\frac{d\omega}{dt}$ = $\frac{a_r}{R}$
Here d$\omega$ is the change in angular velocity,
ar is the linear acceleration
R is the radius of circle
dt is the time taken

The angular acceleration is given by
$\alpha$ = $\frac{\tau}{I}$Here $\tau$ is the torque and I is the moment of inertia or angular mass.

## Average Angular Acceleration

Angular acceleration is the change of angular velocity rate with respect to time. If $\omega_i$ is the initial angular velocity of the body at initial time ti and it undergoes angular displacement and reaches the final angular velocity $\omega_f$ at the end of the action at final time tf. It is given as
$\alpha_{av}$ = $\frac{\omega_f − \omega_i}{t_f − t_i}$where $\omega_f$ = final angular velocity
$\omega_i$ = initial angular velocity
tf = final interval of time or the end of time
ti = initial interval of time
It has the same unit as the angular acceleration, i.e., rad/s2.

## Instantaneous Angular Acceleration

The instantaneous rate at which an object rotates in a circular path is what we call instantaneous angular acceleration. It is the ratio of change in angular velocity over the limit of change in time as it approaches zero ($\Delta$ t -> 0). It is given as
$\alpha$ = lim$\Delta$ t -> 0 $\frac{\Delta \omega}{\Delta t}$ = $\frac{d \omega}{dt}$
Here d$\omega$ is the small change in angular velocity, dt is small change in time and t is time taken.

## Magnitude of Angular Acceleration

The magnitude of angular acceleration is that where you consider the change of angular velocity rate in any direction. It stresses only on magnitude of acceleration irrespective of direction. Here in circular path the direction is limited to two. So any path that the body chooses we just consider only the magnitude.

Let's consider a body moving in a circular path with a velocity v and at a distance r from the center.The magnitude of angular acceleration $\alpha$ is given as
$\alpha$ = $\frac{v^2}{r}$
Here v is the velocity of the rotating body in a circular path and r is the radius from center.

## Torque and Angular Acceleration

Torque is nothing but the ability of a body to rotate about a fixed axis. If the body is in the force that is used to impart the angular acceleration is known as ‘Torque’. It is given as $\tau$. The torque in terms of angular acceleration is
$\tau$ = I $\alpha$Here I is the inertia of the rotating body in circular path.
If a body undergoes rotational motion in a circular path about a fixed axis, it under a constant angular acceleration $\alpha$, its motion can be explained with these equations:
$\omega$ - $\omega_{0}$ = $\alpha$ t.
$\theta$ - $\theta_{0}$ = $\omega_{0}$ t + $\frac{1}{2}$ $\alpha$ t2
$\omega ^{2}$ = $\omega_{0}^{2}$ + 2 $\alpha$ ($\theta$ - $\theta_{0}$)
where $\omega_{0}$ = angular speed of the rigid body at time t = 0
$\omega$ = angular speed of the rigid body at time t
$\alpha$ = angular acceleration.