# Angular acceleration: Wikis

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# Encyclopedia

Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (α).[1]

## Mathematical definition

The angular acceleration can be defined as either:

${\alpha} = \frac{d{\omega}}{dt} = \frac{d^2{\theta}}{dt^2}$ , or
${\alpha} = \frac{\mathbf{a}_{T}}{r}$ ,

where ω is the angular velocity, $\mathbf{a}_{T}$ is the linear tangential acceleration, and r is the radius of curvature.

## Equations of motion

For rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:

${\tau} = I\ {\alpha}$ ,

where τ is the total torque exerted on the body, and I is the mass moment of inertia of the body.

### Constant acceleration

For all constant values of the torque, τ, of an object, the angular acceleration will also be constant. For this special case of constant angular acceleration, the above equation will produce a definitive, constant value for the angular acceleration:

${\alpha} = \frac{\tau}{I}.$

### Non-constant acceleration

For any non-constant torque, the angular acceleration of an object will change with time. The equation becomes a differential equation instead of a constant value. This differential equation is known as the equation of motion of the system and can completely describe the motion of the object. It is also the best way to calculate the angular velocity.

# Study guide

Up to date as of January 14, 2010

## Definition

Angular acceleration is a vector whose magnitude is defined as the change in angular velocity in unit time.

It is in $rad\cdot s^{-2}$ in SI unit.

## Formula

Analogous to translational acceleration, $a=\frac{dv}{dt}$, angular acceleration has the defining formula:

$\alpha=\frac{d\omega}{dt}$

in which $d\omega\,$ represents an instantaneous change in angular velocity,which takes place in $dt\,$, a short flitting time.

Equivalently, think about the limiting case: $\alpha= \lim_{\Delta t\rightarrow 0}\frac{\Delta\omega}{\Delta t}$

## Relationship with Constant Torque

The angular acceleration of an fixed-axis-object is proportional to the net torque applied.

$\alpha=\frac{\tau}{I}$

in which $I\,$ is the Moment of Inertia of the object.

## Angular Kinematics

When an rotation has constant angular acceleration $\alpha\,$, the angle displacement $\theta\,$ covered in a given time $t\,$ is given by an equation that is strikingly similar to the equation for displacement under constant acceleration.

$\theta=\omega_0\,t+\frac{1}{2}\alpha\,t^2$

in which $\omega_0\,$ is the angular velocity at the beginning of the time period $t\,$

$\theta=\frac{1}{2}\alpha\,t^2$

in which case the angular velocity at the beginning $\omega_0\,$ is "zero"

$\theta=\omega_t\,t-\frac{1}{2}\alpha\,t^2$

in which $\omega_t\,$ is the angular velocity at the end of the time period $t\,$