Angular Velocity and Acceleration

Key facts

Angular velocity is the rate of change of the position-specific angle $\theta$ with respect to time:

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

(1)

Angular acceleration is the rate of change of angular velocity with respect to time:

$\alpha = \frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$

(2)

The centripetal acceleration can be defined as:

$a_c = r\omega^2$

(3)

The Coriolis component of acceleration, or compound supplementary acceleration can be defined as:

$a_C = 2v\omega$

(4)

Angular velocity and acceleration, including centripetal and coriolis acceleration.

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Angular velocity and acceleration are vector quantities that describe an object in circular motion. When an object, like a ball attached to a length of string, is rotated at a constant angular velocity, it is said to be in uniform circular motion. However, if the object is rotated at increasing or decreasing speeds, it can be said to be in a state of angular acceleration. The acceleration can have a centripetal component (acting inwards toward the axis of rotation) or a Coriolis component(acting perpendicular to the direction of velocity and the axis of rotation).

Angular Velocity

In order to define angular velocity, consider a particle $P$ moving in a $XOY$ reference plane, as diagramed in Figure 1. Let $x$ be the projection of $P$ on the $OX$ axis, $y$ the projection of $P$ on the $OY$ axis, $r$ the length of $OP$ , and $\theta$ the $\angle{POX}$ angle.

We can then write that:

$x = r\cos\theta$

$y = r\sin\theta$

By differentiating these equations with respect to time, we have:

$\dot x = \frac{dx}{dt} = \dot r \cos\theta - r\dot \theta \sin\theta$

(5)

$\dot y = \frac{dy}{dt} = \dot r \sin\theta + r\dot \theta \cos\theta$

(6)

We can write the tangential component of velocity, $v_T$ , which is the component of velocity perpendicular to $OP$ , as:

$v_T = \dot y \cos\theta - \dot x \sin\theta$

and, by using the expressions of $\dot x$ and $\dot y$ from equations (#1) and (#2) respectively, we obtain:

$v_T = r \dot \theta$

(7)

We define the angular velocity $\omega$ as the rate of change of $\theta$ with respect to time:

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

(8)

Using (#4) in (#3) we get that:

$v_T = r\omega$

from which the angular velocity becomes:

$\omega = \frac{v_T}{r}$

Angular Acceleration

In order to define the acceleration, we first have to calculate $\ddot x$ and $\ddot y$ . To do this we differentiate equations (#1) and (#2) with respect to time, when we obtain that:

$\ddot x = \ddot r \cos\theta - \dot r \sin\theta - r\ddot \theta^2 \cos\theta - r \ddot \theta \sin\theta$

(9)

$\ddot y = \ddot r \sin\theta + \dot r \dot \theta \cos\theta + r \dot \theta^2 \sin\theta +r \ddot \theta \cos\theta$

(10)

We can write the radial component of acceleration, $a_r$ , which is the component of acceleration in the direction of $OP$ , as:

$a_r = \ddot x \cos\theta + \ddot y \sin\theta$

and, by using the expressions of $\ddot x$ and $\ddot y$ from equations (#5) and (#6) respectively, we get:

$a_r = \ddot r - r \dot \theta^2$

(11)

We can also write (#7), by using (#4), as:

$a_r = \dot v - r\omega^2$

where $\dot v$ is the rate of change of velocity, while the $r\omega^2$ term is called the Centripetal Acceleration ( $a_c$ ).

The tangential component of acceleration on the other hand, $a_T$ , can be written as:

$a_T = \ddot y \cos\theta - \ddot x \sin\theta$

and, again by replacing $\ddot x$ and $\ddot y$ from equations (#5) and (#6) respectively, we obtain:

$a_T = r \ddot \theta + 2 \dot r \dot \theta$

(12)

We define the angular acceleration $\alpha$ as the rate of change of the angular velocity $\omega$ with respect to time:

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

(13)

Equation (#8) can also be written, by taking into account (#9), as:

$a_T = r \alpha + 2v\omega$ where the $2v\omega$ term is called the compound supplementary acceleration, or the Coriolis component of acceleration $a_C$ .